ON THE COMPLEXITY OF MATHEMATICAL

PROBLEMS: MEDVEDEV DEGREES AND REVERSE MATHEMATICS

A Dissertation

Presented to the Faculty of the Graduate School

of Cornell University

in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

by Paul Emery Shafer

August 2011 \begin{matrix}c \\ ◯ \end{matrix} 2011 Paul Emery Shafer ALL RIGHTS RESERVED ON THE COMPLEXITY OF MATHEMATICAL PROBLEMS: MEDVEDEV

DEGREES AND REVERSE MATHEMATICS Paul Emery Shafer, Ph.D. Cornell University 2011

We investigate the complexity of mathematical problems from two perspectives: Medvedev degrees and reverse mathematics. In the Medvedev degrees, we calculate the complexity of its first-order theory, and we also calculate the complexities of the first-order theories of several related structures. We characterize the join-irreducible Medvedev degrees and deduce several consequences for the interpretation of propositional logic in the Medvedev degrees. We equate the size of chains of Medvedev degrees with the size of chains of sets of reals under ⊆. In reverse mathematics, we analyze the strength of classical theorems of finite graph theory generalized to the countable. In particular, we consider Menger’s theorem, Birkhoff’s theorem, and unfriendly partitions.

BIOGRAPHICAL SKETCH

Paul was born on February 28, 1983 in Richland, Washington during the final episode Goodbye, Farewell and Amen of the popular television series M*A*S*H. He graduated from Dutch Fork High School in Irmo, South Carolina in 2001. Paul has attended Cornell University in Ithaca, New York since 2001, earning a B.S. in computer science in 2005 and a M.S. in computer science in 2010.

Thanks Mom and Dad and Jessica and Lisey! iv

ACKNOWLEDGEMENTS

Thanks to Andrew Lewis, André Nies, and Andrea Sorbi for their gracious acknowledgement of my work on Theorem 2.3.10 during their presentation of their proof of the theorem at CiE 2009. Thanks to Andrea Sorbi and Sebastiaan Terwijn for suggesting several of the problems considered in Chapter 3. Thanks to Stevo Todorcevic and Bill Mitchell for their fruitful suggestions on how to prove Theorem 4.2.5, and thanks to Justin Moore for patient discussions that helped fill in the details. Thanks to Louis Billera for suggesting the analysis of Birkhoff’s theorem in Chapter 6. Most importantly, many thanks to my advisor Richard A. Shore for introducing me to Medvedev degrees, for introducing me to reverse mathematics, and for many helpful discussions on all the work presented here. This research was partially supported by NSF grants DMS-0554855, DMS-0852811, and DMS-0757507.

TABLE OF CONTENTS

Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction 1 1.1 Basic concepts and notation . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Distributive lattices and Brouwer algebras . . . . . . . . . . . . . . 4 1.3 Mass problems and reducibilities . . . . . . . . . . . . . . . . . . . 5 1.4 Substructures of D_{s} and D_{w} . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Closed degrees . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.2 Effectively closed degrees . . . . . . . . . . . . . . . . . . . 8

1.5 PA^{−} and the standard model of arithmetic . . . . . . . . . . . . . . 10 1.6 Reverse mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Coding true arithmetic in the Medvedev and Muchnik degrees 15 2.1 Interpreting the Medvedev and Muchnik degrees in arithmetic . . 16 2.2 Coding arithmetic in distributive lattices . . . . . . . . . . . . . . . 18

2.2.1 Coding models of relational theories . . . . . . . . . . . . . 18 2.2.2 The finite matching property and the first-order correct-

ness condition . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.3 The coding countable subsets property and the second-

order correctness condition . . . . . . . . . . . . . . . . . . 22 2.2.4 The coding all subsets property and the third-order cor-

rectness condition . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.5 Counting quantifiers . . . . . . . . . . . . . . . . . . . . . . 26

2.3 The complexities of Th(D_{s}; ≤_{s}) and Th(D_{w}; ≤_{w}) . . . . . . . . . . . 28 2.3.1 Defining D_{w} in D_{s} . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Coding third-order arithmetic in D_{w} . . . . . . . . . . . . . 29

2.4 The complexities of Th(D_{w,cl}; ≤_{w}) and Th(D_{w,cl}^{01}; ≤_{w}) . . . . . . . . 32 2.5 D_{s,cl} and D_{s,cl}^{01} have the coding countable subsets property . . . . . 35 2.6 The complexities of Th(D_{s,cl}; ≤_{s}) and Th(D_{s,cl}^{01}; ≤_{s}) . . . . . . . . . 38 2.7 A first-order sentence distinguishing D_{s,cl} and D_{s,cl}^{01} from D_{w,cl} and

D_{w,cl}^{01} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.8 Meet-irreducibles in E_{s} and r.e. separating degrees . . . . . . . . . 41 2.9 The complexity of Th(E_{s}; ≤_{s}) . . . . . . . . . . . . . . . . . . . . . . 44 2.10 The degree of E_{s} is 0′′′ . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.11 The undecidability of Th(E_{w}) and Th(E_{w}; ≤_{w}) . . . . . . . . . . . . 50

3 Join-irreducibles and propositional logics in the Medvedev degrees 54 3.1 Characterizing the join-irreducible Medvedev degrees . . . . . . . 55 3.2 Degrees that bound no join-irreducible degrees >_{s} 0′ . . . . . . . . 56 3.3 New degrees whose corresponding logic is contained in JAN … 61

3.4 𝔉_{cl} is not prime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Forcing no big chains in the power set of the reals 70 4.1 Forcing prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 Forcing no chains in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}} . . . . . . . . . . . 73 4.3 Big chains in the Medvedev degrees . . . . . . . . . . . . . . . . . 77

5 Menger’s theorem in Π_{1}^{1}-CA_{0} 78 5.1 Warps, waves, and alternating walks . . . . . . . . . . . . . . . . . 79 5.2 Menger’s theorem in Π_{1}^{1}-CA_{0} . . . . . . . . . . . . . . . . . . . . . 84 5.3 Extended Menger’s theorem . . . . . . . . . . . . . . . . . . . . . . 90

6 Partial analyses of Birkhoff’s theorem and of unfriendly partitions 92 6.1 Countable Birkhoff’s theorem in WKL_{0} . . . . . . . . . . . . . . . . 92 6.2 Variations on countable Birkhoff’s theorem . . . . . . . . . . . . . 99 6.3 Reverse mathematics and unfriendly partitions . . . . . . . . . . . 101

Bibliography 108

CHAPTER 1 INTRODUCTION

Medvedev degrees describe the relative complexity of subsets of ω^{ω} in a computational sense. A set Y ⊆ ω^{ω} is at least as complicated as a set X ⊆ ω^{ω} if there is a computational procedure for producing a member of X given a member of Y. We interpret a set X ⊆ ω^{ω} as an abstract mathematical problem. Namely, X corresponds to the problem of finding a member of X. Under this interpretation, the Medvedev degrees serve as a model for studying the relative complexity of mathematical problems. We investigate the Medvedev degrees in Chapter 2, Chapter 3, and Chapter 4. See Section 1.3 for a full introduction to the Medvedev degrees and related structures.

Chapter 2 is mainly concerned with calculating the complexities of the firstorder theories of the Medvedev degrees and related structures. The main results are as follows.

• The first-order theories of the Medvedev degrees and the Muchnik degrees are both recursively isomorphic to the third-order theory of arithmetic (Theorem 2.3.10).

• The first-order theories of the closed Medvedev degrees, the compact Medvedev degrees, the closed Muchnik degrees, and the compact Muchnik degrees are all recursively isomorphic to the second-order theory of arithmetic (Theorem 2.4.10 and Theorem 2.6.5).

• Neither the closed Medvedev degrees nor the compact Medvedev degrees is elementarily equivalent to either the closed Muchnik degrees or the compact Muchnik degrees (Theorem 2.7.2).

• The first-order theory of the Medvedev degrees of Π_{1}^{0} classes is recursively isomorphic to the first-order theory of arithmetic (Theorem 2.9.4).

• For any of the above-mentioned degree structures and also for the Muchnik degrees of Π_{1}^{0} classes, the structure’s three-quantifier theory as a lattice is undecidable, and the structure’s four-quantifier theory as a partial order is undecidable (Theorem 2.3.11, Theorem 2.4.11, Theorem 2.6.6, Theorem 2.9.5, and Theorem 2.11.7).

• The degree of the Medvedev degrees of Π_{1}^{0} classes is 0′′′ in the sense that there is a presentation of the Medvedev degrees of Π_{1}^{0} classes recursive in 0′′′ and that 0′′′ is recursive in any such presentation (Theorem 2.10.6).

In Chapter 3, we characterize the join-irreducible Medvedev degrees and investigate the Medvedev degrees as semantics for propositional logic. The main results are as follows.

• A Medvedev degree is join-irreducible if and only if it is the degree of the complement of a Turing ideal (Theorem 3.1.3).

• There is a Medvedev degree greater than the second-least degree that bounds no join-irreducible degree greater than the second-least degree (Theorem 3.2.5).

• The filter in the Medvedev degrees generated by the non-minimum closed degrees is not prime (Theorem 3.4.3).

In Chapter 4, we provide an explicit construction demonstrating that the statement “there is no chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}” is consistent with ZFC (Corollary 4.2.6). We then compare the cardinalities of chains in the Medvedev degrees to the cardinalities of chains in (2^{2^{ω}}, ⊆). The main results are as follows.

• For any cardinal κ, there is a chain of cardinality κ in (2^{2^{ω}}, ⊆) if and only if there is a chain of cardinality κ in the Medvedev degrees (Theorem 4.3.1).

• The statements “there is a chain in the Medvedev degrees of cardinality 2^{2^{ω}},” “there is a chain in the Muchnik degrees of cardinality 2^{2^{ω}},” and “there is a chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}” are equivalent and are independent of ZFC (Corollary 4.3.2).

Reverse mathematics is an analysis of the logical strength of theorems from ordinary mathematics in the context of second-order arithmetic. Given a theorem, we wish to determine the weakest set of axioms required to prove that theorem. Theorems requiring stronger axioms are considered more complicated than theorems requiring weaker axioms. See Section 1.6 for a full introduction to reverse mathematics.

We consider reverse mathematics in Chapter 5 and Chapter 6. The main result is that Menger’s theorem for countable graphs is provable in the system Π_{1}^{1}-CA_{0} (Theorem 5.2.4). We also present several partial results concerning the reverse mathematics of Birkhoff’s theorem and of unfriendly partitions.

1.1 Basic concepts and notation

Let n ∈ ω, σ, τ ∈ ω^{<ω}, f, g ∈ ω^{ω}, and X , Y ⊆ ω^{ω}. Then

• f ↾ n is the initial segment of f of length n,

• |σ| is the length of σ,

• σ ⊆ τ means that σ is an initial segment of τ,

• σ ⊂ f means that σ is an initial segment of f,

• I(σ) = {f ∈ ω^{ω} | σ ⊂ f},

• σ^{⌢}f is the concatenation of σ and f:

(σ^{⌢}f)(n) = \begin{cases}σ(n) & if n < |σ| \\ f(n − |σ|) & if n ≥ |σ|, \end{cases}

• f ⊕ g is the function defined by

(f ⊕ g)(n) = \begin{cases}f(m) & if n = 2m \\ g(m) & if n = 2m + 1, \end{cases}

• σ^{⌢}X = {σ^{⌢}f | f ∈ X },

• X + Y = {f ⊕ g | f ∈ X ∧ g ∈ Y}, and

• X × Y = 0^{⌢}X ∪ 1^{⌢}Y.

The + and × in the last two items above correspond to the lattice-theoretic operations of join and meet in the Medvedev and Muchnik degrees, which is explained in the introduction to these degrees below. These symbols also retain their more common meanings, such as addition for + and product and cartesian product for ×. The meaning of a particular instance of either symbol will be clear from context.

The function ⟨ ·, · ⟩: ω × ω → ω is a fixed recursive bijection. Φ_{e} denotes the e^{th} Turing functional. Φ always denotes a Turing functional, and if f ∈ ω^{ω}, then Φ(f) is the partial function computed when Φ uses f as its oracle. For σ ∈ ω^{<ω}, Φ(σ) is the partial function that, on input n ∈ ω, is computed by running Φ on input n for at most |σ| steps and using σ to answer oracle queries. The restriction on the running time of Φ(σ) ensures that oracle queries are only made of numbers < |σ|. Consequently, if Φ(σ)(n)↓, then Φ(f)(n) = Φ(σ)(n) for all f ⊃ σ.

Let A, B ⊆ ω. A ≤_{1} B if and only if there is a one-to-one recursive function f such that ∀n(n ∈ A ↔ f(n) ∈ B). A and B are recursively isomorphic if and only if there is such an f that is a bijection. The Myhill isomorphism theorem states that A and B are recursively isomorphic if and only if A ≡_{1} B, that is, if and only if A ≤_{1} B and B ≤_{1} A (see [69] Section I.5).

A set X ⊆ ω^{ω} is independent if and only if g ≰_{T} f_{0} ⊕ f_{1} ⊕ ⋯ ⊕ f_{n−1} for any distinct g, f_{0}, …, f_{n−1} ∈ X. Let {f_{n}}_{n∈ω} ⊆ ω^{ω} be a sequence of functions, and let m ∈ ω. Define ⨁_{n∈ω} f_{n} and ⨁_{n∈ω\{m}} f_{n} by

(⨁_{n∈ω} f_{n})(⟨ i, j ⟩) = f_{i}(j) and

( ⨁_{n∈ω\\{m\}} f_{n})(⟨ i, j ⟩) = \begin{cases}f_{i}(j) & if i ≠ m \\ 0 & if i = m. \end{cases}

The sequence {f_{n}}_{n∈ω} ⊆ ω^{ω} is strongly independent if and only if ∀m(f_{m} ≰_{T} ⨁_{n∈ω\{m}} f_{n}). A sequence of sets {X_{n}}_{n∈ω} with X_{n} ⊆ ω^{ω} for each n is strongly independent if and only if {f_{n}}_{n∈ω} is strongly independent whenever ∀n(f_{n} ∈ X_{n}).

We consider Baire space ω^{ω} and Cantor space 2^{ω}, both with their usual product topologies. Basic open sets in ω^{ω} have the form I(σ) for σ ∈ ω^{<ω}. 2^{ω} has the subspace topology. When we are working in Cantor space, we usually write I(σ) for I(σ) ∩ 2^{ω}.

A tree is a set T ⊆ ω^{<ω} closed under initial segments: (∀σ, τ ∈ ω^{<ω})(σ ∈ T ∧ τ ⊆ σ → τ ∈ T). A function f ∈ ω^{ω} is a path through T if and only if (∀n ∈ ω)(f ↾ n ∈ T). If T is a tree, then [T] denotes the set of all paths through T. If X ⊆ ω^{ω} is closed, then X is the set of paths through the tree T = {σ | (∃f ∈ X)(σ ⊂ f)}. Conversely, if T ⊆ ω^{<ω} is a tree, then [T] is a closed subset of ω^{ω}. A set X ⊆ ω^{ω} is compact if and only if it is closed and bounded (i.e., there is a g: ω → ω such that (∀f ∈ X)(∀n ∈ ω)(f(n) ≤ g(n))) if and only if it is the set of paths through a finitely branching tree.

X ⊆ ω^{ω} is a Π_{1}^{0} class if and only if it has a Π_{1}^{0} definition. That is, if and only if X = {f ∈ ω^{ω} | ∀nφ(f, n)} for some recursive predicate φ. A useful characterization of the Π_{1}^{0} classes are as the sets of paths through recursive trees: a set X ⊆ ω^{ω} is a Π_{1}^{0} class if and only if X = [T] for some recursive tree T (see [17] Lemma 2.2). For this reason, the Π_{1}^{0} classes are sometimes called the effectively closed sets. The Π_{1}^{0} classes have been persistent objects of study throughout computability theory, due in no small part to their applications to recursive mathematics and reverse mathematics. The surveys by Cenzer [14] and by Cenzer and Remmel [17] provide an extensive overview of the theory of the Π_{1}^{0} classes, as does the forthcoming book by Cenzer and Remmel [16].

In this work, we only consider Π_{1}^{0} classes that are non-empty subsets of 2^{ω}. Henceforth the term “Π_{1}^{0} class” refers exclusively to a non-empty Π_{1}^{0} subset of 2^{ω}.

1.2 Distributive lattices and Brouwer algebras

A lattice L is distributive if and only if + and × distribute over each other:

• (∀x, y, z ∈ L)(x +(y × z) = (x + y) ×(x + z)) and

• (∀x, y, z ∈ L)(x ×(y + z) = (x × y) +(x × z)).

An element x of a lattice L is join-reducible if and only if (∃y, z < x)(y + z = x). Otherwise x is join-irreducible. Dually, x is meet-reducible if and only if (∃y, z > x)(y × z = x). Otherwise x is meet-irreducible. We frequently use the following well-known characterization without mention.

Lemma 1.2.1 (see [7] Section III.2). If L is a distributive lattice, then x ∈ L is joinirreducible if and only if (∀y, z ∈ L)(x ≤ y + z → x ≤ y ∨ x ≤ z). Dually, x ∈ L is meet-irreducible if and only if (∀y, z ∈ L)(x ≥ y × z → x ≥ y ∨ x ≥ z).

Proof.Suppose x is join-irreducible and x ≤ y + z. Then

x = x ×(y + z) = (x × y) +(x × z).

Thus x = x × y or x = x × z which means x ≤ y or x ≤ z. Conversely, if x is joinreducible, then by definition there are y, z < x with y + z = x. The proof for the meet-irreducible case is obtained by dualizing the proof for the join-irreducible case.

A lattice L is join-complete if and only if every non-empty X ⊆ L has a least upper bound. L is meet-complete if and only if every non-empty X ⊆ L has a greatest lower bound. L is complete if and only if it is both join-complete and meet-complete. Similarly, a lattice L is countably join-complete if and only if every non-empty countable X ⊆ L has a least upper bound. L is countably meetcomplete if and only if every non-empty countable X ⊆ L has a greatest lower bound. L is countably complete if and only if it is both countably join-complete and countably meet-complete. In a lattice L, a set X ⊆ L is called strongly joinincomplete if and only if for every finite {y_{i} | i < n} ⊆ X there is an x ∈ X such that x ≰ ∑_{i<n} y_{i}. Dually, a set X ⊆ L is called strongly meet-incomplete if and only if for every finite {y_{i} | i < n} ⊆ X there is an x ∈ X such that x ≱ ∏_{i<n} y_{i}.

Sometimes we want to ignore the lattice operations of a lattice L and consider L as a partial order. When we do, we write (L; ≤) to indicate that we are considering only the partial order structure on L.

A Brouwer algebra is a distributive lattice B with least element 0 and greatest element 1 such that for all x, y ∈ B there is a least z ∈ B such that x + z ≥ y. This least z is denoted x → y.

Brouwer algebras give semantics for propositional logic. For a Brouwer algebra B, a valuation is a function ν: propositional variables → B. A valuation ν extends to all propositional formulas φ by defining

ν(φ ∧ ψ) = ν(φ) + ν(ψ),
ν(φ ∨ ψ) = ν(φ) × ν(ψ),
ν(φ → ψ) = ν(φ) → ν(ψ), and
ν(¬φ) = ν(φ) → 1.

A propositional formula φ is called valid in B if ν(φ) = 0 for every valuation ν. Let PTh(B)^{1} denote the set of propositional formulas valid in B. The axioms of intuitionistic logic are valid in every Brouwer algebra B, so IPC ⊆ Th(B) ⊆ CPC for every Brouwer algebra B. Here IPC denotes intuitionistic logic and CPC denotes classical logic. Logics L for which IPC ⊆ L ⊆ CPC are called intermediate logics.

1.3 Mass problems and reducibilities

A mass problem is a set of functions X ⊆ ω^{ω}. Mass problem X Medvedev reduces (or strongly reduces) to mass problem Y (written X ≤_{s} Y) if and only if there is a Turing functional Φ such that (∀f ∈ Y)(Φ(f) ∈ X) (written Φ(Y) ⊆ X). Mass problems X and Y are Medvedev equivalent (or strongly equivalent, written X ≡_{s} Y) if and only if X ≤_{s} Y and Y ≤_{s} X. The relation ≡_{s} is an equivalence relation on 2^{ω^{ω}}, and the equivalence class deg_{s}(X) is called the Medvedev degree of X. D_{s} = {deg_{s}(X) | X ⊆ ω^{ω}} denotes the collection of all Medvedev degrees.

Mass problem X Muchnik reduces (or weakly reduces) to mass problem Y (written X ≤_{w} Y) if and only if (∀f ∈ Y)(∃g ∈ X)(g ≤_{T} f). Muchnik reducibility is the non-uniform version of Medvedev reducibility. Mass problems X and Y are Muchnik equivalent (or weakly equivalent, written X ≡_{w} Y) if and only if X ≤_{w} Y and Y ≤_{w} X. The equivalence class deg_{w}(X) is called the Muchnik degree of X. D_{w} = {deg_{w}(X) | X ⊆ ω^{ω}} denotes the collection of all Muchnik degrees.

1Usually what we call PTh(B) is called Th(B). In this work, Th(B) always denotes the set of first-order sentences true in B.

Medvedev introduced D_{s} in [41] as a formalization of Kolmogorov’s ideas of a “calculus of problems” and a “logic of problem solving.” Medvedev’s intuition was that a mass problem X represents a mathematical problem, namely the problem of finding an element of X. For example, if A ⊆ ω, the problem “find an enumeration of A” may be formalized as “find an element of the mass problem X = {f ∈ ω^{ω} | ran(f) = A}.” Under this interpretation, X ≤_{s} Y means that problem X is at least as hard as problem Y in a strongly intuitionistic sense: solutions to Y can be translated to solutions to X by a uniform effective procedure. Muchnik introduced his non-uniform variant in [45].

Medvedev reducibility and Muchnik reducibility induce a partial orders on the corresponding degrees: deg(X) ≤ deg(Y) if and only if X ≤ Y, where deg = deg_{s} and ≤ = ≤_{s} in the Medvedev case, and deg = deg_{w} and ≤ = ≤_{w} in the Muchnik case. D_{s} and D_{w} are distributive lattices. For mass problems X and Y, it is an easy check that in both cases

deg(X) + deg(Y) = deg(X + Y),
deg(X) × deg(Y) = deg(X × Y),

and that join and meet distribute over each other. In the Muchnik case, the equivalence X × Y ≡_{w} X ∪ Y always holds, and deg_{w}(X) × deg_{w}(Y) = deg_{w}(X ∪ Y). This equivalence is not always true in the Medvedev case.

D_{s} and D_{w} have a least element 0 = deg(ω^{ω}) and a greatest element 1 = deg(∅). In both structures, a mass problem has degree 0 if and only if it contains a recursive function, and a mass problem has degree 1 if and only if it is empty. D_{s} and D_{w} also both have a second-least element 0′ = deg({f ∈ ω^{ω}|f >_{T} 0}). The Medvedev degree 0′ and Muchnik degree 0′ have little to do with the Turing degree 0′ (the Turing jump of the Turing degree 0). In this work, 0′ usually refers to the second-least Medvedev or Muchnik degree, and it is clear from context which degree is meant. Finally, both D_{s} and D_{w} are Brouwer algebras. For mass problems X and Y,

deg_{s}(X) → deg_{s}(Y) = deg_{s}({e^{⌢}g | (∀f ∈ X)(Φ_{e}(f ⊕ g) ∈ Y)}) and
deg_{w}(X) → deg_{w}(Y) = deg_{w}({g | (∀f ∈ X)(∃h ∈ Y)(h ≤_{T} f ⊕ g)}).

See Sorbi’s [74] for a good introduction to D_{s} and D_{w}.

1.4 Substructures of D_{s} and D_{w}

Substructures of D_{s} and D_{w} naturally arise by restricting the family of mass problems under consideration. We consider the degrees of closed mass problems and effectively closed mass problems (i.e., Π_{1}^{0} classes).

1.4.1 Closed degrees

A Medvedev or Muchnik degree is closed (compact) if it is of the form deg(X) where X is closed (compact) in ω^{ω}. Let

D_{s,cl} = {deg_{s}(X) | X ⊆ ω^{ω} ∧ X is closed },
D_{s,cl}^{01} = {deg_{s}(X) | X ⊆ ω^{ω} ∧ X is compact},
D_{w,cl} = {deg_{w}(X) | X ⊆ ω^{ω} ∧ X is closed }, and
D_{w,cl}^{01} = {deg_{w}(X) | X ⊆ ω^{ω} ∧ X is compact}.

By inspecting the definitions, one can check that if X and Y are closed (compact), then so are X + Y and X × Y. Thus D_{s,cl} and D_{s,cl}^{01} are distributive sublattices of D_{s}, and D_{w,cl} and D_{w,cl}^{01} are distributive sublattices of D_{w}. Similarly, D_{s,cl}^{01} is a distributive sublattice of D_{s,cl} and D_{w,cl}^{01} is a distributive sublattice of D_{w,cl}. D_{s,cl}, D_{s,cl}^{01}, D_{w,cl}, and D_{w,cl}^{01} all have least element 0 = deg(ω^{ω}) = deg(2^{ω}) and greatest element 1 = deg(∅). Notice that ω^{ω} is not compact, but it has the same degree as 2^{ω}.

The closed subsets of ω^{ω} form the topologically simplest class which yields non-trivial degree structures because every non-empty open set contains a recursive function. As such, closed degrees are worthy objects of study. For example, Bianchini and Sorbi [9] studied the filter in D_{s} generated by the nonminimum closed degrees. Lewis, Shore, and Sorbi [39] have made a recent study of topologically-defined collections of Medvedev degrees.

In general, every X ⊆ ω^{ω} is Medvedev equivalent (and hence also Muchnik equivalent) to some Y ⊆ 2^{ω}.

Lemma 1.4.1. If X ⊆ ω^{ω} then there is a Y ⊆ 2^{ω} with X ≡_{s} Y.

Proof.For f ∈ ω^{ω}, let graph(f) ∈ 2^{ω} denote {⟨ n, m ⟩ | f(n) = m}. Given X, let Y = {graph(f) | f ∈ X }. Let Φ be the functional such that Φ(f)(⟨ n, m ⟩) = 1 if f(n) = m and Φ(f)(⟨ n, m ⟩) = 0 otherwise. Then Φ(f) = graph(f) for all f. Thus Φ(X) = Y. Let Ψ be the functional such that Ψ(g)(n) searches for an m such that g(⟨ n, m ⟩) = 1 and outputs such an m if it is found. If g is the characteristic function of graph(f), then Ψ(g) is total and equals f. Hence Ψ(X) = Y.

If we let D_{s}^{01} denote the Medvedev degrees of mass problems X ⊆ 2^{ω} and let D_{w}^{01} denote the Muchnik degrees of mass problems X ⊆ 2^{ω}, then Lemma 1.4.1 says D_{s} = D_{s}^{01} and D_{w} = D_{w}^{01}. However, if X ⊆ ω^{ω} is closed, then the Y ⊆ 2^{ω} produced by Lemma 1.4.1 need not be. Turing functionals are continuous, but ω^{ω} and 2^{ω} are not homeomorphic. Nevertheless, if X ⊆ ω^{ω} is compact, then Lemma 1.4.1 produces a closed Y ⊆ 2^{ω}. So every compact X ⊆ ω^{ω} is Medvedev equivalent (and hence also Muchnik equivalent) to a closed (hence compact) Y ⊆ 2^{ω}. This explains the notations D_{s,cl}^{01} and D_{w,cl}^{01} for the collections of compact degrees.

Our topological considerations of Medvedev reducibility are consequences of the familiar use property (see [37] section I.3). If Φ(f)(m) = n, then there is a finite σ ⊂ f such that σ contains all the answers to the oracle queries made during the computation of Φ(f)(m) = n. This is written Φ(σ)(m) = n and implies Φ(g)(m) = n for any g ⊃ σ. The starting point is the following simple lemma.

Lemma 1.4.2. Let m, n ∈ ω. For any program Φ, the set {f ∈ ω^{ω} | Φ(f)(m) = n} is open. If Φ(f) is total for all f ∈ X, then {f ∈ X | Φ(f)(m) = n} is clopen in X.

Proof.If Φ(f)(m) = n, then by the use property there is some σ ⊂ f such that Φ(σ)(m) = n. Hence {f ∈ ω^{ω} | Φ(f)(m) = n} = ⋃{I(σ) | Φ(σ)(m) = n}.

If Φ is total on X, then

{f ∈ X | Φ(f)(m) = n} = X ∩ {f ∈ ω^{ω} | Φ(f)(m) = n}

= X ∩ (⋂_{i≠n}{f ∈ ω^{ω} | Φ(f)(m) ≠ i}).

The last equality holds because if Φ(f) is total and Φ(f)(m) ≠ i for all i ≠ n, then it must be that Φ(f)(m) = n.

1.4.2 Effectively closed degrees

Recall our convention that a Π_{1}^{0} class is a non-empty Π_{1}^{0} subset of 2^{ω}. Let

E_{s} = {deg_{s}(X) | X is a Π_{1}^{0} class} and
E_{w} = {deg_{w}(X) | X is a Π_{1}^{0} class}.

By inspecting the definitions, one can check that if X and Y are both Π_{1}^{0} classes, then so are X + Y and X × Y. Thus E_{s} is a distributive sublattice of D_{s} (in fact of D_{s,cl}^{01}) and E_{w} is a distributive sublattice of D_{w} (in fact of D_{w,cl}^{01}). Moreover, given indices for trees T_{0} and T_{1}, we can effectively produce produce indices for trees corresponding to [T_{0}] +[T_{1}] and [T_{0}] ×[T_{1}]. Let T_{0} + T_{1} = {σ ⊕ τ | σ ∈ T_{0} ∧ τ ∈ T_{1} ∧ |τ| ≤ |σ| ≤ |τ|+1}. Then [T_{0}] +[T_{1}] = [T_{0} + T_{1}] and [T_{0}] ×[T_{1}] = [0^{⌢}T_{0} ∪1^{⌢}T_{1}].

E_{s} and E_{w} inherit the least element 0 = deg(2^{ω}) from D_{s} and D_{w}, respectively. The empty set is not a Π_{1}^{0} class, so deg(∅) is not in E_{s} or E_{w}. However, E_{s} and E_{w} still have a greatest element 1. Let DNR_{2} = {f ∈ 2^{ω} | ∀e(f(e) ≠ Φ_{e}(e))} (DNR stands for diagonally non-recursive). Then 1 = deg(DNR_{2}) has greatest degree in both E_{s} and E_{w} (see [65] Lemma 3.20). There are many more natural Π_{1}^{0} classes which also have greatest degree. For example, the class of all (appropriately Gödel numbered) complete consistent extensions of Peano arithmetic has greatest degree in both E_{s} and E_{w}.

E_{s} and E_{w} are the effective counterparts of D_{s,cl}^{01} and D_{w,cl}^{01}. They have enjoyed considerable attention from many authors, beginning with Simpson’s suggestion to the Foundations of Mathematics discussion group that E_{w} is analogous to E_{T}, the Turing degrees of r.e. sets, but with more natural examples [64]. This analogy with E_{T} drives much of the research on E_{s} and E_{w}. For example, every non-minimum member of E_{s} and E_{w} is join-reducible [10], reflecting Sacks’s splitting theorem for E_{T} [51], and E_{s} dense [15], reflecting Sacks’s density theorem for E_{T} [53]. The question of whether E_{w} is dense remains open. See the recent surveys by Simpson [61] and Hinman [26] for an overview of E_{s} and E_{w}.

The interplay between uniformity and compactness gives us a simple description of Medvedev reducibility between two Π_{1}^{0} classes.

Lemma 1.4.3. [T_{0}] ≤_{s}[T_{1}] is Σ_{3}^{0} relative to the trees T_{0} and T_{1}.

Proof.For a given Turing functional Φ, we show that

Φ([T_{1}]) ⊆ [T_{0}] if and only if
(∀n ∈ ω)(∃s ∈ ω)(∀σ ∈ 2^{s})(σ ∈ T_{1} → Φ(σ) ↾ n ∈ T_{0}),

where Φ(σ) ↾ n ∈ T_{0} abbreviates (∀i < n)(Φ(σ)(i)↓) ∧ Φ(σ) ↾ n ∈ T_{0}. It then follows that

[T_{0}] ≤_{s}[T_{1}] if and only if
(∃e ∈ ω)(∀n ∈ ω)(∃s ∈ ω)(∀σ ∈ 2^{s})(σ ∈ T_{1} → Φ_{e}(σ) ↾ n ∈ T_{0}),

which gives our Σ_{3}^{0} definition of ≤_{s}.

For the forward direction, let n ∈ ω be given. Let Σ = {σ ∈ 2^{<ω} | Φ(σ) ↾ n ∈ T_{0}}. The condition Φ([T_{1}]) ⊆ [T_{0}] implies that [T_{1}] ⊆ ⋃_{σ∈Σ} I(σ). By compactness, there is a finite Σ_{0} ⊆ Σ such that [T_{1}] ⊆ ⋃_{σ∈Σ_{0}} I(σ) and an s ∈ ω such that (∀σ ∈ 2^{s})(σ ∈ T_{1} →(∃σ_{0} ∈ Σ_{0})(σ_{0} ⊆ σ)). Then (∀σ ∈ 2^{s})(σ ∈ T_{1} → Φ(σ) ↾ n ∈ T_{0}).

For the reverse direction, consider f ∈ [T_{1}]. Given any n ∈ ω, let s ∈ ω be such that (∀σ ∈ 2^{s})(σ ∈ T_{1} → Φ(σ) ↾ n ∈ T_{0}). Then Φ(f ↾ s) ↾ n ∈ T_{0}, so Φ(f) ↾ n ∈ T_{0}. Thus ∀n(Φ(f) ↾ n ∈ T_{0}). Hence Φ(f) ∈ [T_{0}], and therefore Φ([T_{1}]) ⊆ [T_{0}].

A sequence of trees {T_{n}}_{n∈ω} is uniformly recursive if and only if the set {⟨ n, σ ⟩ | σ ∈ T_{n}} is recursive. A recursive sequence of Π_{1}^{0} classes is a sequence of Π_{1}^{0} classes {X_{n}}_{n∈ω} for which there is a uniformly recursive sequence of trees {T_{n}}_{n∈ω} such that T_{n} ⊆ 2^{<ω} and X_{n} = [T_{n}] for each n ∈ ω. For convenience, we also allow indexing over recursive sets A and consider recursive sequences of Π_{1}^{0} classes of the form {X_{n}}_{n∈A}. Though not strictly necessary for our results, a convenient fact is that there is a recursive sequence of all Π_{1}^{0} classes (with many repetitions). Lemma 1.4.4 (see [16] Chapter XV and [17] Section 2.7). There is a uniformly recursive sequence of infinite trees {T_{e}}_{e∈ω} such that if {Z_{e}}_{e∈ω} is the corresponding recursive sequence of Π_{1}^{0} classes, then for every Π_{1}^{0} class X there is an e ∈ ω such that X = Z_{e}.

Proof.In fact, [17] Lemma 2.2 proves that every Π_{1}^{0} class is of the form [T] for a primitive recursive tree. Let {P_{e}}_{e∈ω} be a recursive sequence of all primitive recursive functions. Then define T′_{e} to be the tree T′_{e} = {σ ∈ 2^{<ω} | (∀τ ⊆ σ)(P_{e}(τ) = 1)}. If P_{e} is the characteristic function of a tree, then T′_{e} is that tree. Thus if X is a Π_{1}^{0} class, then X = [T′_{e}] for some e ∈ ω. We just need to make a final adjustment to ensure that every tree in the sequence is infinite. To this end, let

T_{e} = {σ ∈ 2^{<ω} | σ ∈ T′_{e} ∨(∀m ≤ |σ|)(σ ↾ m ∉ T′_{e} →(∀τ ∈ 2^{m})(τ ∉ T_{e}))}.

1.5 PA^{−} and the standard model of arithmetic

In the next chapter, we code structures that model PA^{−} (Peano arithmetic without induction) in distributive lattices. For reference, we present the axioms of PA^{−} as they appear in [33]. Definition 1.5.1 (see [33] Section 2.1). PA^{−} is the theory axiomatized by the following sentences.

(i) ∀x, y, z((x + y) + z = x + (y + z)) (ii) ∀x, y(x + y = y + x) (iii) ∀x, y, z((x × y) × z = x × (y × z)) (iv) ∀x, y(x × y = y × x)

(v) ∀x, y, z(x × (y + z) = (x × y) + (x × z)) (vi) ∀x(x + 0 = x ∧ x × 0 = 0) (vii) ∀x(x × 1 = x) (viii) ∀x, y, z(x < y ∧ y < z → x < z) (ix) ∀x¬(x < x)

(x) ∀x, y(x < y ∨ x = y ∨ y < x) (xi) ∀x, y, z(x < y → x + z < y + z) (xii) ∀x, y, z(0 < z ∧ x < y → x × z < y × z) (xiii) ∀x, y(x < y → ∃z(x + z = y)) (xiv) 0 < 1 ∧ ∀x(0 < x → x = 1 ∨ 1 < x) (xv) ∀x(x = 0 ∨ 0 < x)

To reduce the quantifier complexity of axiom (xiii) for when we analyze the fragments of Th(L) for various lattices L, we introduce the monus symbol “ ” and Skolemize. We call the resulting theory PA^{} Definition 1.5.2. PA^{}is the theory whose axioms are the same as PA^{−} but with axiom (xiii) replaced by the axiom ∀x, y(x < y → x + (y x) = y).

The standard relational model of arithmetic is the structure N = (ω; < , +, ×, 0, 1), where < is a 2-ary relation on ω, + and × are 3-ary relations on ω, and 0 and 1 are constants in ω interpreted as the usual less-than, plus, times, zero, and one respectively. Th(N) denotes the first-order theory of N. We use the relational versions of + and × instead of the usual functional versions because our coding techniques most naturally code relations. Any formula in which + and × are relation symbols can be trivially translated into an equivalent formula in which + and × are function symbols. Translations in the other direction require unnesting. In general, a formula is said to be unnested if and only if every atomic subformula is of the form x = y, c = y, f(x_{0}, …, x_{n−1}) = y, or R(x_{0}, …, x_{n−1}), where x, y, and the x_{i} for i < n are variables, c is a constant symbol, f is a function symbol, and R is a relation symbol. Every formula can be recursively translated into an equivalent unnested formula (see [28] section 2.6).

When unnesting is applied to a first-order formula in the functional language of arithmetic, we get an equivalent formula in which every atomic subformula is of the form x = y, 0 = y, 1 = y, x < y, x + y = z, or x × y = z. That is, we get an equivalent formula in the relational language of arithmetic. Therefore the relational and functional versions of Th(N) are recursively isomorphic.

Th_{2}(N) denotes the second-order theory of N, in which we allow secondorder variables X, quantification ∃X and ∀X, and second-order membership x ∈ X. Th_{3}(N) denotes the third-order theory of N, in which we allow secondorder variables X, third-order variables X, quantification ∃X, ∀X, ∃X, and ∀X, second-order membership x ∈ X, and third-order membership X ∈ X.

We also make use of the structure N^{}= (ω; <, +, ×, , 0, 1), where <, +, ×, 0, and 1 are as for N, and is the 3-ary relation on ω corresponding to the function

x y = \begin{cases}x − y & if x ≥ y \\ 0 & if x < y. \end{cases}

Clearly, N ⊧ PA^{−}, N^{}⊧ PA^{}, and PA^{}⊢ PA^{−}.

Let M ⊧ PA^{−}. An initial segment of M is a <-downward-closed substructure M′ of M: (∀x ∈ M′)(∀y ∈ M)((M ⊧ y < x) →(y ∈ M′)). An initial interval of M is a subset of M of the form {y ∈ M | M ⊧ y < x ∨ y = x} for some x ∈ M. The reason that we have defined initial intervals to be non-empty is simply for later convenience.

For us, the crucial facts about PA^{−} and its models are the following.

Lemma 1.5.3 (see [33] Theorem 2.2). If M ⊧ PA^{−}, then there is an initial segment of M that is isomorphic to N. In particular, N is the unique model of PA^{−}, up to isomorphism, in which every initial interval is finite.

Lemma 1.5.4 (see [33] Corollary 2.9). If φ is a Σ_{1}^{0} sentence and N ⊧ φ, then PA^{−} ⊢ φ.

1.6 Reverse mathematics

Reverse mathematics, introduced by Friedman [24], is an analysis of the logical strength of the theorems of ordinary mathematics in the context of second-order arithmetic. A result in reverse mathematics typically has the form “T is equivalent to strong stystem over weak system ,” where strong stystem and weak system are subsystems of second-order arithmetic and T is some theorem from ordinary mathematics. This means that T is provable in strong stystem and that all the axioms of strong stystem are provable in weak system ∪{T }. The proof of strong stystem from weak system ∪{T } is called a reversal.

We now describe the axiomatic systems relevant for our work. We follow [67], the standard reference for reverse mathematics. Also see [4] Section 2 for a thorough introduction to most of the systems we consider and for computability-theoretic interpretations of these systems.

Before we describe the systems, we need to know that the basic axioms are the sentences

∀m(m + 1 ≠ 0)
∀m∀n(m + 1 = n + 1 → m = n)
∀m(m + 0 = m)
∀m∀n(m + (n + 1) = (m + n) + 1)
∀m(m × 0 = 0)
∀m∀n(m × (n + 1) = (m × n) + m)
∀m¬(m < 0)
∀m∀n(m < n + 1 ↔(m < n ∨ m = n)),

that the induction axiom is the sentence

∀X((0 ∈ X ∧ ∀n(n ∈ X → n + 1 ∈ X)) → ∀n(n ∈ X)),

and that the comprehension scheme consists of all universal closures of formulas of the form

∃X∀n(n ∈ X ↔ φ(n)),

where φ can be any formula in the language of second-order arithmetic in which X does not occur freely. Full second-order arithmetic consists of the basic axioms, the induction axiom, and the comprehension scheme.

RCA_{0} (for recursive comprehension axiom) consists of the basic axioms, the Σ_{1}^{0} induction scheme, and the Δ_{1}^{0} comprehension scheme. The Σ_{1}^{0} induction scheme consists of all universal closures of formulas of the form

(φ(0) ∧ ∀n(φ(n) → φ(n + 1))) → ∀nφ(n)

where φ is Σ_{1}^{0}. The Δ_{1}^{0} comprehension scheme consists of all universal closures of formulas of the form

∀n(φ(n) ↔ ψ(n)) → ∃X∀n(n ∈ X ↔ φ(n))

where φ is Σ_{1}^{0}, ψ is Π_{1}^{0}, and X does not occur freely in φ. RCA_{0} is the standard weak system for the purpose of reversals. RCA_{0} proves that the function ⟨ i, j ⟩ ↦ (i + j)^{2} + i is injective (see [67] Section II.2). For X ⊆ N and n ∈ N, we define

(X)_{n} = {i | ⟨ i, n ⟩ ∈ X} and
(X)^{n} = {⟨ i, m ⟩ | ⟨ i, m ⟩ ∈ X ∧ m < n}.

RCA_{0} proves that if X exists, then so do (X)_{n} and (X)^{n}. We interpret (X)_{n} as the n^{th} column of X and (X)^{n} as set of the first n columns of X.

WKL_{0} (for weak König’s lemma) consists of RCA_{0} plus the axiom “every infinite subtree of 2^{<ℕ} has an infinite path.” The following equivalent formulation is useful. A tree T ⊆ ℕ^{<ℕ} is called bounded if and only if there is a function g: ℕ → ℕ such that (∀τ ∈ T)(∀m < |τ|)(τ(m) < g(m)). Over RCA_{0}, WKL_{0} is equivalent to the statement “every bounded infinite subtree of ℕ^{<ℕ} has an infinite path.” See [67] Lemma IV.1.4.

ACA_{0} (for arithmetical comprehension axiom) consists of the basic axioms, the induction axiom, and the arithmetical comprehension scheme. The arithmetical comprehension scheme is the restriction of the comprehension scheme to formulas φ that are arithmetical.

ATR_{0} (for arithmetical transfinite recursion) consists of ACA_{0} plus an axiom scheme that says if a set can be constructed by iterating arithmetical comprehension along an existing well-order, then that set exists. Let LO(X, <_{X}) be a formula that says “<_{X} is a linear order on the set X,” and let WO(X, <_{X}) be a formula that says “<_{X} is a well-order on the set X.” Given a formula θ(n, Y), let H_{θ}(X, <_{X}, Y) be a formula that says “LO(X, <_{X}) and Y = {⟨ n, j ⟩ | j ∈ X ∧ θ(n, {⟨ m, i ⟩ ∈ Y | i <_{X} j})}.” The axioms of ATR_{0} consist of those of ACA_{0} plus all universal closures of formulas of the form

∀X∀<_{X} (WO(X, <_{X}) → ∃Y H_{θ}(X, <_{X}, Y))

where θ is arithmetical. An easier-to-understand equivalent of ATR_{0} is the system Σ_{1}^{1} separation, which consists of the axioms of RCA_{0} plus the all universal closures of formulas of the form

¬∃n(φ_{0}(n) ∧ φ_{1}(n)) → ∃Z∀n((φ_{0}(n) → n ∈ Z) ∧(φ_{1}(n) → n ∉ Z)),

where φ_{0} and φ_{1} are Σ_{1}^{1} and Z does not occur freely in either φ_{0} or φ_{1} (see [67] Theorem V.5.1).

Σ_{1}^{1}-DC_{0} (for Σ_{1}^{1} dependent choice) consists of ACA_{0} and the scheme of Σ_{1}^{1} dependent choice. The scheme of Σ_{1}^{1} dependent choice consists of all universal closures of formulas of the form

∀n∀X∃Y η(n, X, Y) → ∃Z∀nη(n, (Z)^{n}, (Z)_{n})

where η is Σ_{1}^{1} and Z does not occur freely in η.

Π_{1}^{1}-CA_{0} (for Π_{1}^{1} comprehension axiom) consists of the basic axioms, the induction axiom, and the Π_{1}^{1} comprehension scheme. The Π_{1}^{1} comprehension scheme is the restriction of the comprehension scheme to formulas φ that are Π_{1}^{1}.

RCA_{0} is strictly weaker than ACA_{0}, which is strictly weaker than both ATR_{0} and Σ_{1}^{1}-DC_{0}. ATR_{0} and Σ_{1}^{1}-DC_{0} are independent over RCA_{0}. However, ATR_{0} proves the consistency of Σ_{1}^{1}-DC_{0}. Both ATR_{0} and Σ_{1}^{1}-DC_{0} are strictly weaker than Π_{1}^{1}-CA_{0}.

Our proof of Menger’s theorem in Π_{1}^{1}-CA_{0} relies on two key metamathematical facts. The first key fact concerns the existence of β-models. The second key fact concerns the existence of models of Σ_{1}^{1}-DC_{0}.

Definition 1.6.1. A countable coded ω-model is a set X ⊆ ℕ viewed as coding the structure M = (ℕ, {(X)_{n} | n ∈ ℕ}, +, ·, 0, 1, <).

We usually identify a countable coded ω-model X with the structure M that it codes.

Definition 1.6.2. A countable coded β-model is a countable coded ω-model M that is absolute for Σ_{1}^{1} formulas with parameters from M. That is, if φ is a Σ_{1}^{1} formula with parameters from M, then M ⊧ φ if and only if φ is true.

Theorem 1.6.3 (see [67] Theorem VII.2.10). The statement “for every X there is a countable coded β-model M with X ∈ M” is equivalent to Π_{1}^{1}-CA_{0} over ACA_{0}.

It is helpful to keep in mind that ACA_{0} proves that every countable coded β-model is a model of ATR_{0} (see [67] Theorem VII.2.7).

Theorem 1.6.4 (see [67] Theorem VIII.4.20). ATR_{0} proves that for every X there is a countable coded ω-model M of Σ_{1}^{1}-DC_{0} with X ∈ M.

The statement “for every X there is a countable coded ω-model M of Σ_{1}^{1}-DC_{0} with X ∈ M” is in fact equivalent to ATR_{0} over RCA_{0}. See [67] Lemma VIII.4.15 for the reversal.

CHAPTER 2

CODING TRUE ARITHMETIC IN THE MEDVEDEV AND MUCHNIK DEGREES

The results of this chapter also appear in [54] and [57], both by the author.

A classic problem in computability theory is to determine the complexity of the first-order theory of a given degree structure, such as D_{T}, D_{s}, or D_{w}. The benchmarks are theories of arithmetic, the comparisons are made via recursive isomorphisms, and the results typically express that the first-order theories of the degree structures are as complicated as possible. The original result of this sort, due to Simpson, is that the first-order theory of D_{T} is recursively isomorphic to the second-order theory of true arithmetic [62]. We show that the firstorder theories of D_{s} and D_{w} are both recursively isomorphic to the third-order theory of true arithmetic (Theorem 2.3.10). This result was obtained independently by Lewis, Nies, and Sorbi [38].

Various substructures arise in the study of degree structures, and the complexities of their first-order theories naturally come into question. In the Turing degrees, two popular substructures are D_{T}(≤_{T} 0′), the Turing degrees below 0′, and E_{T}, the Turing degrees of r.e. sets. Both D_{T}(≤_{T} 0′) and E_{T} have first-order theories that are recursively isomorphic to the first-order theory of true arithmetic. The D_{T}(≤_{T} 0′) case is due to Shore [59]. The E_{T} case is due to unpublished work of Harrington and Slaman (see also [46]). We consider the substructures D_{s,cl}, D_{s,cl}^{01}, and E_{s} of D_{s} and the substructures D_{w,cl}, D_{w,cl}^{01}, and E_{w} of D_{w}. Our results are that the first-order theories of D_{s,cl}, D_{s,cl}^{01}, D_{w,cl}, and D_{w,cl}^{01} are all recursively isomorphic to the second-order theory of true arithmetic (Theorem 2.6.5 and Theorem 2.6.5), that the first-order theory of E_{s} is recursively isomorphic to the first-order theory of true arithmetic (Theorem 2.9.4), and that the first-order theory of E_{w} is undecidable (Theorem 2.11.7). The question of the exact complexity of the first-order theory of E_{w} remains wide open. Cole and Simpson conjecture that the first-order theory of E_{w} is recursively isomorphic to O^{(ω)} (the ω^{th} Turing jump of Kleene’s O), the obvious upper bound [19]. As a bonus, our coding methods also yield that neither D_{s,cl} nor D_{s,cl}^{01} is elementarily equivalent to either D_{w,cl} or D_{w,cl}^{01} (Theorem 2.7.2).

We also consider the decidabilities of fragments of the first-order theories of D_{s}, D_{w}, and their substructures. Our results are that if L is any of D_{s}, D_{w}, D_{s,cl}, D_{s,cl}^{01}, D_{w,cl}, D_{w,cl}^{01}, E_{s}, or E_{w}, then the Σ_{3}^{0}-theory of L as a lattice and the Σ_{4}^{0}-theory of L as a partial order are undecidable (Theorem 2.3.11, Theorem 2.4.11, Theorem 2.6.6, Theorem 2.9.5, and Theorem 2.11.7). In the positive direction, Binns has shown that the Σ_{1}^{0}-theories of E_{s} and E_{w} as a lattices are identical and decidable [10]. Binns’s results also imply that the Σ_{1}^{0}-theories of D_{s}, D_{w}, D_{s,cl}, D_{s,cl}^{01}, D_{w,cl}, and D_{w,cl}^{01} as lattices are decidable. Cole and Kihara have shown that the Σ_{2}^{0}-theory of E_{s} as a partial order is decidable [18]. No further results of this sort are known. There has been a huge amount of difficult work on the decidability of various fragments of the first-order theories of D_{T} and E_{T}. In light of the guiding analogy that E_{s} and E_{w} are like E_{T}, we summarize the results for E_{T} for comparison (see [60] for a survey of this area). The Σ_{1}^{0}-theory of E_{T} as an upper-semilattice is decidable [52]. The decidability of the Σ_{2}^{0}-theory of E_{T} as a partial order and the decidability of the Σ_{2}^{0}-theory of E_{T} as an uppersemilattice remain unknown. However, the Σ_{3}^{0}-theory of E_{T} as a partial order is undecidable [36]. Moreover, if one extends the partial infimum function on E_{T} (as an upper-semilattice) to any total function, then the Σ_{2}^{0}-theory of the resulting structure is undecidable [43]. These two undecidability results for E_{T} suggest by analogy that the Σ_{2}^{0}-theory of E_{s} as a lattice, the Σ_{3}^{0}-theory of E_{s} as a partial order, the Σ_{2}^{0}-theory of E_{w} as a lattice, and the Σ_{3}^{0}-theory of E_{w} as a partial order may all be undecidable.

We prove that E_{s} is as complicated as possible in terms of degree of presentation (Theorem 2.10.6). Specifically, we prove that the degree of E_{s} as a lattice is 0′′′. This means that 0′′′ computes a presentation of E_{s} as a lattice and that 0′′′ is computable in every presentation of E_{s} as a lattice. A corollary is that E_{s} has no recursive presentation as a partial order. The natural presentation of E_{w} has Turing degree O [19], so it is reasonable to expect that E_{w} has degree O, though this question remains open. For comparison, it follows from the results of [46] (though it is not stated explicitly) that the degree of E_{T} as an upper-semilattice is 0^{(4)}.

2.1 Interpreting the Medvedev and Muchnik degrees in arithmetic

Before we code arithmetic into D_{s}, D_{w}, and their substructures, we show how to interpret these structures in arithmetic.

The reductions Th(D_{s}) ≤_{1} Th_{3}(N) and Th(D_{w}) ≤_{1} Th_{3}(N) follow from the fact that every mass problem X is equivalent to some Y ⊆ 2^{ω} (i.e., Lemma 1.4.1) and that the Medvedev and Muchnik reducibilities are definable in third-order arithmetic.

Lemma 2.1.1. Th(D_{s}; ≤_{s}) ≤_{1} Th_{3}(N) and Th(D_{w}; ≤_{w}) ≤_{1} Th_{3}(N).

Proof.The relation R(Y, e, m, n) expressing Φ_{e}(Y)(m) = n is definable by a formula which says “there exists a number s coding a sequence of configurations witnessing the computation Φ_{e}(Y)(m) = n.” The relation S(X, Y, e) expressing Φ_{e}(Y) = X is definable by the formula

∀m((m ∈ X → R(Y, e, m, 1)) ∧(m ∉ X → R(Y, e, m, 0))).

Thus the relation X ≤_{s} Y is definable by the formula

φ(X , Y) = ∃e∀Y (Y ∈ Y → ∃X(X ∈ X ∧ S(X, Y, e))).

Now, given a first-order sentence ψ in the language of partial orders, produce a third-order sentence ψ′ in the language of arithmetic by replacing quantifications ∀x and ∃x with third-order quantifications ∀X and ∃X, by replacing atomic formulas x ≤ y with φ(X , Y), and by replacing atomic formulas x = y with φ(X , Y) ∧ φ(Y, X). Then N ⊧ ψ′ if and only if D_{s} ⊧ ψ.

The reduction Th(D_{w}; ≤_{w}) ≤_{1} Th_{3}(N) is obtained by switching the quantifiers ∃e and ∀Y in the definition of the formula φ above.

The interpretations of D_{s,cl} and D_{w,cl} (D_{s,cl}^{01} and D_{w,cl}^{01}) in second-order arithmetic rely on the fact that X ⊆ ω^{ω} (X ⊆ 2^{ω}) is closed if and only if it is the set of paths through some tree T ⊆ ω^{<ω} (T ⊆ 2^{<ω}). Thus we quantify over all closed mass problems by quantifying over all trees. Fix some definable coding of sequences, trees, and functions in second-order arithmetic. See [67] Section II.2 for a particularly careful method.

Lemma 2.1.2. Th(D_{s,cl}; ≤_{s}) ≤_{1} Th_{2}(N), Th(D_{s,cl}^{01}; ≤_{s}) ≤_{1} Th_{2}(N), Th(D_{w,cl}; ≤_{w}) ≤_{1} Th_{2}(N), and Th(D_{w,cl}^{01}; ≤_{w}) ≤_{1} Th_{2}(N).

Proof.The relation P (f, T) expressing “function f is a path through tree T ” is definable by the formula

∀n∃σ(σ ∈ T ∧ |σ| = n ∧(∀i < |σ|)(σ(i) = f(i))).

Relations R(g, e, m, n) expressing Φ_{e}(g)(m) = n and S(f, g, e) expressing Φ_{e}(g) = f are definable as in Lemma 2.1.1. Thus the relation [T] ≤_{s}[S] is definable by the formula

φ(T, S) = ∃e∀g(P (g, S) → ∃f(P (f, T) ∧ S(f, g, e))).

Now, given a first-order sentence ψ in the language of partial orders, produce a second-order sentence ψ′ in the language of arithmetic by replacing quantifications ∀x and ∃x with second-order quantifications ∀T_{x} and ∃T_{x} quantifying over trees T_{x} ⊆ ω^{<ω}, by replacing atomic formulas x ≤ y with φ(T_{x}, T_{y}), and by replacing atomic formulas x = y with φ(T_{x}, T_{y}) ∧ φ(T_{y}, T_{x}). Then N ⊧ ψ′ if and only if D_{s,cl} ⊧ ψ. This proves Th(D_{s,cl}; ≤_{s}) ≤_{1} Th_{2}(N). The reduction Th(D_{s,cl}^{01}; ≤_{s}) ≤_{1} Th_{2}(N) is exactly the same, except we quantify over trees T ⊆ 2^{<ω}.

The reductions Th(D_{w,cl}; ≤_{w}) ≤_{1} Th_{2}(N) and Th(D_{w,cl}^{01}; ≤_{w}) ≤_{1} Th_{2}(N) are obtained by switching the quantifiers ∃e and ∀g in the definition of the formula φ above.

To interpret E_{s} in first-order arithmetic, we use the recursive sequence containing all Π_{1}^{0} classes {Z_{e}}_{e∈ω} from Lemma 1.4.4 and the fact that Z_{i} ≤_{s} Z_{j} is a Σ_{3}^{0} property of i and j.

Lemma 2.1.3. Th(E_{s}; ≤_{s}) ≤_{1} Th(N).

Proof.Let {Z_{e}}_{e∈ω} be a recursive sequence containing all Π_{1}^{0} classes as in Lemma 1.4.4, and let {T_{e}}_{e∈ω} be the corresponding uniformly recursive sequence of trees. Given a first-order sentence ψ in the language of partial orders, produce an equivalent sentence in the language of partial orders by replacing every atomic formula x = y by the formula x ≤ y ∧ y ≤ x. Then produce a first-order sentence ψ′ in the language of arithmetic by replacing every atomic formula x ≤ y by the Σ_{3}^{0} formula from Lemma 1.4.3 expressing [T_{x}] ≤_{s}[T_{y}]. Then N ⊧ ψ′ if and only if E_{s} ⊧ ψ.

2.2 Coding arithmetic in distributive lattices

We present our scheme for coding arithmetic in distributive lattices. Although the definitions below make sense in any lattice, they were designed with the particular goal of coding N into D_{s}, D_{w}, and their sublattices in mind. For example, meet-irreducible elements play a major role in the coding. One may dualize the coding to replace meet-irreducible by join-irreducible, but this would not suffice for our purposes because all non-zero elements of E_{s} are join-reducible [10]. The coding presented here has been slightly modified from the original version developed in [57] in order to reduce the quantifier complexity of coded relations.

2.2.1 Coding models of relational theories

Definition 2.2.1. For elements s and w of a lattice, s meets to w if and only if ∃y(y > w ∧ s × y = w).

Definition 2.2.2. For a lattice L and a w ∈ L,

E(w) = {s ∈ L | s is meet-irreducible ∧ s meets to w}.

The next two lemmas prove important properties of E in distributive lattices.

Lemma 2.2.3. If L is a distributive lattice and w ∈ L, then E(w) is an antichain.

Proof.Suppose for a contradiction that there are s, s′ ∈ E(w) with s > s′. Let y > w be such that s × y = w. Then s′ ≥ y because s′ is meet-irreducible, s′ ≥ s × y, and s′ ≱ s. Therefore s > s′ ≥ y > w, giving the contradiction s × y = y > w.

Lemma 2.2.4. If L is a distributive lattice and {s_{i}}_{i<n} ⊆ L is a finite non-empty antichain of meet-irreducible, non-maximum elements, then E(∏_{i<n} s_{i}) = {s_{i}}_{i<n}.

Proof.Suppose n = 1. There is a y > s_{0} because s_{0} is not maximum. Thus s_{0} × y = s_{0}, so y witnesses that s_{0} meets to s_{0}. Thus s_{0} ∈ E(s_{0}). On the other hand, if x ∈ E(s_{0}), then there is a y > s_{0} such that x × y = s_{0}. Since s_{0} is meet-irreducible and s_{0} ≱ y, it must be that s_{0} ≥ x. E(s_{0}) is an antichain by Lemma 2.2.3 and s_{0} ∈ E(s_{0}), so it must be that x = s_{0}. Hence E(s_{0}) = {s_{0}}.

Suppose n > 1, and let w = ∏_{i<n} s_{i}. First we show that s_{i} ∈ E(w) for each i < n. Fix i < n and let t_{i} = ∏{s_{j} | j < n ∧ j ≠ i} so that s_{i} × t_{i} = w. Then s_{i} ≱ t_{i} because otherwise the meet-irreducibility of s_{i} implies that s_{i} ≥ s_{j} for some j ≠ i, contradicting that {s_{i}}_{i<n} is an antichain. Thus s_{i} is meet-irreducible and t_{i} witnesses that s_{i} meets to w. Hence s_{i} ∈ E(w). Conversely, if x ∈ E(w), then x is meet-irreducible and x ≥ w. Thus x ≥ s_{i} for some i < n, so x = s_{i} because E(w) is an antichain by Lemma 2.2.3. Thus E(w) = {s_{i}}_{i<n}.

In the above lemma, the non-maximum hypothesis is only for the n = 1 case. If n > 1, then no member of {s_{i}}_{i<n} is maximum because it is an antichain, so the non-maximum hypothesis is automatically satisfied. If L has a maximum element 1, then E(1) = ∅ even though 1 is meet-irreducible. This is by the definition of “meets to,” because there is no y ∈ L with y > 1.

Given an element w of a lattice, we think of w as code for the set E(w). The symbol “E” stands for “elements,” as in the elements of the set coded by w.^{1}

Now we code 2-ary and 3-ary relations on E(w_{0}) for an element w_{0} of a lattice L. The same scheme can code n-ary relations for any n ∈ ω, but we only need to code 2-ary and 3-ary relations to code N. The intuition behind the following definition is that if s_{0}, u_{0} ∈ E(w_{0}), then s_{0} + u_{0} should code the pair (s_{0}, u_{0}). However, this coding makes the pairs (s_{0}, u_{0}) and (u_{0}, s_{0}) indistinguishable because s_{0} + u_{0} = u_{0} + s_{0}. To solve this problem, we fix additional parameters w_{1}, w_{2}, m ∈ L. Once w_{0}, w_{1}, w_{2}, m ∈ L are fixed, any c ∈ L can be interpreted as coding a 2-ary relation R_{c}^{2} on E(w_{0}) and a 3-ary relation R_{c}^{3} on E(w_{0}).

Definition 2.2.5. Let L be a lattice and fix elements w_{0}, w_{1}, w_{2}, m ∈ L. Then any c ∈ L defines a 2-ary relation R_{c}^{2} on E(w_{0}) and a 3-ary relation R_{c}^{3} on E(w_{0}) by

R_{c}^{2}(s_{0}, u_{0}) if and only if s_{0} ∈ E(w_{0}) ∧ u_{0} ∈ E(w_{0})
∧ ∃u_{1}(u_{1} meets to w_{1} ∧ u_{0} + u_{1} ≥ m ∧ s_{0} + u_{1} ≥ c)
R_{c}^{3}(s_{0}, u_{0}, v_{0}) if and only if s_{0} ∈ E(w_{0}) ∧ u_{0} ∈ E(w_{0}) ∧ v_{0} ∈ E(w_{0})
∧ ∃u_{1}∃v_{2}(u_{1} meets to w_{1} ∧ v_{2} meets to w_{2}
∧ u_{0} + u_{1} ≥ m ∧ v_{0} + v_{2} ≥ m ∧ s_{0} + u_{1} + v_{2} ≥ c).

With Definition 2.2.5 in hand, we can define codes for models of various theories. For PA^{−} we have the following definitions.

Definition 2.2.6. In a lattice L, a code (for a structure in the language of arithmetic) is a sequence of elements

w⃗ = (w_{0}, w_{1}, w_{2}, m, ℓ, p, t, z, o)

from L interpreted as coding the structure

Mw⃗ = (E(w_{0}); R_{ℓ}^{2}, R_{p}^{3}, R_{t}^{3}, z, o)

where R_{ℓ}^{2}, R_{p}^{3}, and R_{t}^{3} are the relations on E(w_{0}) defined from ℓ, p, and t, respectively, as in Definition 2.2.5.

In Definition 2.2.6, w is for “ω,” m is for “match,” ℓ is for “less,” p is for “plus,” t is for “times,” z is for “zero,” and o is for “one.”

If w⃗ is a code in a lattice L, then sentences in the language of arithmetic are interpreted in Mw⃗ in the obvious way.

Definition 2.2.7. Let φ be a first-order sentence in the language of arithmetic. The translation of φ is the first-order formula φ′(w_{0}, w_{1}, w_{2}, m, ℓ, p, t, z, o) (with the displayed variables free) in the language of lattices obtained from φ by making the following replacements.

1In [57], E(w) was called Ẽ(w) (see [57] Definition 4.4) and its definition required that the s ∈ Ẽ(w) also be minimal with respect being meet-irreducible and meeting to w. The minimality requirement is unnecessary by Lemma 2.2.3.

• Replace < by the formula defining R_{ℓ}^{2},

• replace + by the formula defining R_{p}^{3},

• replace × by the formula defining R_{t}^{3},

• replace 0 by z,

• replace 1 by o,

• replace ∃x by the formula expressing ∃x ∈ E(w_{0}), and

• replace ∀x by the formula expressing ∀x ∈ E(w_{0}).

If L is a lattice and w⃗ is a code, then Mw⃗ ⊧ φ means that L ⊧ φ′(w⃗).

Definition 2.2.8. In a lattice L, a code for a model of PA^{−} is a code w⃗ such that Mw⃗ ⊧ PA^{−}.

If φ is a first-order sentence in the language of arithmetic, then its translation φ′ is a first-order formula in the language of lattices. Thus for such a sentence φ, the property “w⃗ is a code such that Mw⃗ ⊧ φ” is first-order. The property “w⃗ is a code for a model of PA^{−}” is therefore expressible by a first-order formula in the language of lattices.

To code true first-, second-, or third-order arithmetic in a lattice, we impose an extra first-, second-, or third-order correctness condition on a code w⃗ for a model of PA^{−}, and we impose additional assumptions on the properties of the lattice in which the coding is done. In this way we ensure that coded structures Mw⃗ are isomorphic to N and that first-, second-, and third-order quantification over Mw⃗ can be simulated by first-order quantification over L.

2.2.2 The finite matching property and the first-order correctness condition

In this section we present sufficient conditions for interpreting true first-order arithmetic in a distributive lattice. Our strategy is to recognize the codes w⃗ such that Mw⃗ ≅ N as the codes w⃗ of models of PA^{−} such that every initial interval of Mw⃗ is finite. The following definitions allows us to compare the cardinalities of initial intervals of coded models of PA^{−}.

Definition 2.2.9. Let L be a lattice and let w⃗ be a code for a model of PA^{−}. An a ∈ L codes an initial interval of Mw⃗ if and only if (∃s ∈ E(w_{0}))(∀b ∈ L)(b ∈ E(a) ↔ R_{ℓ}^{2}(b, s) ∨ b = s).

Definition 2.2.10. For a lattice L and elements r, q ∈ L, E(r) matches E(q) if and only if there is a z ∈ L such that

(i) (∀x ∈ E(q))(∃!y ∈ E(r))(x + y ∈ E(z)), and (ii) (∀x ∈ E(r))(∃!y ∈ E(q))(x + y ∈ E(z)).

Clearly if E(r) matches E(q), then |E(r)| = |E(q)|. The next definition enforces a weak converse of this fact.

Definition 2.2.11. A lattice L has the finite matching property if and only if whenever q, q′ ∈ L are such that |E(q)| = |E(q′)| = n for some n ∈ ω then there is an r ∈ L such that E(r) matches both E(q) and E(q′).

We can now define the first-order correctness condition and prove that a code for a model of PA^{−} that satisfies the first-order correctness condition always codes a structure isomorphic to N provided that L is distributive, that L has the finite matching property, and that some code in L codes a structure isomorphic to N. It follows that Th(N) ≤_{1} Th(L).

Definition 2.2.12. In a lattice L, a code w⃗ satisfies the first-order correctness condition if and only if

(i) (∀s ∈ E(w_{0}))(∃a ∈ L)(∀b ∈ L)(b ∈ E(a) ↔ R_{ℓ}^{2}(b, s) ∨ b = s) (that is, every initial interval of Mw⃗ is coded by some a ∈ L), and

(ii) for every a ∈ L that codes an initial interval of Mw⃗ and every code w′ for a model of PA^{−} that satisfies item (i), there is an a′ ∈ L that codes an initial interval of M_{}_{w′} and an r ∈ L such that E(r) matches both E(a) and E(a′).

Observe that the property “w⃗ is a code a model of PA^{−} that satisfies the firstorder correctness condition” can be expressed by a first-order formula in the language of lattices.

Lemma 2.2.13. Let L be a distributive lattice with the finite matching property.

(i) If w⃗ is a code such that Mw⃗ ≅ N, then w⃗ is a code for a model of PA^{−} satisfying the first-order correctness condition.

(ii) If there is a code w⃗ such that Mw⃗ ≅ N, then M_{}_{w′} ≅ N for every w′ that is a that is a code for a model of PA^{−} satisfying the first-order correctness condition.

Proof.For item (i), let w⃗ be a code such that Mw⃗ ≅ N. The code w⃗ is a code for a model of PA^{−} because Mw⃗ ≅ N. For Definition 2.2.12 item (i), let s ∈ E(w_{0}) and notice that {b | R_{ℓ}^{2}(b, s) ∨ b = s} is finite because it is an initial interval of a structure isomorphic to N and that it is an antichain because it is a subset of E(w_{0}) which is an antichain by Lemma 2.2.3. Thus a = ∏{b | R_{ℓ}^{2}(b, s) ∨ b = s} witnesses Definition 2.2.12 item (i) for s because E(a) = {b | R_{ℓ}^{2}(b, s) ∨ b = s} by Lemma 2.2.4. For Definition 2.2.12 item (ii), let a ∈ L code an initial interval of Mw⃗ and let w′ be a code for a model of PA^{−} satisfying Definition 2.2.12 item (i). |E(a)| = n for some n ∈ ω because E(a) is an initial interval of a structure isomorphic to N. M_{}_{w′} ⊧ PA^{−}, so by Lemma 1.5.3 there is an initial interval of M_{}_{w′} of cardinality n and, by Definition 2.2.12 item (i), there is an a′ ∈ L coding this initial interval. Thus |E(a)| = |E(a′)| = n, so by the finite matching property there is an r ∈ L such that E(r) matches both E(a) and E(a′). Thus w⃗ is indeed a code for a model of PA^{−} satisfying the first-order correctness condition.

For item (ii), let w⃗ be a code such that Mw⃗ ≅ N, and let w′ be a code for a model of PA^{−} satisfying the first-order correctness condition. We show that M_{}_{w′} ≅ N. By Lemma 1.5.3, it suffices to show that every initial interval M_{}_{w′} is finite. Thus let s′ ∈ E(w′_{0}), let {b′ | R_{ℓ′}^{2}(b′, s′) ∨ b′ = s′} be the corresponding initial interval, and, by Definition 2.2.12 item (i), let a′ ∈ L code this initial interval. By item (i), w⃗ is a code for a model of PA^{−} satisfying Definition 2.2.12 item (i), so by Definition 2.2.12 item (ii) (applied to w′) there is an a ∈ L coding an initial interval of Mw⃗ ≅ N such that |E(a)| = |E(a′)|. E(a) is finite, hence the initial interval {b′ | R_{ℓ′}^{2}(b′, s′) ∨ b′ = s′} is finite.

Lemma 2.2.14. Let L be a distributive lattice with the finite matching property such that there exists a code w⃗ such that Mw⃗ ≅ N. Then Th(N) ≤_{1} Th(L; ≤).

Proof.It suffices to prove Th(N) ≤_{1} Th(L) because Th(L) ≤_{1} Th(L; ≤) as the lattice operations + and × are first-order definable from the partial order ≤. Let φ be a first-order sentence in the language of arithmetic. Let θ be the first-order sentence

θ = ∃w⃗(w⃗ is a code for a model of PA^{−}
∧ w⃗ satisfies the first-order correctness condition
∧ Mw⃗ ⊧ φ)

in the language of lattices. By Lemma 2.2.13, there are codes in L for models of PA^{−} satisfying the first-order correctness condition, and every code in L for a model of PA^{−} satisfying the first-order correctness condition codes a structure isomorphic to N. Thus N ⊧ φ if and only if L ⊧ θ.

2.2.3 The coding countable subsets property and the second-order correctness condition

In this section we present sufficient conditions for interpreting true secondorder arithmetic in a distributive lattice. Our strategy is to use first-order quantification over the lattice to quantify over all countable subsets of a coded structure. We then recognize the codes w⃗ such that Mw⃗ ≅ N as the codes w⃗ of models of PA^{−} such that Mw⃗ is well-founded.

Consider a code w⃗ for a structure Mw⃗ in a lattice L. Given a ∈ L, let F (a) = {s ∈ L | s ≥ a}. Every element a ∈ L codes the subset F (a) ∩ E(w_{0}) ⊆ E(w_{0}). Using this coding, we extend the translation described in Definition 2.2.7 to second-order sentences φ in the language of arithmetic.

Definition 2.2.15. Let φ be a second-order sentence in the language of arithmetic. The translation of φ is the first-order formula φ′(w_{0}, w_{1}, w_{2}, m, ℓ, p, t, z, o) (with the displayed variables free) in the language of lattices obtained from φ by making the replacements described in Definition 2.2.7 and by also making the following replacements.

• Replace the second-order variable X by the first-order variable v_{X},

• replace x ∈ X by the formula expressing x ∈ F (v_{X}) ∩ E(w_{0}), and

• replace quantifiers ∃X and ∀X by ∃v_{X} and ∀v_{X} respectively.

If L is a lattice and w⃗ is a code, then Mw⃗ ⊧ φ means that L ⊧ φ′(w⃗).

Notice that even though φ is allowed to be a second-order sentence in the language of arithmetic, its translation φ′ is still a first-order formula in the language of lattices.

The next definition ensures that if w⃗ is a code such that Mw⃗ ≅ N, then for every S ⊆ E(w_{0}) there is an a ∈ L such that F (a) ∩ E(w_{0}) = S. That is, every subset of E(w_{0}) has a code, so quantifying over all subsets of E(w_{0}) is the same as quantifying over all coded subsets of E(w_{0}).

Definition 2.2.16. A lattice L has the coding countable subsets property if and only if for every w ∈ L and every countable S ⊆ E(w) there is an a ∈ L such that F (a) ∩ E(w) = S.

The following second-order correctness condition recognizes the structures isomorphic to N among the coded models of PA^{−} in a distributive lattice with the coding countable subsets property.

Definition 2.2.17. In a lattice L, a code w⃗ satisfies the second-order correctness condition if and only if for every a ∈ L, if there is an s ∈ F (a) ∩ E(w_{0}), then there is an R_{ℓ}^{2}-least such s.

Observe that the property “w⃗ is a code a model of PA^{−} that satisfies the second-order correctness condition” can be expressed by a first-order formula in the language of lattices.

Lemma 2.2.18. Let L be a distributive lattice with the coding countable subsets property.

(i) If w⃗ is a code such that Mw⃗ ≅ N, then w⃗ is a code for a model of PA^{−} satisfying the second-order correctness condition.

(ii) If w⃗ is a code for a model of PA^{−} satisfying the second-order correctness condition, then M_{}_{w′} ≅ N.

Proof.Item (i) is true because N is a well-founded model of PA^{−}. For item (ii), let w⃗ be a code for a model of PA^{−} satisfying the second-order correctness condition. If Mw⃗ were not well-founded, then there would be a countable S ⊆ E(w_{0}) with no R_{ℓ}^{2}-least element. By the coding countable subsets property, there would be an a ∈ L such that F (a) ∩ E(w_{0}) = S. This contradicts the second-order correctness condition. Thus Mw⃗ is a well-founded model of PA^{−} and hence is isomorphic to N.

Lemma 2.2.19. Let L be a distributive lattice with the coding countable subsets property such that there exists a code w⃗ such that Mw⃗ ≅ N. Then Th_{2}(N) ≤_{1} Th(L; ≤).

Proof.It suffices to prove Th_{2}(N) ≤_{1} Th(L) because Th(L) ≤_{1} Th(L; ≤) as the lattice operations + and × are first-order definable from the partial order ≤. Let φ be a second-order sentence in the language of arithmetic. Let θ be the first-order sentence

θ = ∃w⃗(w⃗ is a code for a model of PA^{−}
∧ w⃗ satisfies the second-order correctness condition
∧ Mw⃗ ⊧ φ)

in the language of lattices. By Lemma 2.2.18 item (i), there is a code w⃗ for a model of PA^{−} satisfying the second-order correctness condition. Thus to show that N ⊧ φ if and only if L ⊧ θ, it suffices to show that, for every such w⃗, N ⊧ φ if and only if Mw⃗ ⊧ φ. Let w⃗ be such a code. Then Mw⃗ ≅ N by Lemma 2.2.18 item (ii). In this case E(w_{0}) is countable, so if S ⊆ E(w_{0}), then there is an a ∈ L such that F (a) ∩ E(w_{0}) = S by the coding countable subsets property. That is, quantifying over all coded subsets of E(w_{0}) quantifies over all subsets of E(w_{0}). Therefore N ⊧ φ if and only if Mw⃗ ⊧ φ.

2.2.4 The coding all subsets property and the third-order correctness condition

In this section we present sufficient conditions for interpreting true third-order arithmetic in a distributive lattice. Our strategy for recognizing N among coded models of PA^{−} is the same as in the previous section. Given w ∈ L, we show how to simulate quantification over subsets of 2^{E(w)} by quantification over the subsets of E(r) for some other r ∈ L.

Consider a code w⃗ for a structure Mw⃗ in a lattice L. As in the previous section, every a ∈ L codes the subset F (a) ∩ E(w_{0}) ⊆ E(w_{0}). Fix an r ∈ L. Then every b ∈ L also codes a set S_{r}(b) ⊆ 2^{E(w_{0})}, where

S_{r}(b) = {X ⊆ E(w_{0}) | (∃s ∈ F (b) ∩ E(r))(∀u ∈ E(w_{0}))(u ∈ X ↔ u ≤ s)}.

Here “r” stands for “reals,” and we think of E(r) as the set 2^{E(w)}. F (b) ∩ E(r) ranges over subsets of E(r) as b ranges over L, and this is how we simulate third-order quantification over Mw⃗ by first-order quantification over L.

We can express F (a) ∩ E(w_{0}) ∈ S_{r}(b) (that is, “the subset of E(w_{0}) coded by a is an element of the subset of 2^{E(w_{0})} coded by b and r”) by the following first-order formula in the language of lattices:

(∃s ∈ F (b) ∩ E(r))(∀u ∈ E(w_{0}))(a ≤ u ↔ u ≤ s).

Using this coding, we extend the translations described in Definition 2.2.7 and Definition 2.2.15 to third-order sentences φ in the language of arithmetic.

Definition 2.2.20. Let φ be a third-order sentence in the language of arithmetic. The translation of φ is the first-order formula φ′(w_{0}, w_{1}, w_{2}, m, ℓ, p, t, z, o, r) (with the displayed variables free) in the language of lattices obtained from φ by making the replacements described in Definition 2.2.7 and Definition 2.2.15 and by also making the following replacements.

• Replace the third-order variable X by the first-order variable v_{X},

• replace X ∈ X by the formula expressing F (v_{X}) ∩ E(w_{0}) ∈ S_{r}(v_{X}), and

• replace quantifiers ∃X and ∀X by ∃v_{X} and ∀v_{X} respectively.

If L is a lattice, w⃗ is a code, and r ∈ L, then M_{w⃗,r} ⊧ φ means that L ⊧ φ′(w⃗, r).

Again, even though φ is allowed to be a third-order sentence in the language of arithmetic, its translation φ′ is still a first-order formula in the language of lattices. Note the inclusion of the new parameter r in the translation φ′.

The next definitions ensure that if w⃗ is a code such that Mw⃗ ≅ N, then there is an r ∈ L such that for every S ⊆ 2^{E(w_{0})} there is a b ∈ L such that S_{r}(b) = S. That is, every subset of 2^{E(w_{0})} has a code, so quantifying over all subsets of 2^{E(w_{0})} is the same as quantifying over all coded subsets of 2^{E(w_{0})}.

Definition 2.2.21. A lattice L has the coding all subsets property if and only if for every w ∈ L and every S ⊆ E(w) there is an a ∈ L such that F (a) ∩ E(w) = S.

Lemma 2.2.22.

(i) If L is a meet-complete distributive lattice with 1, then L has the coding all subsets property.

(ii) If L is a countably meet-complete distributive lattice with 1, then L has the coding countable subsets property.

Proof.For (i), Let w ∈ L, and let S ⊆ E(w). If S = ∅, then take a = 1. Then F (a) ∩ E(w) = ∅ because by definition 1 does not meet to any element of L. If S ≠ ∅, then let a = ∏ S. Clearly S ⊆ F (a) ∩ E(w). For the converse, suppose for a contradiction that there is an x ∈ (F (a) ∩ E(w)) \ S. This x meets to a because x meets to w and x ≥ a ≥ w. Let y > a be such that x × y = a. Then, for all s ∈ S, we have that s ≥ x × y, that s ≱ x because E(w) is an antichain by Lemma 2.2.3 and x ∉ S, and that s is meet-irreducible. Thus s ≥ y for all s ∈ S which gives the contradiction a ≥ y.

The same proof works for (ii) when S is countable.

Definition 2.2.23. In a lattice L, a code w⃗ satisfies the third-order correctness condition if and only if w⃗ satisfies the second-order correctness condition and there is an r ∈ L such that

(∀a ∈ L)(∃s ∈ E(r))(∀u ∈ E(w_{0}))(a ≤ u ↔ u ≤ s).

Observe that the property “w⃗ is a code a model of PA^{−} that satisfies the thirdorder correctness condition” can be expressed by a first-order formula in the language of lattices.

Lemma 2.2.24. Let L be a distributive lattice with the coding all subsets property such that there exists a code w⃗ such that Mw⃗ ≅ N and such that w⃗ satisfies the third-order correctness condition. Then Th_{3}(N) ≤_{1} Th(L; ≤).

Proof.It suffices to prove Th_{3}(N) ≤_{1} Th(L) because Th(L) ≤_{1} Th(L; ≤) as the lattice operations + and × are first-order definable from the partial order ≤. Let φ be a third-order sentence in the language of arithmetic. Let θ be the first-order sentence

θ = ∃w⃗∃r(w⃗ is a code for a model of PA^{−}
∧ r witnesses that w⃗ satisfies the third-order correctness condition
∧ M_{w⃗,r} ⊧ φ)

in the language of lattices. By hypothesis there exist w⃗ and r such that w⃗ is a code for a model of PA^{−} and r witnesses that w⃗ satisfies the third-order correctness condition. Thus to show that N ⊧ φ if and only if L ⊧ θ, it suffices to show that, for every such w⃗ and r, N ⊧ φ if and only if M_{w⃗,r} ⊧ φ. Fix such a w⃗ and r. Mw⃗ ≅ N by Lemma 2.2.18 item (ii). Every subset of E(w_{0}) is of the form F (a)∩E(w_{0}) for some a ∈ L by the coding all subsets property. We need to show that every subset of 2^{E(w_{0})} is of the form S_{r}(b) for some b ∈ L. To this end, let X ⊆ 2^{E(w_{0})}. For each X ∈ X, let a_{X} ∈ L be such that F (a_{X}) ∩ E(w_{0}) = X by the coding all subsets property, and let s_{X} ∈ E(r) be such that (∀u ∈ E(w_{0}))(a_{X} ≤ u ↔ u ≤ s_{X}) by the third-order correctness condition. Again by the coding all subsets property, let b ∈ L be such that F (b) ∩ E(r) = {s_{X} | X ∈ X }. Then S_{r}(b) = X. We have shown that quantifying over all coded subsets of E(w_{0}) quantifies over all subsets of E(w_{0}) and that quantifying over all coded subsets of 2^{E(w_{0})} quantifies over all subsets of 2^{E(w_{0})}. Therefore N ⊧ φ if and only if M_{w⃗,r} ⊧ φ.

2.2.5 Counting quantifiers

An analysis of the quantifier complexity of our coding scheme shows that to determine the truth of existential sentences in N we only need to determine the truth of Π_{3}^{0} sentences in L.

We switch to coding models of PA^{}because the axioms of PA^{}are all of the form ∀x⃗ψ(x⃗) for quantifier-free ψ. Here code now means a code for a structure in the language of N^{}. A code is now a sequence

w⃗ = (w_{0}, w_{1}, w_{2}, m, ℓ, p, t, d, z, o)

(with “d” for “difference”) interpreted as coding the structure

M ; w⃗ = (E(w_{0}); R_{ℓ}^{2}, R_{p}^{3}, R_{t}^{3}, R_{d}^{3}, z, o).

As in Definition 2.2.7, sentences in the language of N^{}translate to formulas in the language of lattices. The new relation is replaced in the translation by the formula defining R_{d}^{3}. A code for a model of PA^{}is a code w⃗ such that

M ; w⃗ ⊧ PA^{}.

In the language of lattices, “s is meet-irreducible” is a Π_{1}^{0} property and “s meets to w” is a Σ_{1}^{0} property, so “s ∈ E(w)” is a Δ_{2}^{0} property. Hence R_{c}^{2}(s_{0}, u_{1}) and R_{c}^{3}(s_{0}, u_{1}, v_{2}) are both Δ_{2}^{0} properties of s_{0}, u_{1}, v_{2}, and the coding parameters w_{0}, w_{1}, w_{2}, m, and c. Therefore, our coding translates atomic formulas in the language of N^{}to Δ_{2}^{0} properties of lattices. Every Boolean combination of Δ_{2}^{0} properties is again a Δ_{2}^{0} property, so our coding also translates quantifier-free formulas in the language of N^{}to Δ_{2}^{0} properties of lattices. Thus if φ = ∃x⃗ψ(x⃗) is a sentence in the language of N^{}where ψ is quantifier-free, then the translation φ′(w⃗) may be taken to be a Σ_{2}^{0} formula in the language of lattices. Similarly, if φ = ∀x⃗ψ(x⃗), then the translation φ′(w⃗) is Π_{2}^{0}. Thus “M ; w⃗ ⊧ PA^{}” can be expressed by a Π_{2}^{0} formula in the language of lattices. The axioms of PA^{}need to be unnested before they are translated, but this can be done in such a way that they all remain of the form ∀x⃗ψ(x⃗) for quantifier-free ψ.

In a lattice, the relations x + y = z and x × y = z are definable by Π_{1}^{0} formulas in the language of partial orders. This translation increases the quantifiercomplexities calculated in the previous paragraph by one alternation. Existential sentences in the language of N^{}translate to Σ_{3}^{0} formulas in the language of partial orders, and universal sentences in the language of N^{}translate to Π_{3}^{0} formulas in the language of partial orders. The property “M ; w⃗ ⊧ PA^{}” is a Π_{3}^{0} property of w⃗ in the language of partial orders.

Lemma 2.2.25. Let L be a lattice, and let w⃗ be a code such that M ; w⃗ ≅ N^{}. Then Σ_{3}^{0}-Th(L) and Σ_{4}^{0}-Th(L; ≤) are undecidable.

Proof.We prove

{∃x⃗ψ(x⃗) | ψ is quantifier-free ∧ N ⊧ ∃x⃗ψ(x⃗)} ≤_{1} Π_{3}^{0}-Th(E_{s}).

It is well-known that the problem of determining whether N ⊧ ∃x⃗ψ(x⃗) for quantifier-free ψ is undecidable.^{2} Clearly Σ_{3}^{0}-Th(L) ≡_{1} Π_{3}^{0}-Th(L).

Let φ = ∃x⃗ψ(x⃗) be a sentence in the language of arithmetic where ψ is quantifier-free. Let θ be the sentence

θ = ∀w⃗(M ; w⃗ ⊧ PA^{}→ M ; w⃗ ⊧ φ)

in the language of lattices. As calculated above, M ; w⃗ ⊧ PA^{}is a Π_{2}^{0} property of w⃗, and M ; w⃗ ⊧ φ is a Σ_{2}^{0} property of w⃗. Thus θ is a Π_{3}^{0} sentence in the language of lattices. We need to show N ⊧ φ if and only if L ⊧ θ. Suppose N ⊧ φ. Then PA^{}⊢ φ by Lemma 1.5.4, which implies that L ⊧ θ. Suppose N |̸= φ. Then by assumption there is a code w⃗ such that M ; w⃗ ≅ N^{}. For this w⃗, M ; w⃗ ⊧ PA^{}but M ; w⃗ |̸= φ, which implies L |̸= θ.

The proof that Σ_{0}^{4}-Th(L; ≤) is undecidable is the same. The above sentence θ is Π_{4}^{0} in the language of partial orders.

2For example, undecidability is implied by Matiyasevich’s solution to Hilbert’s tenth problem [40]. It is a standard fact in computability theory that determining whether N ⊧ ∃x⃗ψ(x⃗)

is undecidable if ψ is allowed bounded quantifiers, but allowing bounded quantifiers in ψ increases the quantifier complexity of the translated formula.

2.3 The complexities of Th(D_{s}; ≤_{s}) and Th(D_{w}; ≤_{w})

In this section we prove Th_{3}(N) ≤_{1} Th(D_{w}; ≤_{w}) ≤_{1} Th(D_{s}; ≤_{s}), thereby completing the proof that all three theories are pairwise recursively isomorphic.

2.3.1 Defining D_{w} in D_{s}

The Muchnik degrees are definable in the Medvedev degrees [22], thereby giving Th(D_{w}; ≤_{w}) ≤_{1} Th(D_{s}; ≤_{s}).

Definition 2.3.1. For a mass problem A, let C(A) denote the Turing upwardclosure of A: C(A) = {f | (∃g ∈ A)(g ≤_{T} f)}.

Definition 2.3.2. A Medvedev degree s is called a degree of solvability if s = deg_{s}({f}) for some f ∈ ω^{ω}.

Definition 2.3.3. A Medvedev degree m is called a Muchnik degree if m = deg_{s}(C(A)) for some mass problem A.

Notice that C(A) ≤_{s} B if and only if B ⊆ C(A). Medvedev degrees of the form deg_{s}(C(A)) are called Muchnik degrees because A ≤_{w} B if and only if C(B) ⊆ C(A) if and only if C(A) ≤_{s} C(B). The mapping deg_{w}(A) ↦ deg_{s}(C(A)) embeds D_{w} into D_{s} as an upper-semilattice but not as a lattice [71].

Lemma 2.3.4 (Medvedev [41], Dyment [22]). The degrees of solvability and the Muchnik degrees are definable in D_{s}.

The formula defining the degrees of solvability is

θ(x) = ∃y(x < y ∧ ∀z(x < z → y ≤ z)).

For a degree of solvability x = deg_{s}({f}), the witnessing y is the degree deg_{s}({e^{⌢}g | Φ_{e}(g) = f ∧ g ≰_{T} f}). Complete proofs that θ defines the degrees of solvability are found in [22] and [74]. We reproduce the definability of the Muchnik degrees here. The result essentially appears in [22], but is not phrased in terms of definability.

Proof that the Muchnik degrees are definable in D_{s}. The defining formula is χ(x) = ∀y(∀z((θ(z) ∧ y ≤ z) → x ≤ z) → x ≤ y), where θ is the formula defining the degrees of solvability as above. Let deg_{s}(C(A)) be a Muchnik degree. If B satisfies (∀f ∈ ω^{ω})(B ≤_{s}{f} → C(A) ≤_{s}{f}), then in particular we must have C(A) ≤_{s}{f} for all f ∈ B. Hence B ⊆ C(A) and so χ(deg_{s}(C(A))) holds. Conversely, suppose χ(deg_{s}(A)). As (∀f ∈ ω^{ω})(C(A) ≤_{s}{f} → A ≤_{s}{f}), we have A ≤_{s} C(A). Thus A ≡_{s} C(A), so deg_{s}(A) is a Muchnik degree.

Corollary 2.3.5. Th(D_{w}; ≤_{w}) ≤_{1} Th(D_{s}; ≤_{s}).

Proof.Interpret Th(D_{w}) inside Th(D_{s}) by restricting quantification in D_{s} to quantify only over degrees of the form deg_{s}(C(A)). That is, given a sentence ψ in the language of partial orders, generate a sentence ψ′ by inductively replacing subformulas ∃xφ and ∀xφ by formulas ∃x(χ(x) ∧ φ) and ∀x(χ(x) → φ). Then D_{w} ⊧ ψ if and only if D_{s} ⊧ ψ′.

In D_{w}, a degree s is also called a degree of solvability if s = deg_{w}({f}) for some f ∈ ω^{ω}. The formula θ(x) as above defines the degrees of solvability in D_{w}, and the proof is similar to that for D_{s}.

2.3.2 Coding third-order arithmetic in D_{w}

We find a code w⃗ in D_{w} such that Mw⃗ ≅ N. First, it is well-known and an easy observation that D_{w} is a complete lattice. Hence D_{w} has the coding all subsets property by Lemma 2.2.22.

Lemma 2.3.6. D_{w} is a complete lattice.

Proof.Let 𝔛 ⊆ D_{w} be non-empty, and let ⟨ X_{i} | i ∈ I ⟩ be a selection of one representative for each degree in 𝔛. Then the least upper bound of 𝔛 is deg_{w}(⋂_{i∈I} C(X_{i})) and the greatest lower bound of 𝔛 is deg_{w}(⋃_{i∈I} C(X_{i})) (which equals deg_{w}(⋃_{i∈I} X_{i})).

The crucial point is now the existence of the degree r witnessing that our w⃗ satisfies the third-order correctness condition. Lemma 2.3.7. Let W be a ≤_{T}-antichain, and let w = deg_{w}(W).

(i) If x ∈ D_{w} meets to w, then x ≤_{w} deg_{w}({f}) for some f ∈ W. (ii) E(w) = {deg_{w}({f}) | f ∈ W}.

Proof.(i) Let x ∈ D_{w} be such that x meets to w. Suppose for a contradiction that (∀f ∈ W)(x ≰_{w} deg_{w}({f})). Let y ∈ D_{w} witness that x meets to w. That is, y >_{w} w and x × y = w. Then, for all f ∈ W, deg_{w}({f}) is meet-irreducible (as it is the degree of a singleton), x ≰_{w} deg_{w}({f}), and x × y ≤_{w} deg_{w}({f}). Therefore (∀f ∈ W)(y ≤_{w} deg_{w}({f})) which gives the contradiction y ≤_{w} w.

(ii) Given f ∈ W, it is an easy check (using the fact that W is a ≤_{T}-antichain) that deg_{w}(W \{f}) witnesses that deg_{w}({f}) meets to w. Hence {deg_{w}({f}) | f ∈ W} ⊆ E(w). To see equality, let x ∈ E(w). By item (i), x ≤_{w} deg_{w}({f}) for some f ∈ W. We have just shown that deg_{w}({f}) ∈ E(w), and E(w) is an antichain by Lemma 2.2.3. So it must be that x = deg_{w}({f}).

The following lemma is proved using standard recursion theoretic techniques.

Lemma 2.3.8. If A = {f_{i} | i ∈ ω} is a countable independent set, then there exists a Turing antichain R = {g_{X} | X ∈ 2^{ω}} such that {f_{i} | i ∈ X} = {f ∈ A | f ≤_{T} g_{X}} for each X ∈ 2^{ω}.

Proof.We construct partial functions g_{σ}: ω → ω for σ ∈ 2^{<ω} and put g_{X} = ∪_{n∈ω}g_{X↾n}. The g_{σ} will have the following properties.

(i) If σ ⊂ τ then dom g_{σ} ⊆ dom g_{τ} and the two functions agree on their common domain.

(ii) If s < |σ| and σ(s) = 0 then g_{σ}(⟨ s, j ⟩) is defined for all j and equals 0 for all but finitely many j.

(iii) If s < |σ| and σ(s) = 1 then g_{σ}(⟨ s, j ⟩) is defined for all j and equals f_{s}(j) for all but finitely many j.

(iv) g_{σ}(⟨ s, j ⟩) is defined for only finitely many ⟨ s, j ⟩ with s ≥ |σ|.

Items (i)–(iii) ensure that each g_{X} is a total function, and item (iii) ensures f_{s} ≤_{T} g_{X} for all s ∈ X. In addition we satisfy the following requirements for all e, i ∈ ω and all X, Y ⊆ ω.

• R_{e,i}^{X}: i ∉ X → Φ_{e}(g_{X}) ≠ f_{i}

• Q_{e}^{X,Y}: X ≠ Y → Φ_{e}(g_{X}) ≠ g_{Y}

Let g_{∅} = ∅. At stage s we have g_{σ} for all σ of length s.

At stage s = 2 ⟨ e, i ⟩ we handle requirement R_{e,i}^{X}. For each σ of length s do the following. If σ(i) = 0, if there is a finite partial function h_{σ} with domain disjoint from g_{σ}, and if there is a number n such that Φ_{e}(g_{σ} ∪ h_{σ})(n)↓ ≠ f_{i}(n), then redefine g_{σ} to be g_{σ} ∪h_{σ}. Then for each σ of length s put g_{σ^{⌢}0} = g_{σ} ∪{⟨ ⟨ s, j ⟩, 0 ⟩ | ⟨ s, j ⟩ ∉ dom g_{σ}} and put g_{σ^{⌢}1} = g_{σ} ∪ {⟨ ⟨ s, j ⟩, f_{s}(j) ⟩ | ⟨ s, j ⟩ ∉ dom g_{σ}}.

At stage s = 2e + 1 we handle requirement Q_{e}^{X,Y}. List the pairs (σ, τ) with |σ| = |τ| = s and σ ≠ τ. For each such (σ, τ) do the following. Let n be least such that n ∉ dom g_{τ}. If there is a finite partial function h_{σ} with domain disjoint from g_{σ} and if there is a number m such that Φ_{e}(g_{σ} ∪ h_{σ})(n)↓ = m, then redefine g_{σ} to be g_{σ} ∪h_{σ} and redefine g_{τ} to be g_{τ} ∪{⟨ n, m+1 ⟩}. After these extensions are made for each pair (σ, τ), then for each σ of length s put g_{σ^{⌢}0} = g_{σ} ∪{⟨ ⟨ s, j ⟩, 0 ⟩ | ⟨ s, j ⟩ ∉ dom g_{σ}} and put g_{σ^{⌢}1} = g_{σ} ∪ {⟨ ⟨ s, j ⟩, f_{s}(j) ⟩ | ⟨ s, j ⟩ ∉ dom g_{σ}}.

We verify i ∉ X → f_{i} ≰_{T} g_{X}. Suppose that i ∉ X and Φ_{e}(g_{X}) = f_{i}. Consider stage s = 2 ⟨ e, i ⟩ of the construction. Let σ = X ↾ s and let f = ⨁{f_{t} | t < s ∧ σ(t) = 1}. The function f computes the graph of the partial function g_{σ}. Thus we can use f to simulate the computation Φ_{e}(g_{σ}∪h)(n) for any finite partial function h with domain disjoint from g_{σ}. We now have the contradiction f_{i} ≤_{T} f as follows. Given input n, use f to search for a finite partial function h with domain disjoint from g_{σ} such that Φ_{e}(g_{σ} ∪ h)(n)↓ = m for some m. There must be such an h because g_{X} extends g_{σ} and Φ_{e}(g_{X})(n)↓. Moreover, we must have m = f_{i}(n). Otherwise at stage s we would have been able to find an h_{σ} such that Φ_{e}(g_{σ} ∪ h_{σ})(n)↓ ≠ f_{i}(n), and this would imply Φ_{e}(g_{X}) ≠ f_{i}.

We verify X ≠ Y → g_{Y} ≰_{T} g_{X}. Suppose for a contradiction that Φ(g_{X}) = g_{Y}. Choose an index e for Φ greater than the least e such that X(e) ≠ Y (e), put s = 2e + 1, and let σ = X ↾ s, τ = Y ↾ s. Consider the g_{σ} and g_{τ} we have right before we process the pair (σ, τ) in stage s. Let n be least such that n ∉ dom g_{τ}. Since g_{X} extends g_{σ} and Φ_{e}(g_{X}) is total, we must have found a finite h_{σ} and number m such that Φ_{e}(g_{σ} ∪ h_{σ})(n)↓ = m. But then we extended g_{τ} so that g_{τ}(n) = m + 1. Thus Φ_{e}(g_{X})(n) = m ≠ g_{Y} (n), a contradiction.

Lemma 2.3.9. There is a code w⃗ in D_{w} such that M ; w⃗ ≅ N^{}and such that w⃗ satisfies the third-order correctness condition.

Proof.Let W_{0} = {f_{0,n}}_{n∈ω}, W_{1} = {f_{1,n}}_{n∈ω}, and W_{2} = {f_{2,n}}_{n∈ω} be such that W_{0} ∪ W_{1} ∪ W_{2} is independent. Then let

w_{0} = deg_{w}(W_{0}),
w_{1} = deg_{w}(W_{1}),
w_{2} = deg_{w}(W_{2}),
m = deg_{w}(M)  for  M = {f_{0,n} ⊕ f_{1,n}}_{n∈ω} ∪ {f_{0,n} ⊕ f_{2,n}}_{n∈ω},
ℓ = deg_{w}(L)  for  L = {f_{0,i} ⊕ f_{1,j} | i < j},
p = deg_{w}(P)  for  P = {f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i + j = k},
t = deg_{w}(T )  for  T = {f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i × j = k},
d = deg_{w}(D)  for  D = {f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i j = k},
z = deg_{w}({f_{0,0}}),  and
o = deg_{w}({f_{0,1}}).

These degrees give the code w⃗. By Lemma 2.3.8, let R = {g_{X} | X ∈ 2^{ω}} be a Turing antichain such that {f_{0,i} ∈ W_{0} | i ∈ X} = {f_{0,i} ∈ W_{0} | f_{0,i} ≤_{T} g_{X}} for each X ∈ 2^{ω}. Let r = deg_{w}(R).

By Lemma 2.3.7 item (ii), E(w_{0}) = {deg_{w}({f_{0,n}})}_{n∈ω}. The map deg_{w}({f_{0,n}}) ↦ n is the isomorphism witnessing M ; w⃗ ≅ N^{}. Clearly z ↦ 0 and o ↦ 1. We show that the map preserves <. The proofs that the map preserves +, ×, and are similar. Let i, j ∈ ω. If i < j, then deg_{w}({f_{1,j}}) meets to w_{1} by Lemma 2.3.7 item (ii), and it is easy to see that deg_{w}({f_{0,j}}) + deg_{w}({f_{1,j}}) ≥_{w} m and that deg_{w}({f_{0,i}}) + deg_{w}({f_{1,j}}) ≥_{w} ℓ. Thus R_{ℓ}^{2}(deg_{w}({f_{0,i}}), deg_{w}({f_{0,j}})). Conversely, suppose that R_{ℓ}^{2}(deg_{w}({f_{0,i}}), deg_{w}({f_{0,j}})). Let u_{1} ∈ D_{w} be such that u_{1} meets to w_{1}, deg_{w}({f_{0,j}}) + u_{1} ≥_{w} m, and deg_{w}({f_{0,i}}) + u_{1} ≥_{w} ℓ. Since u_{1} meets to w_{1}, it must be that u_{1} ≤_{w} deg_{w}({f_{1,k}}) for some k ∈ ω by Lemma 2.3.7 item (i). Thus deg_{w}({f_{0,j}}) + deg_{w}({f_{1,k}}) ≥_{w} m. However, if k ≠ j, then f_{0,j} ⊕f_{1,k} does not compute any member of M by independence. Thus u_{1} ≤_{w} deg_{w}({f_{1,j}}), which implies that deg_{w}({f_{0,i}}) + deg_{w}({f_{1,j}}) ≥_{w} ℓ. Again by independence, if i ≮ j, then f_{0,i} ⊕ f_{1,j} does not compute any member of L. Hence i < j.

We now check the third-order correctness condition. To do this, we first need to check the second-order correctness condition. Let a ∈ D_{s} be such that F (a) ∩ E(w_{0}) is non-empty. Let n ∈ ω be least such that deg_{w}({f_{0,n}}) ∈ F (a) ∩ E(w_{0}). Then this deg_{w}({f_{0,n}}) is the R_{ℓ}^{2}-least element of F (a) ∩ E(w_{0}). Thus the second-order correctness condition holds. Now, given an a ∈ D_{w}, let X = {i | deg_{w}({f_{0,i}}) ∈ F (a) ∩ E(w_{0})}, and let s = deg_{w}({g_{X}}). Then s ∈ E(r) by Lemma 2.3.7 item (ii) because R is a ≤_{T}-antichain. Also, for all deg_{w}({f_{0,i}}) ∈ E(w_{0}), a ≤_{w} deg_{w}({f_{0,i}}) if and only if i ∈ X if and only if deg_{w}({f_{0,i}}) ≤_{w} s. Thus the third-order correctness condition holds.

The following theorem was proved independently by Lewis, Nies, and Sorbi [38].

Theorem 2.3.10. Th(D_{w}; ≤_{w}) ≡_{1} Th(D_{s}; ≤_{s}) ≡_{1} Th_{3}(N).

Proof.Th(D_{s}; ≤_{s}) ≤_{1} Th_{3}(N) by Lemma 2.1.1. Th(D_{w}; ≤_{w}) ≤_{1} Th(D_{s}; ≤_{s}) by Corollary 2.3.5. For Th_{3}(N) ≤_{1} Th(D_{w}; ≤_{w}), by Lemma 2.3.9 let w⃗ be a code in D_{w} such that M ; w⃗ ≅ N^{}and such that w⃗ satisfies the third-order correctness condition. Removing the degree d from the code w⃗ gives a code v⃗ such that Mv⃗ ≅ N and such that v⃗ satisfies the third-order correctness condition. D_{w} has the coding all subsets property by Lemma 2.2.22 because D_{w} has a greatest element and is meet-complete by Lemma 2.3.6. Hence Th_{3}(N) ≤_{1} Th(D_{w}; ≤_{w}) by Lemma 2.2.24.

Theorem 2.3.11. Σ_{3}^{0}-Th(D_{w}) and Σ_{4}^{0}-Th(D_{w}; ≤_{w}) are both undecidable.

Proof.By Lemma 2.3.9, there is a code w⃗ in D_{w} such that M ; w⃗ ≅ N^{}. The results then follow from Lemma 2.2.25.

The undecidability of Σ_{3}^{0}-Th(D_{s}) and the undecidability of Σ_{4}^{0}-Th(D_{s}; ≤_{s}) are proved in Theorem 2.6.6.

2.4 The complexities of Th(D_{w,cl}; ≤_{w}) and Th(D_{w,cl}^{01}; ≤_{w})

In this section, we prove Th_{2}(N) ≤_{1} Th(D_{w,cl}; ≤_{w}), Th(D_{w,cl}^{01}; ≤_{w}), and in the next section we prove Th_{2}(N) ≤_{1} Th(D_{s,cl}; ≤_{s}), Th(D_{s,cl}^{01}; ≤_{s}). Recall from Lemma 2.3.4 that the degrees of solvability are definable in D_{s} and D_{w}. The definability of the degrees of solvability in any of L = D_{s,cl}, D_{s,cl}^{01}, D_{w,cl}, D_{w,cl}^{01} would give an immediate proof of Th_{2}(N) ≤_{1} Th(L; ≤) for that L. This is because the Turing degrees are isomorphic to the degrees of solvability and because the first-order theory of the Turing degrees is recursively isomorphic to Th_{2}(N) [62]. Singleton mass problems {f} are compact, so the degrees of solvability are in all these lattices. However, we do not know if the degrees of solvability are definable in any of these lattices.

Question 2.4.1. Are the degrees of solvability definable in D_{s,cl}, D_{s,cl}^{01}, D_{w,cl}, or D_{w,cl}^{01}?

We show that D_{w,cl} and D_{w,cl}^{01} are countably meet-complete. Hence both lattices have the coding countable subsets property by Lemma 2.2.22.

Lemma 2.4.2. Both D_{w,cl} and D_{w,cl}^{01} are countably meet-complete.

Proof.Let 𝔛 = {x_{i} | i ∈ ω} be a countable set of degrees in D_{w,cl}. Let X_{i} ⊆ ω^{ω} be a closed representative of x_{i} for each i. The degree x = deg_{w}(⋃_{i∈ω} i^{⌢}X_{i}) is the greatest lower bound of 𝔛 in D_{w} and ⋃_{i∈ω} i^{⌢}X_{i} is closed. Hence x ∈ D_{w,cl}, so x is the greatest lower bound of 𝔛 in D_{w,cl}.

The above proof does not work for D_{w,cl}^{01} because ⋃_{i∈ω} i^{⌢}X_{i} is not compact. We provide a modified proof for D_{w,cl}^{01}. Let 𝔛 = {x_{i} | i ∈ ω} be a countable set of degrees in D_{w,cl}^{01}. Let X_{i} ⊆ 2^{ω} be a closed representative of x_{i} for each i. Choose any g in any non-empty X_{i} (if all the X_{i} are empty, then 1 is the greatest lower bound). Let σ_{i} = (g ↾ i)^{⌢}(1 − g(i)) for each i ∈ ω. The set X = {g} ∪ (⋃_{i∈ω} σ_{i}^{⌢}X_{i}) is closed in 2^{ω}, so let x = deg_{w}(X). Then x ∈ D_{w,cl}^{01}, and it is easy to see that x = deg_{w}(⋃_{i∈ω} i^{⌢}X_{i}). Hence x is the greatest lower bound of 𝔛 in D_{w,cl}^{01} (and in D_{w}).

Question 2.4.3. Are D_{w,cl} and D_{w,cl}^{01} countably join-complete?

Lemma 2.4.4. Let W ⊆ ω^{ω} be a closed ≤_{T}-antichain, and let w = deg_{w}(W).

(i) If x ∈ D_{w,cl} meets to w, then x ≤_{w} deg_{w}({f}) for some f ∈ W. (ii) E(w) = {deg_{w}({f}) | f ∈ W}.

The same is true with 2^{ω} in place of ω^{ω} and D_{w,cl}^{01} in place of D_{w,cl}.

Proof.(i) The proof of Lemma 2.3.7 item (i) works in both the D_{w,cl} case and the D_{w,cl}^{01} case.

(ii) First consider the D_{w,cl} case. Let f ∈ W. Let T be a tree such that W = [T]. Let ⟨ τ_{i} | i ∈ ω ⟩ list the sequences in T that are not initial segments of f (so that (∀g ∈ W)(g ≠ f ↔ ∃i(τ_{i} ⊂ g))). Let T_{i} denote the full subtree of T rooted at τ_{i}: T_{i} = {σ ∈ ω^{<ω} | τ_{i}^{⌢}σ ∈ T }. Let R be the tree ⋃_{i∈ω} i^{⌢}T_{i} where i^{⌢}T_{i} = {i^{⌢}σ | σ ∈ T_{i}} for each i. Let Y = [R], and let y = deg_{w}(Y). If, for a mass problem A, we let deg_{T}(A) = {deg_{T}(f) | f ∈ A} denote the set of Turing degrees of the members of A, we see that deg_{T}(Y) = deg_{T}(W)\{deg_{T}(f)}. From this and the fact that W is a ≤_{T}-antichain, it follows that y >_{w} w and deg_{w}({f}) × y = w. Hence deg_{w}({f}) is meet-irreducible (because it is the degree of a singleton) and meets to w. Thus {deg_{w}({f}) | f ∈ W} ⊆ E(w). To see equality, let x ∈ E(w). By item (i), x ≤_{w} deg_{w}({f}) for some f ∈ W. We have just shown that deg_{w}({f}) ∈ E(w), and E(w) is an antichain by Lemma 2.2.3. So it must be that x = deg_{w}({f}).

For the D_{w,cl}^{01} case, as before let f ∈ W, and let T be a tree such that W = [T]. Let ⟨ τ_{i} | i ∈ ω ⟩ list the sequences in T that are not initial segments of f. Let T_{i} denote the full subtree of T rooted at τ_{i}. Choose any g ∈ W \ {f} (if W = {f}, then the lemma is trivial). Let σ_{i} = (g ↾ i)^{⌢}(1 − g(i)) for each i ∈ ω. Let R be the tree ⋃_{i∈ω} σ_{i}^{⌢}T_{i}. Let Y = [R]. Then deg_{T}(Y) = deg_{T}(W) \ {deg_{T}(f)}. The proof now proceeds exactly as in the D_{w,cl} case.

Definition 2.4.5 (Dyment [22]). W ⊆ ω^{ω} is called effectively discrete if (∀f ∈ W)(∀g ∈ W)(f ≠ g → f(0) ≠ g(0)).

An effectively discrete mass problem is closed and at most countable.

Lemma 2.4.6. There is a code w⃗ in D_{w,cl} such that M ; w⃗ ≅ N^{}.

Proof.Let W_{0} = {n^{⌢}f_{0,n}}_{n∈ω}, W_{1} = {n^{⌢}f_{1,n}}_{n∈ω}, and W_{2} = {n^{⌢}f_{2,n}}_{n∈ω} be such that W_{0} ∪ W_{1} ∪ W_{2} is independent. Then let

w_{0} = deg_{w}(W_{0}),
w_{1} = deg_{w}(W_{1}),
w_{2} = deg_{w}(W_{2}),
m = deg_{w}(M)  for M = {(2n)^{⌢}(f_{0,n} ⊕ f_{1,n})}_{n∈ω}
∪ {(2n + 1)^{⌢}(f_{0,n} ⊕ f_{2,n})}_{n∈ω},
ℓ = deg_{w}(L)  for L = {⟨ i, j ⟩^{⌢}(f_{0,i} ⊕ f_{1,j}) | i < j},
p = deg_{w}(P)  for P = {⟨ ⟨ i, j ⟩, k ⟩^{⌢}(f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k}) | i + j = k},
t = deg_{w}(T )  for T = {⟨ ⟨ i, j ⟩, k ⟩^{⌢}(f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k}) | i × j = k},
d = deg_{w}(D)  for D = {⟨ ⟨ i, j ⟩, k ⟩^{⌢}(f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k}) | i j = k},
z = deg_{w}({0^{⌢}f_{0,0}}), and
o = deg_{w}({1^{⌢}f_{0,1}}).

The above mass problems are all effectively discrete, so their degrees are all in D_{w,cl}. These degrees give the code w⃗. The proof that M ; w⃗ ≅ N^{}is the same as in the proof of Lemma 2.3.9. Use Lemma 2.4.4 in place of Lemma 2.3.7.

An infinite effectively discrete Turing antichain is not compact, so we can no longer rely on them to provide codes. Instead we use the following definitions.

Definition 2.4.7. Let g ∈ 2^{ω}. A set X ⊆ 2^{ω} is called a g-spine (or just a spine) if it is of the form {g} ∪ {σ_{i}^{⌢}f_{i} | i ∈ A} where A ⊆ ω is infinite, σ_{i} = (g ↾ i)^{⌢}(1 − g(i)) for each i ∈ A, and f_{i} ∈ 2^{ω} for each i ∈ A.

Definition 2.4.8. Let g ∈ 2^{ω} and let X ⊆ 2^{ω} be countable. Fix an enumeration ⟨ f_{i} | i ∈ ω ⟩ of X. We denote by spine(g, X) the g-spine {g} ∪ {σ_{i}^{⌢}f_{i} | i ∈ ω} where σ_{i} = (g ↾ i)^{⌢}(1 − g(i)) for each i ∈ ω. We denote by spine(X) the f_{0}-spine spine(f_{0}, X \ {f_{0}}).

Notice that a spine is a closed subset of 2^{ω}.

Lemma 2.4.9. There is a code w⃗ in D_{w,cl}^{01} such that M ; w⃗ ≅ N^{}.

Proof.Let W′_{0} = {f_{0,n}}_{n∈ω}, W′_{1} = {f_{1,n}}_{n∈ω}, and W′_{2} = {f_{2,n}}_{n∈ω} be such that W′_{0} ∪ W′_{1} ∪ W′_{2} ⊆ 2^{ω} is independent. Then let

w_{0} = deg_{w}(W_{0})  for W_{0} = spine(W′_{0}),
w_{1} = deg_{w}(W_{1})  for W_{1} = spine(W′_{1}),
w_{2} = deg_{w}(W_{2})  for W_{2} = spine(W′_{2}),
m = deg_{w}(M)  for M = spine({f_{0,n} ⊕ f_{1,n}}_{n∈ω} ∪ {f_{0,n} ⊕ f_{2,n}}_{n∈ω}),
ℓ = deg_{w}(L)  for  L = spine({f_{0,i} ⊕ f_{1,j} | i < j}),
p = deg_{w}(P)  for  P = spine({f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i + j = k}),
t = deg_{w}(T )  for  T = spine({f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i × j = k}),
d = deg_{w}(D)  for  D = spine({f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i j = k}),
z = deg_{w}({f_{0,0}}), and
o = deg_{w}({f_{0,1}}).

These degrees give the code w⃗. The proof that M ; w⃗ ≅ N^{}is the same as in the proof of Lemma 2.3.9. Use Lemma 2.4.4 in place of Lemma 2.3.7.

Theorem 2.4.10. Th(D_{w,cl}; ≤_{w}) ≡_{1} Th(D_{w,cl}^{01}; ≤_{w}) ≡_{1} Th_{2}(N).

Proof.Th(D_{w,cl}; ≤_{w}) ≤_{1} Th_{2}(N) and Th(D_{w,cl}^{01}; ≤_{w}) ≤_{1} Th_{2}(N) by Lemma 2.1.2. For Th_{2}(N) ≤_{1} Th(D_{w,cl}; ≤_{w}), by Lemma 2.4.6 let w⃗ be a code in D_{w,cl} such that M ; w⃗ ≅ N^{}. Removing the degree d from the code w⃗ gives a code v⃗ such that Mv⃗ ≅ N. D_{w,cl} has the coding countable subsets property by Lemma 2.2.22 because D_{w,cl} has a greatest element and is countably meetcomplete by Lemma 2.4.2. Hence Th_{2}(N) ≤_{1} Th(D_{w}; ≤_{w}) by Lemma 2.2.19. The proof that Th_{2}(N) ≤_{1} Th(D_{w,cl}^{01}; ≤_{w}) is the same. Use Lemma 2.4.9 in place of Lemma 2.4.6.

Theorem 2.4.11. The fragments Σ_{3}^{0}-Th(D_{w,cl}), Σ_{4}^{0}-Th(D_{w,cl}; ≤_{w}), Σ_{3}^{0}-Th(D_{w,cl}^{01}), and Σ_{4}^{0}-Th(D_{w,cl}^{01}; ≤_{w}) are all undecidable.

Proof.By Lemma 2.4.6, there is a code w⃗ in D_{w,cl} such that M ; w⃗ ≅ N^{}. By Lemma 2.4.9, there is a code w⃗ in D_{w,cl}^{01} such that M ; w⃗ ≅ N^{}. The results then follow from Lemma 2.2.25.

2.5 D_{s,cl} and D_{s,cl}^{01} have the coding countable subsets property

D_{s,cl} and D_{s,cl}^{01} are not countably meet-complete by Lemma 2.7.1, so Lemma 2.2.22 does not apply to them. We need to prove that both D_{s,cl} and D_{s,cl}^{01} have the coding countable subsets property. The next lemma is a clarifying example. It implies that a closed (compact) W has meet-reducible degree in D_{s,cl} (D_{s,cl}^{01}) if and only if it has meet-reducible degree in D_{s}.

Lemma 2.5.1 (Dyment [22]). If W ≡_{s} X × Y, then W = X̂ ∪ Ŷ where X̂ and Ŷ are disjoint and clopen in W, X̂ ≥_{s} X, Ŷ ≥_{s} Y, and W ≡_{s} X̂ × Ŷ.

Proof.Let Φ be such that Φ(W) ⊆ 0^{⌢}X ∪ 1^{⌢}Y. Put X̂ = {f ∈ W | Φ(f)(0) = 0} and put Ŷ = {f ∈ W | Φ(f)(0) = 1}. By Lemma 1.4.2, X̂ and Ŷ are clopen in W, and it is easily checked that X̂ ≥_{s} X and Ŷ ≥_{s} Y (hence W ≤_{s} X̂ × Ŷ). We have W ≥_{s} 0^{⌢}X̂ ∪ 1^{⌢}Ŷ by the reduction which sends f to 0^{⌢}f if Φ(f)(0) = 0 and sends f to 1^{⌢}f if Φ(f)(0) = 1.

For comparison with the Muchnik case, if a closed (compact) W has meetreducible degree in D_{w,cl} (D_{w,cl}^{01}), then it has meet-reducible degree in D_{w}. However, we do not know the converse. Question 2.5.2. If W is closed (compact) and W ≡_{w} X × Y for X , Y >_{w} W, then are there closed (compact) such X and Y?

If the X in Lemma 2.5.1 has meet-irreducible degree, then we have the following refinement.

Lemma 2.5.3. If W ≡_{s} X × Y where X has meet-irreducible degree and Y >_{s} W, then W = X̂ ∪ Ŷ where X̂ and Ŷ are disjoint and clopen in W, X̂ ≡_{s} X, and X̂ ≱_{s} Ŷ.

Proof.As in Lemma 2.5.1, let Φ be such that Φ(W) ⊆ 0^{⌢}X ∪ 1^{⌢}Y, put X̂ = {f ∈ W | Φ(f)(0) = 0}, and put Ŷ = {f ∈ W | Φ(f)(0) = 1}. Then W = X̂ ∪ Ŷ, X̂ ∩ Ŷ = ∅, X̂ and Ŷ are clopen in W, X̂ ≥_{s} X, Ŷ ≥_{s} Y, and W ≡_{s} X̂ × Ŷ. To see that X ≥_{s} X̂, observe that X ≥_{s} W ≡_{s} X̂ × Ŷ. X has meet-irreducible degree, so X ≥_{s} X̂ or X ≥_{s} Ŷ. We cannot have X ≥_{s} Ŷ because Ŷ ≥_{s} Y and this would imply W ≡_{s} X × Y ≡_{s} Y >_{s} W. Thus X ≥_{s} X̂. Similarly X̂ ≱_{s} Ŷ for otherwise W ≡_{s} Ŷ ≥_{s} Y >_{s} W. Corollary 2.5.4. For all w ∈ D_{s}, E(w) is at most countable.

Proof.Fix a representative W for w. Lemma 2.5.3 shows that if x ∈ E(w), then x has a representative of the form {f ∈ W | Φ(f)(0) = 0} for some program Φ. There are only countably many programs, so there can be at most countably many x ∈ E(w).

Notice that Corollary 2.5.4 is in contrast to the Muchnik case, in which a degree may have uncountably many meet-irreducibles that meet to it. For example, if W is a ≤_{T}-antichain, then, in D_{w}, |E(deg_{w}(W))| = |W| by Lemma 2.3.7. There exist uncountable ≤_{T}-antichains, and there even exist uncountable closed ≤_{T}-antichains (see [69] Section VI.1). Also notice that if w is closed (compact) and x is meet-irreducible and meets to w, then Lemma 2.5.3 produces a closed (compact) representative for x. Thus for a closed (compact) degree w, the meetirreducible degrees that meet to w are the same whether they are computed in D_{s} or in D_{s,cl} (D_{s,cl}^{01}).

Lemma 2.5.5. Let W be a mass problem such that E(deg_{s}(W)) is countable, and let ⟨ X_{i} | i ∈ ω ⟩ be a list of representatives for the degrees in E(deg_{s}(W)). Then there are mass problems ⟨ X̂_{i} | i ∈ ω ⟩ such that:

(i) X̂_{i} ⊆ W is clopen in W for each i, (ii) X̂_{i} ∩ X̂_{j} = ∅ for i ≠ j, (iii) X̂_{i} ≡_{s} X_{i} for each i, (iv) X̂_{i} ≱_{s} W \ X̂_{i} for each i.

Proof.Inductively construct the sequence ⟨ X̂_{i} | i ∈ ω ⟩. At the start of step n + 1 we have ⟨ X̂_{i} | i ≤ n ⟩ satisfying (i)–(iv) for i, j ≤ n, and we have indices e_{0}, …, e_{n} such that, for i ≤ n, X̂_{i} = {f ∈ W \ ⋃_{j<i} X̂_{j} | Φ_{e_{i}}(f)(0) = 0} and W \ ⋃_{j≤i} X̂_{j} = {f ∈ W \ ⋃_{j<i} X̂_{j} | Φ_{e_{i}}(f)(0) = 1}.

We first show W ≡_{s} X̂_{0} × ⋯ × X̂_{n} ×(W \ ⋃_{i≤n} X̂_{i}). The meet is ≥_{s} W because each term is a subset of W. To see the reverse inequality, write the meet as ⋃_{i≤n} i^{⌢}X̂_{i} ∪ (n + 1)^{⌢}(W \ ⋃_{i≤n} X̂_{i}). Then apply the following reduction. For each i ≤ n in order, check if Φ_{e_{i}}(f)(0) is 0 or 1. If it is 0, send f to i^{⌢}f. If it is 1, go to the next i. If Φ_{e_{i}}(f)(0) = 1 for each i ≤ n, then send f to (n + 1)^{⌢}f.

We now have X_{n+1} ≥_{s} W ≡_{s} X̂_{0} × ⋯ × X̂_{n} ×(W \ ⋃_{i≤n} X̂_{i}). We cannot have X_{n+1} ≥_{s} X̂_{i} for any i ≤ n because X̂_{i} ≡_{s} X_{i} and the X_{i}’s are incomparable because E(deg_{s}(W)) is an antichain by Lemma 2.2.3. However, X_{n+1} has meetirreducible degree. Therefore X_{n+1} ≥_{s} W \ ⋃_{i≤n} X̂_{i}. Moreover, by distributivity deg_{s}(X_{n+1}) meets to deg_{s}(W \⋃_{i≤n} X̂_{i}) because deg_{s}(X_{n+1}) meets to deg_{s}(W) and deg_{s}(X_{n+1}) ≥_{s} deg_{s}(W \ ⋃_{i≤n} X̂_{i}) ≥_{s} deg_{s}(W). Thus, as in Lemma 2.5.3, there is an X̂_{n+1} ⊆ W \ ⋃_{i≤n} X̂_{i} clopen in W \ ⋃_{i≤n} X̂_{i} and an e_{n+1} such that

X̂_{n+1} = {f ∈ W \ ⋃_{i≤n} X̂_{i} | Φ_{e_{n+1}}(f)(0) = 0},
W \ ⋃_{i≤n+1} X̂_{i} ={f ∈ W \ ⋃_{i≤n} X̂_{i} | Φ_{e_{n+1}}(f)(0) = 1},
X̂_{n+1} ≡_{s} X_{n+1}, and
X̂_{n+1} ≱_{s} W \ ⋃_{i≤n+1} X̂_{i}.

Clearly X̂_{n+1} is disjoint from X̂_{i} for i ≤ n. X̂_{n+1} is clopen in W because it is clopen in W \ ⋃_{i≤n} X̂_{i} which is clopen in W. Finally, X̂_{n+1} ≱_{s} W \ X̂_{n+1} because X̂_{n+1} has meet-irreducible degree, X̂_{n+1} ≱_{s} X̂_{i} for i ≤ n, X̂_{n+1} ≱_{s} W \ ⋃_{i≤n+1} X̂_{i}, and W \ X̂_{n+1} ≡_{s} X̂_{0} × ⋯ × X̂_{n} ×(W \ ⋃_{i≤n+1} X̂_{i}).

Lemma 2.5.6. Let W be a mass problem. Then for any S ⊆ E(deg_{s}(W)) there is an A ⊆ W closed in W such that F (deg_{s}(A)) ∩ E(deg_{s}(W)) = S. In particular, if W is closed, then so is A. Thus D_{s,cl} and D_{s,cl}^{01} have the coding countable subsets property.

Proof.We only consider the case in which E(w) is infinite. By Corollary 2.5.4, E(w) is countable. Let ⟨ X_{i} | i ∈ ω ⟩ be a list of representatives for the degrees in E(w). Apply Lemma 2.5.5 to W and ⟨ X_{i} | i ∈ ω ⟩ to get a new set of representatives ⟨ X̂_{i} | i ∈ ω ⟩ disjoint and clopen in W with X̂_{i} ≱_{s} W \ X̂_{i} for each i. Put A = W \ ⋃{X̂_{i} | deg_{s}(X̂_{i}) ∉ S}, and note that A is closed in W. We show X̂_{i} ≥_{s} A if and only if deg_{s}(X̂_{i}) ∈ S. If deg_{s}(X̂_{i}) ∈ S then X̂_{i} ⊆ A and so X̂_{i} ≥_{s} A. If deg_{s}(X̂_{i}) ∉ S then A ⊆ W \ X̂_{i} and so A ≥_{s} W \ X̂_{i}. Thus X̂_{i} ≱_{s} A because X̂_{i} ≱_{s} W \ X̂_{i}.

2.6 The complexities of Th(D_{s,cl}; ≤_{s}) and Th(D_{s,cl}^{01}; ≤_{s})

We can now prove that Th_{2}(N) ≤_{1} Th(D_{s,cl}; ≤_{s}) and Th_{2}(N) ≤_{1} Th(D_{s,cl}^{01}; ≤_{s}).

Lemma 2.6.1. Let W ⊆ ω^{ω} be an effectively discrete ≤_{T}-antichain, and let w = deg_{s}(W).

(i) If x ∈ D_{s,cl} meets to w, then x ≤_{s} deg_{s}({f}) for some f ∈ W. (ii) E(w) = {deg_{s}({f}) | f ∈ W}.

Proof.(i) Let x ∈ D_{s,cl} be such that x meets to w. Suppose for a contradiction that (∀f ∈ W)(x ≰_{s} deg_{s}({f})). Let y ∈ D_{s,cl} witness that x meets to w. That is, y ≥_{s} w and x × y = w. Let X be a representative for x, and let Y be a representative for y. Then X × Y ≡_{s} W, so let Φ be a Turing functional such that Φ(W) ⊆ 0^{⌢}X ∪ 1^{⌢}Y. If Φ(f) ∈ 0^{⌢}X for some f ∈ W, then x ≤_{s} deg_{s}({f}) for this f, contrary to assumption. Thus it must be that Φ(f) ∈ 1^{⌢}Y for all f ∈ W. That is, Φ witnesses that w ≥_{s} y, a contradiction.

(ii) Given f ∈ W, it is an easy check (using the fact that W is an effectively discrete ≤_{T}-antichain) that deg_{s}(W \ {f}) witnesses that deg_{s}({f}) meets to w. Hence {deg_{s}({f}) | f ∈ W} ⊆ E(w). To see equality, let x ∈ E(w). By item (i), x ≤_{s} deg_{s}({f}) for some f ∈ W. We have just shown that deg_{s}({f}) ∈ E(w), and E(w) is an antichain by Lemma 2.2.3. So it must be that x = deg_{s}({f}).

Lemma 2.6.2. There is a code w⃗ in D_{s,cl} such that M ; w⃗ ≅ N^{}.

Proof.If W_{0}, W_{1}, W_{2}, M, L, P, T , D, {0^{⌢}f_{0,0}}, and {1^{⌢}f_{0,1}} are the mass problems defined in the proof of Lemma 2.4.6, then their Medvedev degrees give a code w⃗ in D_{s,cl} such that M ; w⃗ ≅ N^{}. The proof that M ; w⃗ ≅ N^{}is the same as in the proof of Lemma 2.3.9. Use Lemma 2.6.1 in place of Lemma 2.3.7.

To code N in D_{s,cl}^{01}, we need to reprove Lemma 2.6.1 for spines.

Lemma 2.6.3. Let W = {g} ∪ {σ_{i}^{⌢}f_{i} | i ∈ X} ⊆ 2^{ω} be a g-spine that is a ≤_{T}- antichain, and let w = deg_{s}(W).

(i) If x ∈ D_{s,cl}^{01} meets to w, then x ≤_{s} deg_{s}({f_{i}}) for some i ∈ X.

(ii) E(w) = {deg_{s}({f_{i}}) | i ∈ X}.

Proof.(i) Let x ∈ D_{s,cl}^{01} be such that x meets to w. Suppose for a contradiction that (∀f ∈ W)(x ≰_{s} deg_{s}({f})). Let y ∈ D_{s,cl}^{01} witness that x meets to w. That is, y >_{s} w and x × y = w. Let X be a representative for x, and let Y be a representative for y. Then X × Y ≡_{s} W, so let Φ be a Turing functional such that Φ(W) ⊆ 0^{⌢}X ∪ 1^{⌢}Y. If Φ(σ_{i}^{⌢}f_{i}) ∈ 0^{⌢}X for some i ∈ X, then x ≤_{s} deg_{s}({f_{i}}) for this i, contrary to assumption. Thus it must be that Φ(σ_{i}^{⌢}f_{i}) ∈ 1^{⌢}Y for all i ∈ X. It must also be that Φ(g) ∈ 1^{⌢}Y. If not, then Φ(g)(0) = 0 and there is some τ ⊂ g such that Φ(τ)(0) = 0. Choose i ∈ X with i > |τ|. Then τ ⊂ σ_{i}, giving the contradiction Φ(σ_{i}^{⌢}f_{i})(0) = 0. Therefore Φ(W) ⊆ 1^{⌢}Y. Thus w ≥_{s} y, a contradiction.

(ii) Given i ∈ X, it is an easy check (using the fact that W is a g-spine that is a ≤_{T}-antichain) that deg_{s}(W \ {σ_{i}^{⌢}f_{i}}) witnesses that deg_{s}({f_{i}}) meets to w. Hence {deg_{s}({f_{i}}) | i ∈ X} ⊆ E(w). To see equality, let x ∈ E(w). By item (i), x ≤_{s} deg_{s}({f_{i}}) for some i ∈ X. We have just shown that deg_{s}({f_{i}}) ∈ E(w), and E(w) is an antichain by Lemma 2.2.3. So it must be that x = deg_{s}({f_{i}}).

Notice the difference between Lemma 2.4.4 and Lemma 2.6.3. If W is a gspine that is a ≤_{T}-antichain, then in D_{w,cl}^{01} we have deg_{w}({g}) ∈ E(deg_{w}(W)), but in D_{s,cl}^{01} we have deg_{s}({g}) ∉ E(deg_{s}(W)).

Lemma 2.6.4. There is a code w⃗ in D_{s,cl}^{01} such that M ; w⃗ ≅ N^{}.

Proof.Let g, W′_{0} = {f_{0,n}}_{n∈ω}, W′_{1} = {f_{1,n}}_{n∈ω}, and W′_{2} = {f_{2,n}}_{n∈ω} be such that {g} ∪ W′_{0} ∪ W′_{1} ∪ W′_{2} ⊆ 2^{ω} is independent. Then let

w_{0} = deg_{s}(W_{0})  for W_{0} = spine(g, W′_{0}),
w_{1} = deg_{s}(W_{1})  for W_{1} = spine(g, W′_{1}),
w_{2} = deg_{s}(W_{2})  for W_{2} = spine(g, W′_{2}),
m = deg_{s}(M)  for M = spine(g, {f_{0,n} ⊕ f_{1,n}}_{n∈ω} ∪ {f_{0,n} ⊕ f_{2,n}}_{n∈ω}),
ℓ = deg_{s}(L)  for  L = spine(g, {f_{0,i} ⊕ f_{1,j} | i < j}),
p = deg_{s}(P)  for  P = spine(g, {f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i + j = k}),
t = deg_{s}(T )  for  T = spine(g, {f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i × j = k}),
d = deg_{s}(D)  for  D = spine(g, {f_{0,i} ⊕ f_{1,j} ⊕ f_{2,k} | i j = k}),
z = deg_{s}({f_{0,0}}), and
o = deg_{s}({f_{0,1}}).

These degrees give the code w⃗. The proof that M ; w⃗ ≅ N^{}is the same as in the proof of Lemma 2.3.9. Use Lemma 2.6.3 in place of Lemma 2.3.7.

Theorem 2.6.5. Th(D_{s,cl}; ≤_{s}) ≡_{1} Th(D_{s,cl}^{01}; ≤_{s}) ≡_{1} Th_{2}(N).

Proof.Th(D_{s,cl}; ≤_{s}) ≤_{1} Th_{2}(N) and Th(D_{s,cl}^{01}; ≤_{s}) ≤_{1} Th_{2}(N) by Lemma 2.1.2. For Th_{2}(N) ≤_{1} Th(D_{s,cl}; ≤_{s}), by Lemma 2.6.2 let w⃗ be a code in D_{s,cl} such that M ; w⃗ ≅ N^{}. Removing the degree d from the code w⃗ gives a code v⃗ such that Mv⃗ ≅ N. D_{s,cl} has the coding countable subsets property by Lemma 2.5.6. Hence Th_{2}(N) ≤_{1} Th(D_{s}; ≤_{s}) by Lemma 2.2.19. The proof that Th_{2}(N) ≤_{1} Th(D_{s,cl}^{01}; ≤_{s}) is the same. Use Lemma 2.6.4 in place of Lemma 2.6.2.

Theorem 2.6.6. Σ_{3}^{0}-Th(D_{s}), Σ_{4}^{0}-Th(D_{s}; ≤_{s}), Σ_{3}^{0}-Th(D_{s,cl}), Σ_{4}^{0}-Th(D_{s,cl}; ≤_{s}), Σ_{3}^{0}- Th(D_{s,cl}^{01}), and Σ_{4}^{0}-Th(D_{s,cl}^{01}; ≤_{s}) are all undecidable.

Proof.By Lemma 2.6.2, there is a code w⃗ in D_{s,cl} such that M ; w⃗ ≅ N^{}. One readily checks that this w⃗ is also satisfies M ; w⃗ ≅ N^{}when M ; w⃗ is interpreted in D_{s} instead of D_{s,cl} because Lemma 2.6.1 is valid in D_{s}. By Lemma 2.6.4, there is a code w⃗ in D_{s,cl}^{01} such that M ; w⃗ ≅ N^{}. The results then follow from Lemma 2.2.25.

2.7 A first-order sentence distinguishing D_{s,cl} and D_{s,cl}^{01} from D_{w,cl} and D_{w,cl}^{01}

D_{w,cl} and D_{w,cl}^{01} are countably meet-complete by Lemma 2.4.2. In contrast, if 𝔛 ⊆ D_{s,cl} or 𝔛 ⊆ D_{s,cl}^{01} is countable and strongly meet-incomplete, then 𝔛 does not have a greatest lower bound by the following lemma. Recall that a subset X of a lattice L is strongly meet-incomplete if and only if for every finite {y_{i} | i < n} ⊆ X there is an x ∈ X such that x ≱ ∏_{i<n} y_{i}.

Lemma 2.7.1 (Dyment [23]; See also [74]). No countable strongly meet-incomplete 𝔛 ⊆ D_{s} has a greatest lower bound.

The proof of Lemma 2.7.1 works in D_{s,cl}, and it only requires a slight modification for D_{s,cl}^{01}.

If w⃗ is a code for a model of PA^{−} that satisfies the second-order correctness condition in any of D_{s,cl}, D_{s,cl}^{01}, D_{w,cl}, D_{w,cl}^{01}, then E(w_{0}) is countable. This observation gives us the following theorem.

Theorem 2.7.2. Neither D_{s,cl} nor D_{s,cl}^{01} is elementarily equivalent to either D_{w,cl} or D_{w,cl}^{01}.

Proof.Let φ be the first-order sentence “for all w⃗, if w⃗ is a code for a model of PA^{−} that satisfies the second-order correctness condition, then E(w_{0}) has a greatest lower bound.” The sentence φ is true in both D_{w,cl} and D_{w,cl}^{01} because such an E(w_{0}) is countable and these lattices are countably meet-complete by Lemma 2.4.2. On the other hand, φ fails in both D_{s,cl} and D_{s,cl}^{01}. If w⃗ is the code produced in either Lemma 2.6.2 or Lemma 2.6.4, then E(w_{0}) = {deg_{s}({f_{i}}) | i ∈ ω} where {f_{i} | i ∈ ω} is a ≤_{T}-antichain. It is then easy to check that E(w_{0}) is strongly meet-incomplete and hence has no greatest lower bound by Lemma 2.7.1.

The relationship between D_{s,cl} and D_{s,cl}^{01} and the relationship between D_{w,cl} and D_{w,cl}^{01} require further study.

Question 2.7.3.

• Is every closed X ⊆ ω^{ω} Medvedev equivalent to some closed Y ⊆ 2^{ω}? If not, are D_{s,cl} and D_{s,cl}^{01} isomorphic? If not, are D_{s,cl} and D_{s,cl}^{01} elementarily equivalent?

• Is every closed X ⊆ ω^{ω} Muchnik equivalent to some closed Y ⊆ 2^{ω}? If not, are D_{w,cl} and D_{w,cl}^{01} isomorphic? If not, are D_{w,cl} and D_{w,cl}^{01} elementarily equivalent?

2.8 Meet-irreducibles in E_{s} and r.e. separating degrees

In this section we present facts about the meet-irreducibles in E_{s} that allow us to implement our coding in E_{s}. We begin with a characterization of the meetirreducibles in E_{s}.

Lemma 2.8.1 ([6] Corollary 3.5). Let Q be a Π_{1}^{0} class. Then deg_{s}(Q) is meetirreducible if and only if for every clopen C ⊆ 2^{ω} either Q ∩ C ≡_{s} Q or Q ∩ C^{c} ≡_{s} Q.

Proof.We prove the contrapositive in both directions. First, suppose C ⊆ 2^{ω} is clopen, Q ∩ C ≢_{s} Q, and Q ∩ C^{c} ≢_{s} Q. Q ∩ C ≥_{s} Q and Q ∩ C^{c} ≥_{s} Q by the identity functional, so it must be that Q ∩ C >_{s} Q and Q ∩ C^{c} >_{s} Q. C is clopen, so there is a finite set of strings {σ_{i}}_{i<n} ⊆ 2^{<ω} such that C = ⋃_{i<n} I(σ). Then 0^{⌢}(Q ∩ C) ∪ 1^{⌢}(Q ∩ C^{c}) ≤_{s} Q by the functional

f ↦ \begin{cases}0^{⌢}f & if (∃i < n)(σ_{i} ⊂ f) \\ 1^{⌢}f & otherwse . \end{cases}

What we have shown is deg_{s}(Q ∩ C) >_{s} deg_{s}(Q), deg_{s}(Q ∩ C^{c}) >_{s} deg_{s}(Q), and deg_{s}(Q ∩ C) × deg_{s}(Q ∩ C^{c}) ≤_{s} deg_{s}(Q). Thus deg_{s}(Q) is meet-reducible.

Conversely, suppose deg_{s}(Q) is meet-reducible, and let X and Y be Π_{1}^{0} classes such that X >_{s} Q, Y >_{s} Q, and Q ≡_{s} 0^{⌢}X ∪1^{⌢}Y. Let Φ be such that Φ(Q) ⊆ 0^{⌢}X ∪ 1^{⌢}Y. Consider the set X̂ = {f ∈ Q | Φ(f)(0) = 0}. Φ(f) is total for all f ∈ Q, so we can write X̂ = Q ∩ {f ∈ 2^{ω} | Φ(f)(0) ≠ 1} (where Φ(f)(0) ≠ 1 includes the possibility that Φ(f)(0) diverges), which is the intersection of two closed subsets of 2^{ω}. Hence X̂ is compact. Let Σ = {σ ∈ 2^{<ω} | Φ(σ)(0) = 0}. Then X̂ ⊆ ⋃_{σ∈Σ} I(σ), so by compactness there is a finite Σ_{0} ⊆ Σ such that X̂ ⊆ ⋃_{σ∈Σ_{0}} I(σ). Let C = ⋃_{σ∈Σ_{0}} I(σ) be this clopen set. Φ witnesses that Q ∩ C ≥_{s} 0^{⌢}X and that Q∩C^{c} ≥_{s} 1^{⌢}Y. As 0^{⌢}X ≡_{s} X >_{s} Q and 1^{⌢}Y ≡_{s} Y >_{s} Q, we have the desired clopen set C ⊆ 2^{ω} such that Q ∩ C ≢_{s} Q and Q ∩ C^{c} ≢_{s} Q.

Degrees of r.e. separating classes are the main examples of meet-irreducibles in E_{s}.

Definition 2.8.2. For A, B ⊆ ω, define

S(A, B) = {f ∈ 2^{ω} | ∀n((n ∈ A → f(n) = 1) ∧(n ∈ B → f(n) = 0))}.

An f ∈ S(A, B) is said to separate A from B. S ⊆ 2^{ω} is an r.e. separating class if and only if there are disjoint r.e. sets A, B ⊆ ω such that S = S(A, B).

From the definition, an r.e. separating class is always a Π_{1}^{0} class. An s ∈ E_{s} is an r.e. separating degree if and only if s = deg_{s}(S) for an r.e. separating class S.

Lemma 2.8.3 ([15] Lemma 6). If S is an r.e. separating class and C ⊆ 2^{ω} is a clopen set such that S ∩ C ≠ ∅, then S ∩ C ≡_{s} S.

Proof.Let S = S(A, B) be an r.e. separating class and let C ⊆ 2^{ω} be a clopen set such that S ∩ C ≠ ∅. S ≤_{s} S ∩ C by the identity functional. To see S ≥_{s} S ∩ C, let σ be such that I(σ) ⊆ C and S ∩ I(σ) ≠ ∅. For any f ∈ 2^{ω}, let f_{σ} be the function obtained from f by replacing the initial segment of f of length |σ| by σ:

f_{σ}(n) = \begin{cases}σ(n) & if n < |σ| \\ f(n) & if n ≥ |σ|. \end{cases}

The condition S ∩ I(σ) ≠ ∅ implies that σ separates {n ∈ A | n < |σ|} from {n ∈ B | n < |σ|}. Thus if f separates A from B, then so does f_{σ}. Therefore the functional f ↦ f_{σ} witnesses S ≥_{s} S ∩ C.

Lemma 2.8.1 and Lemma 2.8.3 imply that every r.e. separating degree is meet-irreducible. It is important to note (as in [15]) that the r.e. separating classes are closed under + and consequently that the r.e. separating degrees are closed under join: if S(A_{0}, B_{0}) and S(A_{1}, B_{1}) are r.e. separating classes then S(A_{0}, B_{0}) + S(A_{1}, B_{1}) = S(A_{0} ⊕ A_{1}, B_{0} ⊕ B_{1}). Thus the join of two r.e. separating degrees is meet-irreducible. In fact, the join of any r.e. separating degree and any meet-irreducible degree is again meet-irreducible.

Lemma 2.8.4. Let q ∈ E_{s} be meet-irreducible and let s ∈ E_{s} be an r.e. separating degree. Then q + s is meet-irreducible.

Proof.Suppose q + s ≥_{s} x × y for some x, y ∈ E_{s}. We show q + s ≥_{s} x or q + s ≥_{s} y. Let Q, X, and Y be Π_{1}^{0} classes such that deg_{s}(Q) = q, deg_{s}(X) = x, and deg_{s}(Y) = y respectively, and let S be an r.e. separating class such that deg_{s}(S) = s. Let Φ be such that Φ(Q + S) ⊆ 0^{⌢}X ∪ 1^{⌢}Y. By compactness, choose a σ ∈ 2^{<ω} such that S ∩ I(σ) ≠ ∅ and an n ∈ ω such that

(∀τ ∈ 2^{n})((∃f ∈ Q)(τ ⊂ f) → Φ(τ ⊕ σ)(0)↓).

Let C = ⋃{I(τ) | τ ∈ 2^{n} ∧ Φ(τ ⊕ σ)(0) = 0}. Then C is clopen, and Φ witnesses that (Q ∩ C) +(S ∩ I(σ)) ≥_{s} 0^{⌢}X ≡_{s} X and that (Q ∩ C^{c}) +(S ∩ I(σ)) ≥_{s} 1^{⌢}Y ≡_{s} Y. Since S ∩ I(σ) ≡_{s} S by Lemma 2.8.3 and either Q ∩ C ≡_{s} Q or Q ∩ C^{c} ≡_{s} Q by Lemma 2.8.1, we have either Q + S ≥_{s} X or Q + S ≥_{s} Y as desired.

Our proof that E_{s} has the finite matching property uses the following lemma of Cole and Kihara. It is the main tool in their proof that the Σ_{2}^{0}-theory of E_{s} as a partial order is decidable.

Lemma 2.8.5 ([18] Lemma 1). Let {q_{i}}_{i<n} ⊆ E_{s} and let m ∈ ω. Then there is a set {r_{i}}_{i<m} ⊆ E_{s} such that

(∀I ⊆ m)(∀J, K ⊆ n)

(J ∩ K = ∅ ∧ ∑_{i∈J} q_{i} ≱_{s} ∏_{i∈K} q_{i} →(∑_{i∈J} q_{i} + ∑_{i∈I} r_{i}) ≱_{s}(∏_{i∈K} q_{i} × ∏_{i∉I} r_{i})),

where ∑_{i∈∅} x_{i} = 0 and ∏_{i∈∅} x_{i} = 1 = deg_{s}(DNR_{2}).

Cole and Kihara note that the {r_{i}}_{i<m} that they construct in Lemma 2.8.5 are all r.e. separating degrees. Their proof of Lemma 2.8.5 is an elaboration of Cenzer and Hinman’s proof that E_{s} is dense [15]. Cenzer and Hinman prove that if p, q ∈ E_{s} are such that q ≰_{s} p, then there is an r.e. separating degree r ∈ E_{s} such that q × r ≰_{s} p and q ≰_{s} p + r. Thus if p <_{s} q, then p <_{s}(p + r) × q <_{s} q, yielding density. To make Lemma 2.8.5 somewhat easier to read and apply, we note that we only need the following special case.

Lemma 2.8.6. Let {q_{i}}_{i<n} ⊆ E_{s} \{1} and let m ∈ ω. Then there is a set of r.e. separating degrees {r_{i}}_{i<m} ⊆ E_{s} such that

(i) (∀i, i′ < n)(∀j < m)(q_{i} ≱_{s} q_{i′} → q_{i} + r_{j} ≱_{s} q_{i′}),

(ii) (∀i < n)(∀j, j′ < m)(j ≠ j′ → q_{i} + r_{j} ≱_{s} r_{j′}), and

(iii) (∀i < n)(∀j < m)(q_{i} + r_{j} ≱_{s} 1).

Notice that item (iii) follows from item (ii) unless m = 1.

We can now show that E_{s} has the finite matching property.

Lemma 2.8.7. E_{s} has the finite matching property. That is, if q, q′ ∈ E_{s} are such that |E(q)| = |E(q′)| = n for some n ∈ ω, then there is an r ∈ E_{s} such that E(r) matches both E(q) and E(q′).

Proof.If n = 0, then let r = q. Any degree z vacuously witnesses that E(r) matches E(q) and that E(r) matches E(q′). So suppose n > 0, let E(q) = {q_{i}}_{i<n}, and let E(q′) = {q′_{i}}_{i<n}. Apply Lemma 2.8.6 to {q_{i}}_{i<n} ∪ {q′_{i}}_{i<n} with m = n, noting that {q_{i}}_{i<n} and {q′_{i}}_{i<n} are both antichains by Lemma 2.2.3, to get r.e. separating degrees {r_{i}}_{i<n} such that

(i) q_{i} + r_{j} ≱_{s} q_{k} and q′_{i} + r_{j} ≱_{s} q′_{k} whenever i, j, k < n are such that i ≠ k, and

(ii) q_{i} + r_{j} ≱_{s} r_{k} and q′_{i} + r_{j} ≱_{s} r_{k} whenever i, j, k < n are such that j ≠ k.

(iii) q_{i} + r_{j} ≱_{s} 1 and q′_{i} + r_{j} ≱_{s} 1 whenever i, j < n.

(Lemma 2.8.6 applies because, by definition, 1 does not meet to any degree and so cannot be in E(q) or E(q′).)

Put r = ∏_{i<n} r_{i}, z = ∏_{i<n}(q_{i} + r_{i}), and z′ = ∏_{i<n}(q′_{i} + r_{i}). We show that z witnesses that E(r) matches E(q). The proof that z′ witnesses that E(r) matches E(q′) is similar. Item (ii) implies that {r_{i}}_{i<n} and {q_{i} + r_{i}}_{i<n} are both antichains. Lemma 2.8.4 implies that q_{i} + r_{i} is meet-irreducible for each i < n. Therefore E(r) = {r_{i}}_{i<n} and E(z) = {q_{i} + r_{i}}_{i<n} by Lemma 2.2.4. Suppose q_{i} + r_{j} ≥_{s} z for some i, j < n. Then q_{i} + r_{j} ≥_{s} q_{k} + r_{k} for some k < n because q_{i} + r_{j} is meetirreducible by Lemma 2.8.4. Item (i) implies that i = k, and item (ii) implies that j = k. Thus for each i < n, r_{i} is the unique y ∈ E(r) such that q_{i} + y ∈ E(z), and q_{i} is the unique y ∈ E(q) such that r_{i} + y ∈ E(z). Thus z witnesses that E(r) matches E(q).

We need one last fact about the r.e. separating classes to implement our coding in E_{s}.

Lemma 2.8.8 ([32] Theorem 4.1). There is a recursive sequence {S_{n}}_{n∈ω} r.e. separating classes that is strongly independent.

2.9 The complexity of Th(E_{s}; ≤_{s})

In this section we prove that Th(E_{s}; ≤_{s}) ≡_{1} Th(N) and that Π_{3}^{0}-Th(E_{s}) and Π_{4}^{0}- Th(E_{s}; ≤_{s}) are undecidable. By Lemma 2.2.14 and Lemma 2.2.25, it suffices to find a code w⃗ in E_{s} such that M ; w⃗ ≅ N^{}.

Definition 2.9.1. Let Q be a Π_{1}^{0} class with no recursive member. Let A be an infinite recursive set, and let {σ_{n}}_{n∈A} be a recursive sequence of pairwise incomparable strings such that ⋃_{n∈A} I(σ_{n}) = 2^{ω} \Q (for example, let T be a recursive tree such that Q = [T] and let {σ_{n}}_{n∈A} be the strings σ ∉ T of minimal length). Let {S_{n}}_{n∈A} be an infinite recursive sequence of Π_{1}^{0} classes. Define spine(Q, {S_{n}}_{n∈A}) to be the Π_{1}^{0} class

spine(Q, {S_{n}}_{n∈A}) = Q ∪ ⋃_{n∈A} σ_{n}^{⌢}S_{n}.

Lemma 2.9.2. Let Q be a Π_{1}^{0} class with no recursive member. Let {S_{n}}_{n∈A} be an infinite recursive sequence of r.e. separating classes (indexed by a recursive set A) that is an antichain and is such that Q ≰_{s} S_{n} for all n ∈ A. Let w = deg_{s}(spine(Q, {S_{n}}_{n∈A})).

(i) If x ∈ E_{s} meets to w, then x ≤_{s} deg_{s}(S_{n}) for some n ∈ A. (ii) E(w) = {deg_{s}(S_{n}) | n ∈ A}.

Proof.Let W = spine(Q, {S_{n}}_{n∈A}).

(i) Let x ∈ E_{s} be such that x meets to w. Suppose for a contradiction that x ≰_{s} deg_{s}(S_{n}) for all n ∈ A. Let X be a Π_{1}^{0} class such that x = deg_{s}(X), and let Y be a Π_{1}^{0} class such that deg_{s}(Y) witnesses that x meets to w. That is, Y >_{s} W and W ≡_{s} 0^{⌢}X ∪ 1^{⌢}Y. Let Φ be such that Φ(Q ∪ ⋃_{n∈A} σ_{n}^{⌢}S_{n}) ⊆ 0^{⌢}X ∪ 1^{⌢}Y.

Claim.

(a) Φ(σ_{n}^{⌢}S_{n}) ⊆ 1^{⌢}Y for all n ∈ A and (b) Φ(Q) ⊆ 1^{⌢}Y.

Proof of claim.If item (a) fails, then for some n ∈ A there is a clopen C ⊆ 2^{ω} such that (σ_{n}^{⌢}S_{n})∩C ≠ ∅ and Φ((σ_{n}^{⌢}S_{n})∩C) ⊆ 0^{⌢}X. So (σ_{n}^{⌢}S_{n})∩C ≥_{s} 0^{⌢}X ≡_{s} X. The class σ_{n}^{⌢}S_{n} is an r.e. separating class because S_{n} is, so (σ_{n}^{⌢}S_{n})∩C ≡_{s} σ_{n}^{⌢}S_{n} ≡_{s} S_{n}, where the first equivalence is by Lemma 2.8.3. Thus the contradiction X ≤_{s} S_{n}.

If item (b) fails, then there is an f ∈ Q and a σ ⊂ f such that Φ(σ)(0)↓ = 0. Since I(σ) ⊈ Q, there is an n ∈ A such that σ_{n} ⊇ σ. Hence Φ(σ_{n}^{⌢}S_{n}) ⊈ 1^{⌢}Y, contradicting item (a).

The claim shows that Φ(Q ∪ ⋃_{n∈ω} σ_{n}^{⌢}S_{n}) ⊆ 1^{⌢}Y. Thus Y ≤_{s} W, which contradicts Y >_{s} W.

(ii) Let n ∈ A. To see that deg_{s}(S_{n}) ∈ E(w), let Y = Q ∪ ⋃_{i∈A\{n}} σ_{i}^{⌢}S_{i}.

Claim. S_{n} ≱_{s} Y

Proof of claim.Suppose for a contradiction that Φ is such that Φ(S_{n}) ⊆ Y. If there is an i ∈ A \ {n} such that Φ(S_{n}) ∩ (σ_{i}^{⌢}S_{i}) ≠ ∅, then there is a clopen C ⊆ 2^{ω} such that S_{n} ∩ C ≠ ∅ and Φ(S_{n} ∩ C) ⊆ σ_{i}^{⌢}S_{i}. S_{n} ≡_{s} S_{n} ∩ C by Lemma 2.8.3, and S_{n} ∩ C ≥_{s} σ_{i}^{⌢}S_{i} ≡_{s} S_{i}. This contradicts that {S_{n}}_{n∈A} is an antichain. Thus Φ(S_{n}) ∩ (σ_{i}^{⌢}S_{i}) = ∅ for all n ∈ A. Therefore Φ(S_{n}) ⊆ Q. This contradicts Q ≰_{s} S_{n}.

It is easy to check that W ≡_{s} 0^{⌢}S_{n} ∪ 1^{⌢}Y, so, by the claim, deg_{s}(Y) witnesses that deg_{s}(S_{n}) meets to w. The degree deg_{s}(S_{n}) is meet-irreducible because it is an r.e. separating degree. Thus deg_{s}(S_{n}) ∈ E(w).

We have shown that {deg_{s}(S_{n}) | n ∈ A} ⊆ E(w). To see equality, let x ∈ E(w). By item (i), x ≤_{s} deg_{s}(S_{n}) for some n ∈ A. E(w) is an antichain by Lemma 2.2.3 and deg_{s}(S_{n}) ∈ E(w), so it must be that x = deg_{s}(S_{n}).

We now have all the ingredients to find a code for N in E_{s}.

Lemma 2.9.3. There is a code w⃗ in E_{s} such that M ; w⃗ ≅ N^{}.

Proof.By Lemma 2.8.8, let Q be an r.e. separating class and let {S_{0,n}}_{n∈ω}, {S_{1,n}}_{n∈ω}, and {S_{2,n}}_{n∈ω} be recursive sequences of r.e. separating classes such that {Q} ∪ {S_{0,n}}_{n∈ω} ∪ {S_{1,n}}_{n∈ω} ∪ {S_{2,n}}_{n∈ω} is strongly independent. Then let

w_{0} = deg_{s}(W_{0}) for W_{0} = spine(Q, {S_{0,n}}_{n∈ω}),
w_{1} = deg_{s}(W_{1}) for W_{1} = spine(Q, {S_{1,n}}_{n∈ω}),
w_{2} = deg_{s}(W_{2}) for W_{2} = spine(Q, {S_{2,n}}_{n∈ω}),
m = deg_{s}(M) for M = spine(Q, {S_{0,n} + S_{1,n}}_{n∈ω} ∪ {S_{0,n} + S_{2,n}}_{n∈ω}),
ℓ = deg_{s}(L)  for  L = spine(Q, {S_{0,i} + S_{1,j} | i < j}),
p = deg_{s}(P)  for  P = spine(Q, {S_{0,i} + S_{1,j} + S_{2,k} | i + j = k}),
t = deg_{s}(T )  for  T = spine(Q, {S_{0,i} + S_{1,j} + S_{2,k} | i × j = k}),

d = deg_{s}(D)  for  D = spine(Q, {S_{0,i} + S_{1,j} + S_{2,k} | i j = k}),
z = deg_{s}(S_{0,0}), and
o = deg_{s}(S_{0,1}).

By Lemma 2.9.2 item (ii), E(w_{0}) = {deg_{s}(S_{0,n})}_{n∈ω}. The map deg_{s}(S_{0,n}) ↦ n is the isomorphism witnessing M ; w⃗ ≅ N^{}. Clearly z ↦ 0 and o ↦ 1. We show that the map preserves <. The proofs that the map preserves +, ×, and are similar. Let i, j ∈ ω. If i < j, then deg_{s}(S_{1,j}) meets to w_{1} by Lemma 2.9.2 item (ii), and it is easy to see that deg_{s}(S_{0,j}) + deg_{s}(S_{1,j}) ≥_{s} m and that deg_{s}(S_{0,i}) + deg_{s}(S_{1,j}) ≥_{s} ℓ. Thus R_{ℓ}^{2}(deg_{s}(S_{0,i}), deg_{s}(S_{0,j})). Conversely, suppose that R_{ℓ}^{2}(deg_{s}(S_{0,i}), deg_{s}(S_{0,j})). Let u_{1} ∈ E_{s} be such that u_{1} meets to w_{1}, deg_{s}(S_{0,j}) + u_{1} ≥_{s} m, and deg_{s}(S_{0,i}) + u_{1} ≥_{s} ℓ. Since u_{1} meets to w_{1}, it must be that u_{1} ≤_{s} deg_{s}(S_{1,k}) for some k ∈ ω by Lemma 2.9.2 item (i). Thus deg_{s}(S_{0,j}) + deg_{s}(S_{1,k}) ≥_{s} m. However, if k ≠ j, then no member of S_{0,j} + S_{1,k} computes any member of M by strong independence. Thus u_{1} ≤_{s} deg_{s}(S_{1,j}), which implies that deg_{s}(S_{0,i}) + deg_{s}(S_{1,j}) ≥_{s} ℓ. Again by strong independence, if i ≮ j, then no member of S_{0,i} + S_{1,j} computes any member of L. Hence i < j.

Higuchi also used spines of recursive sequences of independent r.e. separating classes to prove that E_{s} is not a Brower algebra [25].

Theorem 2.9.4. Th(E_{s}; ≤_{s}) ≡_{1} Th(N).

Proof.Th(E_{s}; ≤_{s}) ≤_{1} Th(N) by Lemma 2.1.3. For Th(N) ≤_{1} Th(E_{s}; ≤_{s}), by Lemma 2.9.3 let w⃗ be a code in E_{s} such that M ; w⃗ ≅ N^{}. Removing the degree d from the code w⃗ gives a code v⃗ such that Mv⃗ ≅ N. E_{s} has the finite matching property by Lemma 2.8.7, thus Th(N) ≤_{1} Th(E_{s}; ≤_{s}) by Lemma 2.2.14.

Theorem 2.9.5. Σ_{3}^{0}-Th(E_{s}) and Σ_{4}^{0}-Th(E_{s}; ≤_{s}) are undecidable.

Proof.There is a code w⃗ in E_{s} such that M ; w⃗ ≅ N^{}by Lemma 2.9.3. The results then follow from Lemma 2.2.25.

2.10 The degree of E_{s} is 0′′′

In this section, we consider the complexities of presentations of E_{s}.

Definition 2.10.1. A presentation of E_{s} as a partial order consists of a relation ≤_{P}⊆ ω ×ω such that the structure P = (ω; ≤_{P}) is isomorphic to (E_{s}; ≤_{s}). A presentation of E_{s} as a lattice consists of a relation ≤_{L}⊆ ω ×ω and functions +_{L}: ω ×ω → ω and ×_{L}: ω × ω → ω such that the structure L = (ω; ≤_{L}, +_{L}, ×_{L}) is isomorphic to E_{s}.

We measure the complexities of presentations by their Turing degrees.

Definition 2.10.2. The degree of a presentation P of E_{s} as a partial order is deg_{T}(P) = deg_{T}(≤_{P}). The degree of a presentation L of E_{s} as a lattice is deg_{T}(L) = deg_{T}(≤_{L} ⊕ +_{L} ⊕ ×_{L}).

Equivalently, the degree of a presentation is the Turing degree of its atomic diagram, suitably Gödel numbered.

Lemma 2.10.3. There is a presentation L of E_{s} as a lattice with deg_{T}(L) ≤_{T} 0′′′.

Proof.Let {Z_{e}}_{e∈ω} be a recursive sequence containing all Π_{1}^{0} classes as in Lemma 1.4.4, and let {T_{e}}_{e∈ω} be the corresponding uniformly recursive sequence of trees. Since [T_{i}] ≤_{s}[T_{j}] is a Σ_{3}^{0} property of ⟨ i, j ⟩ by Lemma 1.4.3, we can use 0′′′ to make a new sequence of trees {T′_{e}}_{e∈ω} such that {[T′_{e}]}_{e∈ω} contains exactly one representative for each degree in E_{s}. Inductively, let T′_{e} be T_{i} for the least i ∈ ω such that (∀j < e)([T_{i}] ≢_{s}[T′_{j}]). Again using 0′′′, for i, j ∈ ω define i ≤_{L} j if and only if [T′_{i}] ≤_{s}[T′_{j}], define +_{L}(i, j) to be the k ∈ ω such that [T′_{k}] ≡_{s}[T′_{i} + T′_{j}], and define ×_{L}(i, j) to be the k ∈ ω such that [T′_{k}] ≡_{s}[0^{⌢}T′_{i} ∪ 1^{⌢}T′_{j}]. Then L ≅ E_{s} and deg_{T}(L) ≤_{T} 0′′′.

We prepare to show that every presentation of E_{s} as a lattice computes 0′′′. Let {X_{n}}_{n∈ω} be a recursive sequence of Π_{1}^{0} classes, and let m ∈ ω. Define ∑_{n∈ω} X_{n} and ∑_{n∈ω\{m}} X_{n} by

∑_{n∈ω} X_{n} = {⨁_{n∈ω} f_{n} | ∀n(f_{n} ∈ X_{n})} and

∑_{n∈ω\{m}} X_{n} = { ⨁_{n∈ω\{m}} f_{n} | ∀n(n ≠ m → f_{n} ∈ X_{n})}.

The predicates ∀n(f_{n} ∈ X_{n}) and ∀n(n ≠ m → f_{n} ∈ X_{n}) are Π_{1}^{0} because the sequence {X_{n}}_{n∈ω} is recursive. Hence ∑_{n∈ω} X_{n} and ∑_{n∈ω\{m}} X_{n} are Π_{1}^{0} classes. If {S(A_{n}, B_{n})}_{n∈ω} is a recursive sequence of r.e. separating classes, then one checks that

∑_{n∈ω} S(A_{n}, B_{n}) = S(⨁_{n∈ω} A_{n}, ⨁_{n∈ω} B_{n}) and

∑_{n∈ω\{m}} S(A_{n}, B_{n}) = S( ⨁_{n∈ω\{m}} A_{n}, ( ⨁_{n∈ω\{m}} B_{n}) ∪ {⟨ m, k ⟩ | k ∈ ω}).

These two Π_{1}^{0} classes are in fact r.e. separating classes because any Π_{1}^{0} class that is a separating class must be an r.e. separating class. If T is a recursive tree such that [T] = S(A, B) for A, B ⊆ ω, then A = {n | (∃s > n)(∀σ ∈ 2^{s})(σ ∈ T → σ(n) = 1)} and B = {n | (∃s > n)(∀σ ∈ 2^{s})(σ ∈ T → σ(n) = 0)}, both of which are r.e.

Lemma 2.10.4. Let Q be an r.e. separating class, and let φ(e, m, k, ℓ) be a recursive predicate. Then there is a recursive sequence of Π_{1}^{0} classes {X_{⟨ e,m ⟩}}_{⟨ e,m ⟩∈ω} such that for all e, m ∈ ω

deg_{s}(X_{⟨ e,m ⟩}) = \begin{cases}0 & if ∀k∃ℓφ(e, m, k, ℓ) \\ deg_{s}(Q) & if ∃k∀ℓ¬φ(e, m, k, ℓ). \end{cases}

Proof.Let A and B be disjoint r.e. sets such that Q = S(A, B). Let {A_{s}}_{s∈ω} and {B_{s}}_{s∈ω} be recursive stage enumerations of A and B respectively. For e, m ∈ ω, let X_{⟨ e,m ⟩} be the r.e. separating class X_{⟨ e,m ⟩} = S(C_{⟨ e,m ⟩}, D_{⟨ e,m ⟩}) where

C_{⟨ e,m ⟩} = {⟨ k, x ⟩ | ∃s(x ∈ A_{s} ∧(∀ℓ ≤ s)(¬φ(e, m, k, ℓ)))} and
D_{⟨ e,m ⟩} = {⟨ k, x ⟩ | ∃s(x ∈ B_{s} ∧(∀ℓ ≤ s)(¬φ(e, m, k, ℓ)))}.

For all k ∈ ω, the k^{th} column of C_{⟨ e,m ⟩} is a subset of A, and the k^{th} column of D_{⟨ e,m ⟩} is a subset of B. Thus C_{⟨ e,m ⟩} and D_{⟨ e,m ⟩} are disjoint. The sequences {C_{⟨ e,m ⟩}}_{⟨ e,m ⟩∈ω} and {D_{⟨ e,m ⟩}}_{⟨ e,m ⟩∈ω} are uniformly r.e., which implies that the sequence {X_{⟨ e,m ⟩}}_{⟨ e,m ⟩∈ω} is a recursive sequence of Π_{1}^{0} classes.

To see that X_{⟨ e,m ⟩} has the desired degree, first suppose that ∀k∃ℓφ(e, m, k, ℓ). In this case, the set C_{⟨ e,m ⟩} is recursive. To determine if ⟨ k, x ⟩ ∈ C_{⟨ e,m ⟩}, search for the least ℓ such that φ(e, m, k, ℓ), which must exist by assumption. Once ℓ is found, enumerate A up to stage ℓ. Then ⟨ k, x ⟩ ∈ C_{⟨ e,m ⟩} if and only if x ∈ A_{ℓ}. X_{⟨ e,m ⟩} contains the characteristic function of C_{⟨ e,m ⟩}, which we have just shown is recursive, so deg_{s}(X_{⟨ e,m ⟩}) = 0. On the other hand, if ∃k∀ℓ¬φ(e, m, k, ℓ), then fix a witnessing k. In this case, the k^{th} column of C_{⟨ e,m ⟩} is A, and the k^{th} column of D_{⟨ e,m ⟩} is B. Given f ∈ 2^{ω}, let f_{k} be the function f_{k}(x) = f(⟨ k, x ⟩). If f separates C_{⟨ e,m ⟩} from D_{⟨ e,m ⟩}, then f_{k} separates A from B. Thus the functional f ↦ f_{k} witnesses X_{⟨ e,m ⟩} ≥_{s} Q. The functional f ↦ g where g(⟨ i, x ⟩) = f(x) always witnesses Q ≥_{s} X_{⟨ e,m ⟩}. Hence deg_{s}(X_{⟨ e,m ⟩}) = deg_{s}(Q).

Lemma 2.10.5. If L is a presentation of E_{s} as a lattice, then 0′′′ ≤_{T} deg_{T}(L).

Proof.Let L = (ω; ≤_{L}, +_{L}, ×_{L}) be a presentation of E_{s}. Let f: E_{s} → L be an isomorphism. Fix a Σ_{3}^{0}-complete set C ⊆ ω. We show how to compute C from deg_{T}(L).

By Lemma 2.8.8, let Q be an r.e. separating class and let {S_{0,n}}_{n∈ω} and {S_{1,n}}_{n∈ω} be recursive sequences of r.e. separating classes such that {Q} ∪ {S_{0,n}}_{n∈ω} ∪ {S_{1,n}}_{n∈ω} is strongly independent. Then let

w_{0} = deg_{s}(W_{0}) for W_{0} = spine(Q, {S_{0,n}}_{n∈ω}),
w_{1} = deg_{s}(W_{1}) for W_{1} = spine(Q, {S_{1,n}}_{n∈ω}),
v = deg_{s}(V) for V = ∑_{n∈ω} S_{0,n},
r = deg_{s}(R) for R = spine(Q, {R_{n}}_{n∈ω}), where R_{n} = ∑_{m∈ω\{n}} S_{0,m},
m = deg_{s}(M) for M = spine(Q, {S_{0,n} + S_{1,n}}_{n∈ω}), and
p = deg_{s}(P) for P = spine(Q, {S_{0,n} + S_{1,n+1}}_{n∈ω}).

Let {Z_{e}}_{e∈ω} be a recursive sequence containing all Π_{1}^{0} classes as in Lemma 1.4.4. Let D ⊆ ω be the set

D = {e | ∃n(n ∈ C ∧ Z_{e} ≤_{s} S_{0,n} ∧ V ≤_{s} Z_{e} + R_{n})}.

D is Σ_{3}^{0} because C is Σ_{3}^{0}, the sequences {Z_{e}}_{e∈ω}, {S_{0,n}}_{n∈ω}, and {R_{n}}_{n∈ω} are recursive, and ≤_{s} is Σ_{3}^{0} by Lemma 1.4.3. Let φ(e, m, k, ℓ) be a recursive predicate such that D = {e | ∃m∀k∃ℓφ(e, m, k, ℓ)}. By Lemma 2.10.4, let {X_{⟨ e,m ⟩}}_{⟨ e,m ⟩∈ω} be a recursive sequence of Π_{1}^{0} classes such that for all e, m ∈ ω

deg_{s}(X_{⟨ e,m ⟩}) = \begin{cases}0 & if ∀k∃ℓφ(e, m, k, ℓ) \\ deg_{s}(Q) & if ∃k∀ℓ¬φ(e, m, k, ℓ). \end{cases}

Let x = deg_{s}(X) for X = spine(Q, {Z_{e} + X_{⟨ e,m ⟩}}_{⟨ e,m ⟩∈ω}).

The procedure for determining whether n ∈ C from deg_{T}(L) uses the fixed parameters f(w_{0}), f(w_{1}), f(v), f(r), f(m), f(p), f(deg_{s}(S_{0,0})), and f(x). Given n ∈ ω search L for elements a_{i,j} for i < 2 and 1 ≤ j ≤ n and for an element b satisfying the conditions

(i) a_{i,j} meets to f(w_{i}) for all i < 2 and all 1 ≤ j ≤ n, (ii) a_{0,j} +_{L} a_{1,j} ≥_{L} f(m) for all 1 ≤ j ≤ n, (iii) a_{0,j} +_{L} a_{1,j+1} ≥_{L} f(p) for all 0 ≤ j ≤ n − 1 (where a_{0,0} = f(deg_{s}(S_{0,0}))), (iv) b meets to f(r), and

(v) a_{0,n} +_{L} b ≥_{L} f(v).

When the search is completed, output “yes” if f(x) ≤_{L} a_{0,n} and output “no” otherwise.

First, observe that the above search is recursive in deg_{T}(L) because the “meets to” relation is r.e. in deg_{T}(L). Furthermore, the search will always terminate because the elements a_{i,j} = f(deg_{s}(S_{i,j})) for all i < 2 and all 1 ≤ j ≤ n and the element b = f(deg_{s}(R_{n})) satisfy conditions (i)–(v), and the search will eventually find them. Conditions (i) and (iv) follow from Lemma 2.9.2 item (ii). Notice that Q and {R_{j}}_{j∈ω} satisfy the hypothesis of Lemma 2.9.2 because {Q} ∪ {S_{0,j}}_{j∈ω} is strongly independent. Conditions (ii) and (iii) are easy to see. For condition (v), it is also easy to see that S_{0,n} + R_{n} ≡_{s} V.

We need to show that the procedure outputs “yes” on input n if and only if n ∈ C. Let a_{i,j} for i < 2 and 1 ≤ j ≤ n and b be the elements found in the search performed on input n.

Claim. For all i < 2 and all 1 ≤ j ≤ n, a_{i,j} ≤_{L} f(deg_{s}(S_{i,j})).

Proof of claim.For each i < 2 and each 1 ≤ j ≤ n, let A_{i,j} be a Π_{1}^{0} class such that deg_{s}(A_{i,j}) = f^{−1}(a_{i,j}). By condition (i) of the search and Lemma 2.9.2 item (i), A_{0,1} ≤_{s} S_{0,m} and A_{1,1} ≤_{s} S_{1,k} for some m, k ∈ ω. Condition (iii) implies that S_{0,0} + S_{1,k} ≥_{s} P, which is false by strong independence unless k = 1. So A_{1,1} ≤_{s} S_{1,1}. Knowing this, condition (ii) implies that S_{0,m} + S_{1,1} ≥_{s} M, which is false by strong independence unless m = 1. So A_{0,1} ≤_{s} S_{0,1}. Now proceed by induction. Let 1 ≤ j < n and assume that A_{0,j} ≤_{s} S_{0,j} and that A_{1,j} ≤_{s} S_{1,j}. Just as in the argument for the base case, A_{0,j+1} ≤_{s} S_{0,m} and A_{1,j+1} ≤_{s} S_{1,k} for some m, k ∈ ω. S_{0,j} + S_{1,k} ≥_{s} P by condition (iii), which implies that k = j + 1. S_{0,m} + S_{1,j+1} ≥_{s} M by condition (ii), which implies that m = j + 1.

At the end of the search, a_{0,n} ≤_{L} f(deg_{s}(S_{0,n})) by the claim, b meets to f(r) by condition (iv), and a_{0,n} +_{L} b ≥_{L} f(v) by condition (v). By Lemma 2.9.2 item (ii), b ≤_{L} f(deg_{s}(R_{m})) for some m ∈ ω. However, if m ≠ n, then S_{0,n} ≤_{s} R_{m}, in which case S_{0,n} + R_{m} ≡_{s} R_{m} ≱_{s} V. Thus it must be that b ≤_{L} f(deg_{s}(R_{n})).

Suppose n ∈ C. Since {Z_{e}}_{e∈ω} lists all the Π_{1}^{0} classes, there is an e ∈ ω such that deg_{s}(Z_{e}) = f^{−1}(a_{0,n}). This e satisfies ∃n(n ∈ C ∧ Z_{e} ≤_{s} S_{0,n} ∧ V ≤_{s} Z_{e} + R_{n}). Thus e ∈ D, which means ∃m∀k∃ℓφ(e, m, k, ℓ). If m witnesses this property for e, then deg_{s}(X_{⟨ e,m ⟩}) = 0 and Z_{e} + X_{⟨ e,m ⟩} ≡_{s} Z_{e}. Thus X ≤_{s} Z_{e}, which means f(x) ≤_{L} a_{0,n}. Thus “yes” was the output.

Suppose n ∉ C. We show X ≰_{s} S_{0,n}.

Claim. For all e, m ∈ ω, S_{0,n} ≱_{s} Z_{e} + X_{⟨ e,m ⟩}.

Proof of claim.If deg_{s}(X_{⟨ e,m ⟩}) = deg_{s}(Q), then S_{0,n} ≱_{s} Z_{e} + X_{⟨ e,m ⟩} because S_{0,n} ≱_{s} Q by strong independence. If deg_{s}(X_{⟨ e,m ⟩}) = 0, then ∀k∃ℓφ(e, m, k, ℓ). Therefore e ∈ D, so there is an n′ such that n′ ∈ C, Z_{e} ≤_{s} S_{0,n′}, and V ≤_{s} Z_{e} + R_{n′}. Notice that n ≠ n′ because n ∉ C and n′ ∈ C. Therefore S_{0,n} ≤_{s} R_{n′}. So if S_{0,n} ≥_{s} Z_{e}, then R_{n′} ≥_{s} Z_{e}. So V ≰_{s} R_{n′} ≡_{s} Z_{e} + R_{n′}, a contradiction. Hence S_{0,n} ≱_{s} Z_{e} + X_{⟨ e,m ⟩}.

Suppose for a contradiction that Φ is such that Φ(S_{0,n}) ⊆ X. If there are n, m ∈ ω such that Φ(S_{0,n}) ∩ (σ_{⟨ e,m ⟩}^{⌢}(Z_{e} + X_{⟨ e,m ⟩})) ≠ ∅, then there is a clopen C ⊆ 2^{ω} such that S_{0,n} ∩ C ≠ ∅ and Φ(S_{0,n} ∩ C) ⊆ σ_{⟨ e,m ⟩}^{⌢}(Z_{e} + X_{⟨ e,m ⟩}). S_{0,n} ≡_{s} S_{0,n} ∩ C by Lemma 2.8.3, and S_{0,n} ∩ C ≥_{s}(σ_{⟨ e,m ⟩}^{⌢}(Z_{e} + X_{⟨ e,m ⟩})) ≡_{s} Z_{e} + X_{⟨ e,m ⟩}. This contradicts the claim. Thus Φ(S_{0,n}) ∩ (σ_{⟨ e,m ⟩}^{⌢}(Z_{e} + X_{⟨ e,m ⟩})) = ∅ for all e, m ∈ ω. Therefore Φ(S_{0,n}) ⊆ Q. This contradicts Q ≰_{s} S_{0,n}. Hence X ≰_{s} S_{0,n}. It follows that f(x) ≰_{L} a_{0,n} because a_{0,n} ≤_{L} f(deg_{s}(S_{0,n})). Thus “no” was the output.

Theorem 2.10.6. The degree of E_{s} as a lattice is 0′′′. That is, there is a presentation of E_{s} as a lattice recursive in 0′′′ and 0′′′ is recursive in every presentation of E_{s} as a lattice.

Proof.Lemma 2.10.3 proves that there is a presentation recursive in 0′′′, and Lemma 2.10.5 proves that 0′′′ is recursive in every presentation.

Corollary 2.10.7. E_{s} has no recursive presentation as a partial order.

Proof.In any lattice, the relations x + y = z and x × y = z are definable from the partial order by Π_{1}^{0} formulas. Thus if E_{s} had a recursive presentation as a partial order, it would have a presentation as a lattice recursive in 0′. This contradicts the theorem.

2.11 The undecidability of Th(E_{w}) and Th(E_{w}; ≤_{w})

In this section, we code N^{}in E_{w}, thereby showing that Σ_{3}^{0}-Th(E_{w}) and Σ_{4}^{0}- Th(E_{w}; ≤_{w}) are undecidable. In place of separating classes, our coding of N^{}in E_{w} uses Simpson’s Σ_{3}^{0} embedding lemma and his embedding of E_{T} into E_{w}. A set X ⊆ ω^{ω} is Σ_{3}^{0} if and only if it is of the form X = {f ∈ ω^{ω} | ∃m∀k∃ℓφ(f, m, k, ℓ)} for some recursive predicate φ.

Lemma 2.11.1 (Σ_{3}^{0} embedding lemma [66] Lemma 3.3). Let S ⊆ ω^{ω} be Σ_{3}^{0} and let P ⊆ 2^{ω} be a Π_{1}^{0} class. Then there is a Π_{1}^{0} class Q ⊆ 2^{ω} such that Q ≡_{w} S ∪ P.

In the Muchnik case, deg_{w}(S) × deg_{w}(P) = deg_{w}(S ∪ P) for any S, P ⊆ ω^{ω}. For this reason, the Σ_{3}^{0} embedding lemma may be phrased as “if S is Σ_{3}^{0} and P is a Π_{1}^{0} class then deg_{w}(S) × deg_{w}(P) ∈ E_{w}.” For our purposes, P is always DNR_{2}, so deg_{w}(P) = deg_{w}(DNR_{2}) = 1, the greatest element of E_{w}.

If A is an r.e. set, then {A} is a Σ_{3}^{0} (in fact a Π_{2}^{0}) subset of 2^{ω}. One of Simpson’s original applications of his Σ_{3}^{0} embedding lemma is to show that the map deg_{T}(A) ↦ deg_{w}({A}) × 1 is an upper-semilattice embedding of E_{T} into E_{w} preserving the least and greatest elements [66]. To show that this map is indeed an embedding, Simpson uses the following variant of the Arslanov completeness criterion, which we also employ.

Lemma 2.11.2 (see [31] Lemma 4.1 and [69] Theorem V.5.1). If A is an r.e. set, then DNR_{2} ≤_{w}{A} if and only if A ≡_{T} 0′.

Proof.It is easy to compute a function in DNR_{2} from 0′. Conversely, if A computes a function in DNR_{2}, then A computes a function f such that ∀e(W_{f(e)} ≠ W_{e}), where here {W_{e}}_{e∈ω} is the standard enumeration of the r.e. sets (such an f is called fixed-point free; see [31] Lemma 4.1). Thus A ≡_{T} 0′ by the Arslanov completeness criterion (see [69] Theorem V.5.1).

For comparison, it is not known whether E_{T} embeds into E_{s}. See [11] for further results concerning embedding distributive lattices in E_{s} and E_{w}.

For us, the key property of the degrees deg_{w}({A}) × 1 for r.e. sets A is that they are all meet-irreducible in E_{w} (of course these degrees are generally meetreducible in D_{w}).

Lemma 2.11.3. If A is an r.e. set, then deg_{w}({A}) × 1 is meet-irreducible in E_{w}.

Proof.Suppose that x, y ∈ E_{w} are such that deg_{w}({A}) × 1 ≥_{w} x × y. Either deg_{w}({A}) ≥_{w} x or deg_{w}({A}) ≥_{w} y because deg_{w}({A}) is the degree of a singleton. As 1 ≥_{w} x and 1 ≥_{w} y, either deg_{w}({A}) × 1 ≥_{w} x or deg_{w}({A}) × 1 ≥_{w} y.

If {A_{n}}_{n∈B} is a uniformly r.e. sequence of r.e. sets indexed by a recursive set B (i.e., the set {⟨ n, m ⟩ | n ∈ B ∧ m ∈ A_{n}} is r.e.), then {A_{n}}_{n∈B} is a Σ_{3}^{0} subset of 2^{ω} and deg_{w}({A_{n}}_{n∈B}) × 1 ∈ E_{w}. In place of Lemma 2.8.8, we use the following simpler fact.

Lemma 2.11.4 (see [69] Section VII.2). There is a uniformly r.e. sequence of r.e. sets {A_{n}}_{n∈ω} that is strongly independent.

Notice that Lemma 2.11.4 is also a consequence of Lemma 2.8.8. If {S(A_{n}, B_{n})}_{n∈ω} is a recursive sequence of r.e. separating classes that is strongly independent, then {A_{n}}_{n∈ω} and {B_{n}}_{n∈ω} are both uniformly r.e. sequences of r.e. sets that are strongly independent.

Lemma 2.11.5. Let {A_{n}}_{n∈B} be an infinite uniformly r.e. sequence of r.e. sets (indexed by a recursive set B) that is a ≤_{T}-antichain. Let w = deg_{w}({A_{n}}_{n∈B}) × 1.

(i) If x ∈ E_{w} meets to w, then x ≤_{w} deg_{w}({A_{n}}) × 1 for some n ∈ B. (ii) E(w) = {deg_{w}({A_{n}}) × 1 | n ∈ B}.

Proof.(i) Let x ∈ E_{w} be such that x meets to w, and suppose that x ≰_{w} deg_{w}({A_{n}}) × 1 for all n ∈ B for a contradiction. Since x ≤_{w} 1, it must be that x ≰_{w} deg_{w}({A_{n}}) for all n ∈ B. Let y ∈ E_{w} witness that x meets to w. That is, y >_{w} w and x × y = w. Let X and Y be Π_{1}^{0} classes such that x = deg_{w}(X) and y = deg_{w}(Y). Then X ∪ Y ≤_{w}{A_{n}} for all n ∈ B. Thus Y ≤_{w}{A_{n}} for all n ∈ B because X ≰_{w}{A_{n}} for all n ∈ B. Therefore Y ≤_{w}{A_{n}}_{n∈B}, which implies that y ≤_{w} w, a contradiction.

(ii) Let n ∈ B. To see that deg_{w}({A_{n}}) × 1 ∈ E(w), let y = deg_{w}({A_{i}}_{i∈B\{n}}) × 1. It is easy to check that (deg_{w}({A_{n}}) × 1) × y = w. Moreover, deg_{w}({A_{n}}) × 1 ≱_{w} y. This is because {A_{n}} ≱_{w}{A_{i}}_{i∈B\{n}} as {A_{i}}_{i∈B} is a ≤_{T}-antichain and because {A_{n}} ≱_{w} DNR_{2} by Lemma 2.11.2 (note that A_{n} <_{T} 0′ because {A_{i}}_{i∈B} is a ≤_{T}-antichain). Thus y >_{w} w, and therefore y witnesses that deg_{w}({A_{n}}) × 1 meets to w. The degree deg_{w}({A_{n}}) × 1 is meet-irreducible in E_{w} by Lemma 2.11.3. Thus deg_{w}({A_{n}}) × 1 ∈ E(w).

We have shown that {deg_{w}({A_{n}}) × 1 | n ∈ B} ⊆ E(w). To see equality, let x ∈ E(w). By item (i), x ≤_{w} deg_{w}({A_{n}}) × 1 for some n ∈ B. E(w) is an antichain by Lemma 2.2.3 and deg_{w}({A_{n}}) × 1 ∈ E(w), so it must be that x = deg_{w}({A_{n}}) × 1.

We are now able to code N^{}in E_{w}. Lemma 2.11.6. There is a code w⃗ in E_{w} such that M ; w⃗ ≅ N^{}.

Proof.By Lemma 2.11.4, let {A_{0,n}}_{n∈ω}, {A_{1,n}}_{n∈ω}, and {A_{2,n}}_{n∈ω} be uniformly r.e. sequences of r.e. sets such that {A_{0,n}}_{n∈ω} ∪ {A_{1,n}}_{n∈ω} ∪ {A_{2,n}}_{n∈ω} is strongly independent. Let

w_{0} = deg_{w}(W_{0}) × 1  for W_{0} = {A_{0,n}}_{n∈ω},
w_{1} = deg_{w}(W_{1}) × 1  for W_{1} = {A_{1,n}}_{n∈ω},
w_{2} = deg_{w}(W_{2}) × 1  for W_{2} = {A_{2,n}}_{n∈ω},
m = deg_{w}(M) × 1  for M = {A_{0,n} ⊕ A_{1,n}}_{n∈ω} ∪ {A_{0,n} ⊕ A_{2,n}}_{n∈ω},
ℓ = deg_{w}(L) × 1  for  L = {A_{0,i} ⊕ A_{1,j} | i < j},
p = deg_{w}(P) × 1  for  P = {A_{0,i} ⊕ A_{1,j} ⊕ A_{2,k} | i + j = k},
t = deg_{w}(T ) × 1  for  T = {A_{0,i} ⊕ A_{1,j} ⊕ A_{2,k} | i × j = k},
d = deg_{w}(D) × 1  for  D = {A_{0,i} ⊕ A_{1,j} ⊕ A_{2,k} | i j = k},
z = deg_{w}({A_{0,0}}) × 1, and
o = deg_{w}({A_{0,1}}) × 1 .

To aid readability, let a_{i,j} = deg_{w}({A_{i,j}}) for all i < 3 and j ∈ ω. By Lemma 2.11.5 item (ii), E(w_{0}) = {a_{0,n} × 1}_{n∈ω}. The map a_{0,n} × 1 ↦ n is the isomorphism witnessing M ; w⃗ ≅ N^{}. Clearly z ↦ 0 and o ↦ 1. We show that the map preserves <. The proofs that the map preserves +, ×, and are similar. Let i, j ∈ ω. If i < j, then a_{1,j} × 1 meets to w_{1} by Lemma 2.11.5 item (ii), and by distributivity

(a_{0,j} × 1) +(a_{1,j} × 1) = (a_{0,j} + a_{1,j}) × 1
= deg_{w}({A_{0,j} ⊕ A_{1,j}}) × 1
≥_{w} m, and
(a_{0,i} × 1) +(a_{1,j} × 1) = (a_{0,i} + a_{1,j}) × 1
= deg_{w}({A_{0,i} ⊕ A_{1,j}}) × 1
≥_{w} ℓ .

Thus R_{ℓ}^{2}(a_{0,i} × 1, a_{0,j} × 1). Conversely, suppose that R_{ℓ}^{2}(a_{0,i} × 1, a_{0,j} × 1). Let u_{1} ∈ E_{w} be such that u_{1} meets to w_{1}, (a_{0,j} × 1) + u_{1} ≥_{w} m, and (a_{0,i} × 1) + u_{1} ≥_{w} ℓ. Since u_{1} meets to w_{1}, it must be that u_{1} ≤_{w} a_{1,k} × 1 for some k ∈ ω by Lemma 2.11.5 item (i). Thus (a_{0,j} × 1) +(a_{1,k} × 1) ≥_{w} m, so by distributivity

deg_{w}({A_{0,j} ⊕ A_{1,k}}) × 1 = (a_{0,j} + a_{1,k}) × 1 = (a_{0,j} × 1) +(a_{1,k} × 1) ≥_{w} m.

However, if k ≠ j, then {A_{0,j} ⊕ A_{1,k}} ≱_{w} M by strong independence and {A_{0,j} ⊕ A_{1,k}} ≱_{w} DNR_{2} by Lemma 2.11.2. This implies that deg_{w}({A_{0,j} ⊕ A_{1,k}}) × 1 ≱_{w} m, so it must be that k = j. Thus u_{1} ≤_{w} a_{1,j} × 1, which implies that (a_{0,i} × 1) +(a_{1,j} × 1) ≥_{w} ℓ. Then

deg_{w}({A_{0,i} ⊕ A_{1,j}}) × 1 = (a_{0,i} + a_{1,j}) × 1 = (a_{0,i} × 1) +(a_{1,j} × 1) ≥_{w} ℓ .

So if i ≮ j, then {A_{0,i} ⊕ A_{1,j}} ≱_{w} L by strong independence and {A_{0,i} ⊕ A_{1,j}} ≱_{w} DNR_{2} by Lemma 2.11.2, giving the contradiction deg_{w}({A_{0,i} ⊕ A_{1,j}}) × 1 ≱_{w} ℓ. Hence i < j.

Theorem 2.11.7. Σ_{3}^{0}-Th(E_{w}) and Σ_{4}^{0}-Th(E_{w}; ≤_{w}) are undecidable.

Proof.There is a code w⃗ in E_{w} such that M ; w⃗ ≅ N^{}by Lemma 2.11.6. The results then follow from Lemma 2.2.25.

Clearly then Th(E_{w}; ≤_{w}) is undecidable. Unfortunately we do not yet know how to prove anything like the finite matching property for E_{w} to obtain Th(N) ≤_{1} Th(E_{w}; ≤_{w}). The proof of the finite matching property for E_{s} (Lemma 2.8.7) appeals to a lemma of Cole and Kihara that grew out of Cenzer and Hinman’s proof that E_{s} is dense. By analogy, perhaps progress must be made on the density of E_{w} before further progress is made on the complexity of Th(E_{w}; ≤_{w}).

CHAPTER 3 JOIN-IRREDUCIBLES AND PROPOSITIONAL LOGICS IN THE MEDVEDEV DEGREES

The results of this chapter also appear in [56] by the author.

We present solutions to three problems concerning the Medvedev degrees. First, Dyment characterized the meet-reducible elements of D_{s} in the following theorem. Its corollary helps identify meet-irreducible Medvedev degrees.

Theorem 3.0.8 ([22]). A Medvedev degree a is meet-reducible if and only if a = deg_{s}(A) for a mass problem A for which there are r.e. sets V_{0}, V_{1} ⊆ ω^{<ω} such that

• (∀f ∈ A)(∃σ ∈ V_{0} ∪ V_{1})(σ ⊂ f),

• The following mass problems are ≤_{s}-incomparable:

{f ∈ A | (∃σ ∈ V_{0})(σ ⊂ f)} and{f ∈ A | (∃σ ∈ V_{1})(σ ⊂ f)}

Corollary 3.0.9 ([22]). If A is a mass problem such that σ^{⌢}A ⊆ A for all σ ∈ ω^{<ω}, then deg_{s}(A) is meet-irreducible.

In particular, 0′ is meet-irreducible because σ^{⌢}f >_{T} 0 whenever σ ∈ ω^{<ω} and f >_{T} 0.

The problem of characterizing the join-irreducible elements of D_{s} was posed in [74]. We prove that a ∈ D_{s} is join-irreducible if and only if a = deg_{s}(ω^{ω} \ I) for some Turing ideal I (Theorem 3.1.3).

Second, providing semantics for propositional logic was one of Medvedev’s main motivations behind introducing D_{s}, and he proved PTh(D_{s}) = JAN in Medvedev [42]. JAN denotes the logic IPC +¬p ∨ ¬¬p named after Jankov who studied it in Jankov [30]. In any Brouwer algebra B, the quotient of B by the principal filter generated by a ∈ B is denoted B /a. The quotient B /a is isomorphic to the interval [0, a] which is a Brouwer algebra under the operations inherited from B. Logics of the form PTh(D_{s} /a) have been studied in Skvortsova [68]^{1}, Sorbi [73], and Sorbi and Terwijn [75]. Our work is motivated by the following question which remains open. Question 3.0.10 ([75]). Is PTh(D_{s} /a) ⊆ JAN for all a >_{s} 0′?

Sorbi and Terwijn’s study of Question 3.0.10 in [75] lead them to ask whether every degree >_{s} 0′ bounds a join-irreducible degree >_{s} 0′ because a “yes” answer to this question implies a “yes” answer to Question 3.0.10. However, Sorbi and Terwijn conjectured that there is a degree >_{s} 0′ that bounds no join-irreducible degree >_{s} 0′, and we prove that this is correct (Theorem 3.2.5). We also provide slight extensions to some of the results in [73], thereby widening the class of degrees a for which PTh(D_{s} /a) ⊆ JAN is known.

Third, we use techniques similar to those used to characterize the joinirreducible degrees to prove that the filter generated by the non-minimum degrees of closed mass problems is not prime (Theorem 3.4.3). This problem was posed in Bianchini and Sorbi [9] and in Sorbi [74].

1Skvortsova and Dyment are the same person. Dyment got married and became Skvortsova.

3.1 Characterizing the join-irreducible Medvedev degrees

A Turing ideal is a set I ⊆ ω^{ω} that is closed downward under ≤_{T} (i.e., f ∈ I ∧ g ≤_{T} f → g ∈ I) and closed under ⊕ (i.e., f, g ∈ I → f ⊕ g ∈ I). We prove that a ∈ D_{s} is join-irreducible if and only if a = deg_{s}(ω^{ω} \ I) for some Turing ideal I.

For a mass problem A, let C(A) denote the Turing upward-closure of A: C(A) = {f | (∃g ∈ A)(f ≥_{T} g)}. A mass problem A is called Turing upwardclosed if A = C(A). The identity functional witnesses C(A) ≤_{s} A for any mass problem A, and if A and B are mass problems such that A is Turing upwardclosed, then A ≤_{s} B if and only if B ⊆ A. Our starting point is the following observation.

Lemma 3.1.1 ([74]). If A is a mass problem such that deg_{s}(A) is join-irreducible, then ω^{ω} \ C(A) is a Turing ideal.

Proof.We prove the contrapositive. If ω^{ω} \ C(A) is not a Turing ideal, then there are f, g ∉ C(A) with f ⊕ g ∈ C(A). This means that {f}, {g} ≱_{s} A but {f} +{g} ≥_{s} A. Thus deg_{s}(A) is join-reducible.

The next lemma is the main step in our characterization.

Lemma 3.1.2. If A is a mass problem such that deg_{s}(A) is join-irreducible, then A ≡_{s} C(A)

Proof.We prove the contrapositive. Suppose A ≢_{s} C(A). Then it must be that A ≰_{s} C(A). We find mass problems X and Y such that X , Y ≱_{s} A but X + Y ≥_{s} A. Thus deg_{s}(A) is join-reducible.

To find X and Y, first find a sequence (h_{n} | n ∈ ω) of functions and a sequence (e_{n} | n ∈ ω) of indices such that

(i) Φ_{e_{n}}(h_{n}) ∈ A for all n ∈ ω, (ii) Φ_{n}(h_{2n}) ∉ A and Φ_{n}(h_{2n+1}) ∉ A for all n ∈ ω, and (iii) h_{n}(0) = ⟨ n, e_{0}, e_{1}, …, e_{n−1} ⟩ for all n ∈ ω.

We find the desired sequences by iterating the following claim.

Claim. If A ≰_{s} C(A), then for every e, m ∈ ω there is an h ∈ C(A) such that h(0) = m and Φ_{e}(h) ∉ A.

Proof of claim.Suppose not. Then there are e, m ∈ ω such that h(0) = m implies Φ_{e}(h) ∈ A for all h ∈ C(A). Thus h ↦ Φ_{e}(m^{⌢}h) is a reduction witnessing A ≤_{s} C(A), a contradiction.

Suppose we have h_{i} and e_{i} for all i < n. To find h_{n} and e_{n}, let e = ⌊n/2⌋ and let m = ⟨ n, e_{0}, e_{1}, …, e_{n−1} ⟩. By the claim, there is an h_{n} ∈ C(A) such that h_{n}(0) = m and Φ_{e}(h_{n}) ∉ A. The fact that h_{n} ∈ C(A) means that there is an e_{n} such that Φ_{e_{n}}(h_{n}) ∈ A.

Now set X = {h_{2n} | n ∈ ω} and Y = {h_{2n+1} | n ∈ ω}. Then Φ_{e}(X) ⊈ A and Φ_{e}(Y) ⊈ A for each e by item (ii). Hence X , Y ≱_{s} A. The following reduction witnesses X + Y ≥_{s} A.

Given h, decompose h as h = f ⊕ g and decode f(0) and g(0) as f(0) = ⟨ 2n, x_{0}, x_{1}, …, x_{2n−1} ⟩ and g(0) = ⟨ 2m + 1, y_{0}, y_{1}, …, y_{2m} ⟩. If either f(0) or g(0) is not of the required form, then output the 0 function (as such an h cannot be in X + Y). Otherwise output Φ_{x_{2m+1}}(g) if 2n > 2m + 1 and output Φ_{y_{2n}}(f) if 2m + 1 > 2n.

Suppose this reduction is applied to some h = h_{2n} ⊕ h_{2m+1} ∈ X + Y. In this case f = h_{2n}, g = h_{2m+1}, and f(0) and g(0) are of the required form by item (iii). So if 2n > 2m + 1 we output Φ_{e_{2m+1}}(h_{2m+1}) and if 2m + 1 > 2n we output Φ_{e_{2n}}(h_{2n}). Both alternatives are in A by item (i). Thus X + Y ≥_{s} A.

Theorem 3.1.3. A Medvedev degree a is join-irreducible if and only if a = deg_{s}(ω^{ω}\I) for some Turing ideal I.

Proof.Suppose a is join-irreducible, and let A be a mass problem such that a = deg_{s}(A). Then I = ω^{ω} \ C(A) is a Turing ideal by Lemma 3.1.1, A ≡_{s} C(A) by Lemma 3.1.2, and therefore A ≡_{s} C(A) = ω^{ω} \ I. Hence a = deg_{s}(ω^{ω} \ I) for the Turing ideal I.

Conversely, suppose I is a Turing ideal and let X and Y be mass problems such that X , Y ≱_{s} ω^{ω} \ I. We show that X + Y ≱_{s} ω^{ω} \ I. Observe X , Y ⊈ ω^{ω} \ I for otherwise the identity functional would witness X , Y ≥_{s} ω^{ω} \I. Let f ∈ X ∩I and let g ∈ Y ∩ I, thereby making f ⊕ g ∈ (X + Y) ∩ I. The function f ⊕ g is in X + Y, but it does not compute any member of ω^{ω} \I. Therefore X + Y ≱_{s} ω^{ω} \I. Hence deg_{s}(ω^{ω} \ I) is join-irreducible.

Theorem 3.1.3 is also valid with D_{w} in place of D_{s}, a fact first noticed by Terwijn [76]. The proof of Lemma 3.1.1 also works for D_{w}, and it is easy to check that A ≡_{w} C(A) for any mass problem A (i.e., the D_{w} analogue of Lemma 3.1.2 is trivial). This gives the forward direction of Theorem 3.1.3 for D_{w}. The proof of the reverse direction of Theorem 3.1.3 also works for D_{w}.

3.2 Degrees that bound no join-irreducible degrees >_{s} 0′

Recall that JAN is the intermediate logic IPC +¬p ∨ ¬¬p. The results of this section and the next are motivated by Question 3.0.10: is PTh(D_{s} /a) ⊆ JAN for every a >_{s} 0′?

PTh(D_{s} /0′) = CPC because D_{s} /0′ ≅ [0, 0′] = {0, 0′}. In fact, 0′ is the only degree for which PTh(D_{s} /0′) = CPC. This is because if a >_{s} 0′, then 0′ → a = a, hence 0′ ×(0′ → a) = 0′. Thus let p = 0′ to see that the formula p ∨ ¬p is not valid in PTh(D_{s} /a).

Furthermore, if a >_{s} 0′, then we cannot have PTh(D_{s} /a) ⊋ JAN. It is an easy check that in any Brouwer algebra B with meet-irreducible 0 (such as the algebras D_{s} /a), ¬p ∨ ¬¬p ∈ PTh(B) if and only if 1 is join-irreducible. However, if a >_{s} 0′ is join-irreducible, then PTh(D_{s} /a) = JAN [73]. Thus if a >_{s} 0′ and PTh(D_{s} /a) ⊇ JAN, then ¬p ∨ ¬¬p ∈ PTh(D_{s} /a) which implies that a is join-irreducible which implies that PTh(D_{s} /a) = JAN. Thus a “no” answer to Question 3.0.10 must yield a degree a such that PTh(D_{s} /a) ⊈ JAN and JAN ⊈ PTh(D_{s} /a).

The following theorem shows that to give a “yes” answer to Question 3.0.10 it suffices to show that every a >_{s} 0′ bounds a finite meet of join-irreducible degrees >_{s} 0′.

Theorem 3.2.1 ([73]). If a is a degree such that a ≥_{s} ∏_{i=0}^{n} d_{i} for join-irreducible degrees d_{i} >_{s} 0′, i ≤ n, then PTh(D_{s} /a) ⊆ JAN.

(The above theorem is stated more generally in [73]. Each degree d_{i} for i ≤ n is allowed to be either join-irreducible or De-irreducible. See the parenthetical discussion following Theorem 3.3.1 for the definition of De-irreducible and an explanation of why we do not consider such degrees here. Theorem 3.3.1 is a restatement of [73] Theorem 2.11 which is the main tool used to prove Theorem 3.2.1.)

The degrees of the mass problems B_{f} = {g | g ≰_{T} f} play an important role in the study of Question 3.0.10. The following lemma from Sorbi [72] encapsulates the properties of the deg_{s}(B_{f})’s that we need in this section and the next.

Lemma 3.2.2 ([72]).

(i) Every deg_{s}(B_{f}) is join-irreducible. (ii) Every ∑_{i=0}^{n} deg_{s}(B_{f_{i}}) is meet-irreducible. (iii) Let V and J be finite sets and let U_{v} and I_{j} be finite sets for each v ∈ V and j ∈ J.

Let x_{u}^{v} and y_{i}^{j} be degrees of the form deg_{s}(B_{f}) for every v ∈ V, u ∈ U_{v}, j ∈ J, and i ∈ I_{j}. Let a = ∑_{v∈V} ∏_{u∈U_{v}} x_{u}^{v} and b = ∑_{j∈J} ∏_{i∈I_{j}} y_{i}^{j}. Then a ≤_{s} b if and only if

(∀v ∈ V )(∃j ∈ J)(∀i ∈ I_{j})(∃u ∈ U_{v})(x_{u}^{v} ≤_{s} y_{i}^{j}).

(iv) In the notation of item (iii),

a → b = ∑{∏_{i∈I_{j}} y_{i}^{j} | (∀v ∈ V )(∏_{i∈I_{j}} y_{i}^{j} ≰_{s} ∏_{u∈U_{v}} x_{u}^{v})}

(where the empty join is 0).

Proof.Item (i) is by Theorem 3.1.3 and item (ii) is by Corollary 3.0.9. Item (iv) is proved in [72]. Item (iii) follows from item (iv) because a ≤_{s} b if and only if b → a = 0.

In [75] it is asked if every degree a >_{s} 0′ bounds a join-irreducible degree >_{s} 0′, and it is conjectured that this is not the case based on the evidence provided by the following theorem.

Theorem 3.2.3 ([75]). There is a degree a >_{s} 0′ such that a ≱_{s} deg_{s}(B_{f}) for every f >_{T} 0.

Our characterization of the join-irreducible degrees implies that every joinirreducible degree >_{s} 0′ bounds some degree deg_{s}(B_{f}) with f >_{T} 0. Thus the conjecture is correct.

Corollary 3.2.4 (to Theorem 3.1.3). If a >_{s} 0′ is join-irreducible, then a ≥_{s} deg_{s}(B_{f}) for some f >_{T} 0.

Proof.If a is join-irreducible, then, by Theorem 3.1.3, a = deg_{s}(ω^{ω} \ I) for some Turing ideal I. If deg_{s}(ω^{ω} \ I) >_{s} 0′, then I contains some function f >_{T} 0. Thus ω^{ω} \ I ⊆ B_{f}. Hence a = deg_{s}(ω^{ω} \ I) ≥_{s} deg_{s}(B_{f}).

Theorem 3.2.5. There is a degree a >_{s} 0′ such that every degree x with 0′ <_{s} x ≤_{s} a is join-reducible.

Proof.By Theorem 3.2.3, let a >_{s} 0′ be such that a ≱_{s} deg_{s}(B_{f}) for every f >_{T} 0. This a is the desired degree because, by Corollary 3.2.4, if a ≥_{s} x for some joinirreducible x >_{s} 0′, then a ≥_{s} deg_{s}(B_{f}) for some f >_{T} 0.

The degree a satisfying Theorem 3.2.3 was constructed by diagonalization in [75]. We can give somewhat more concrete examples of degrees satisfying Theorem 3.2.3 and Theorem 3.2.5. Recall the following definitions. Functions f, g >_{T} 0 are a Turing minimal pair if, for all h, h ≤_{T} f, g implies h ≤_{T} 0. A function f has minimal Turing degree if, for all h, h <_{T} f implies h ≤_{T} 0. Minimal pairs and minimal degrees exist. In fact, there are continuum many distinct minimal Turing degrees. See Lerman [37] Section II.4 and Section V.2.

Theorem 3.2.6. If f and g are a minimal pair, then the degree a = deg_{s}(B_{f}) × deg_{s}(B_{g}) witnesses Theorem 3.2.5.

Proof.Let f and g be a minimal pair. Then deg_{s}(B_{f}), deg_{s}(B_{g}) >_{s} 0′ because f, g >_{T} 0. Thus deg_{s}(B_{f}) × deg_{s}(B_{g}) >_{s} 0′ because 0′ is meet-irreducible by Corollary 3.0.9. To show that deg_{s}(B_{f}) × deg_{s}(B_{g}) bounds no join-irreducible degree >_{s} 0′, it suffices by Corollary 3.2.4 to show that deg_{s}(B_{f}) × deg_{s}(B_{g}) bounds no deg_{s}(B_{h}) for h >_{T} 0. This is true because f, g is a minimal pair, so for any h >_{T} 0, either h ≰_{T} f or h ≰_{T} g. Thus either h ∈ B_{f} or h ∈ B_{g} which means B_{f} × B_{g} contains a function ≡_{T} h. B_{h} contains no function ≤_{T} h, therefore B_{f} × B_{g} ≱_{s} B_{h}.

We can extend the idea behind Theorem 3.2.6 to find a degree a >_{s} 0′ that does not bound any finite meet of join-irreducible degrees >_{s} 0′. Several of our examples in this section and the next are of the form deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) for mass problems D_{i}, i ∈ ω.

Lemma 3.2.7. Let d = deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) where deg_{s}(D_{i}) >_{s} 0′ for each i ∈ ω. Then d >_{s} 0′.

Proof.Suppose for a contradiction that Φ is a reduction witnessing d ≤_{s} 0′ (i.e., Φ(f) ∈ ⋃_{i∈ω} i^{⌢}D_{i} for all f >_{T} 0). Let σ ∈ ω^{<ω} be such that Φ(σ)(0)↓ and let i = Φ(σ)(0). Then f ↦ Φ(σ^{⌢}f) is a reduction witnessing 0′ ≥_{s} deg_{s}(D_{i}), contradicting deg_{s}(D_{i}) >_{s} 0′.

Theorem 3.2.8. There is a degree a >_{s} 0′ such that no degree x with 0′ <_{s} x ≤_{s} a is of the form ∏_{i=0}^{n} d_{i} for join-irreducible degrees d_{i} >_{s} 0′, i ≤ n.

Proof.By Corollary 3.2.4, it suffices to find a degree a >_{s} 0′ which is not above any degree of the form ∏_{i=0}^{n} deg_{s}(B_{f_{i}}) where f_{i} >_{T} 0 for each i ≤ n. Let {g_{i} | i ∈ ω} be a countable collection of functions all of distinct minimal Turing degree. Let A = ⋃_{i∈ω} i^{⌢}B_{g_{i}} and put a = deg_{s}(A). Lemma 3.2.7 proves that a >_{s} 0′.

Now consider any degree ∏_{i=0}^{n} deg_{s}(B_{f_{i}}), where f_{i} >_{T} 0 for each i ≤ n. There is a j ∈ ω such that, for each i ≤ n, g_{j} ≱_{T} f_{i}. Thus for this j, deg_{s}(B_{g_{j}}) ≱_{s} deg_{s}(B_{f_{i}}) for each i ≤ n. Therefore deg_{s}(B_{g_{j}}) ≱_{s} ∏_{i=0}^{n} deg_{s}(B_{f_{i}}) because deg_{s}(B_{g_{j}}) is meetirreducible. Clearly deg_{s}(B_{g_{j}}) ≥_{s} a, so a ≱_{s} ∏_{i=0}^{n} deg_{s}(B_{f_{i}}) as well.

For mass problems A_{i}, i ∈ ω, the Medvedev degree deg_{s}(⋃_{i∈ω} i^{⌢}A_{i}) is not in general the greatest lower bound of the degrees deg_{s}(A_{i}), i ∈ ω. Such greatest lower bounds need not even exist. For example, the degrees deg_{s}(B_{g_{i}}), i ∈ ω from Theorem 3.2.8 do not have a greatest lower bound. This follows from results in Dyment [23] (specifically, see Lemma 2.7.1), which studies when countable collections of degrees have least upper bounds and greatest lower bounds.

If a is a degree such that a ≱_{s} d for all join-irreducible d >_{s} 0′, then a → d = d for all join-irreducible d >_{s} 0′. The degree a constructed in Theorem 3.2.8 enjoys a similar property. Theorem 3.2.9. There is a degree a >_{s} 0′ such that a → ∏_{i=0}^{n} d_{i} = ∏_{i=0}^{n} d_{i} whenever d_{i} >_{s} 0′ and is join-irreducible for each i ≤ n.

Proof.As in Theorem 3.2.8, let {g_{i} | i ∈ ω} be a countable collection of functions all of distinct minimal Turing degree, let A = ⋃_{i∈ω} i^{⌢}B_{g_{i}}, and put a = deg_{s}(A). Suppose d_{i} >_{s} 0′ and is join-irreducible for each i ≤ n. By Theorem 3.1.3, for each i ≤ n let I_{i} ⊆ ω^{ω} be a Turing ideal such that d_{i} = deg_{s}(ω^{ω} \ I_{i}). Thus ∏_{i=0}^{n} d_{i} = deg_{s}(⋃_{i=0}^{n} i^{⌢}(ω^{ω} \ I_{i})) and

a → ∏_{i=0}^{n} d_{i} = deg_{s}({e^{⌢}g | (∀f ∈ A)(Φ_{e}(f ⊕ g) ∈ ⋃_{i=0}^{n} i^{⌢}(ω^{ω} \ I_{i}))}).

We now describe a reduction witnessing a → ∏_{i=0}^{n} d_{i} ≥_{s} ∏_{i=0}^{n} d_{i}.

Given e^{⌢}g, for each i ≤ n + 1 search for a string i^{⌢}σ_{i} such that Φ_{e}((i^{⌢}σ_{i}) ⊕ g)(0)↓. If there is a k ≤ n such that

Φ_{e}((i^{⌢}σ_{i}) ⊕ g)(0) = Φ_{e}((j^{⌢}σ_{j}) ⊕ g)(0) = k

for two distinct i, j ≤ n + 1, choose the least such k and output k^{⌢}g. Otherwise output 0.

Suppose we apply this reduction to e^{⌢}g ∈ A → ⋃_{i=0}^{n} i^{⌢}(ω^{ω} \ I_{i}). Φ_{e}(f ⊕ g) must be total for each f ∈ A, and for each i ∈ ω there is an f ∈ A with f(0) = i. Thus for each i ≤ n + 1 the search finds a string i^{⌢}σ_{i} such that Φ_{e}((i^{⌢}σ_{i}) ⊕ g)(0)↓. Moreover, each i^{⌢}σ_{i} can be extended to a function in A, so Φ_{e}((i^{⌢}σ_{i}) ⊕ g)(0) ≤ n for each i ≤ n + 1. Therefore there is a least k ≤ n for which there are distinct i, j ≤ n+1 with Φ_{e}((i^{⌢}σ_{i})⊕g)(0) = Φ_{e}((j^{⌢}σ_{j})⊕g)(0) = k. The reduction outputs k^{⌢}g, so we must show that k^{⌢}g ∈ ⋃_{i=0}^{n} i^{⌢}(ω^{ω} \ I_{i}) which means we must show that g ∉ I_{k}. Suppose for a contradiction that g ∈ I_{k}. The functions g_{i} and g_{j} have distinct minimal degree, so either g ≰_{T} g_{i} or g ≰_{T} g_{j} (g >_{T} 0 because a ≱_{s} ∏_{i=0}^{n} d_{i} by Theorem 3.2.8). For the sake of argument, suppose g ≰_{T} g_{i}. Then σ_{i}^{⌢}g ≰_{T} g_{i} as well, so σ_{i}^{⌢}g ∈ B_{g_{i}} and i^{⌢}σ_{i}^{⌢}g ∈ A. However, Φ_{e}((i^{⌢}σ_{i}^{⌢}g) ⊕ g) ∈ k^{⌢}(ω^{ω} \ I_{k}) by the choice of i^{⌢}σ_{i}. This cannot be because (i^{⌢}σ_{i}^{⌢}g) ⊕ g ∈ I_{k}, thus anything it computes is also in I_{k}.

By Corollary 3.3.6, the degree a = deg_{s}(⋃_{i∈ω} i^{⌢}B_{g_{i}}) used to witness Theorem 3.2.8 and Theorem 3.2.9 satisfies PTh(D_{s} /a) ⊆ JAN and so does any degree that bounds it. There are, however, degrees >_{s} 0′ that do not bound any degree of the form deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) where deg_{s}(D_{i}) >_{s} 0′ and is join-irreducible for each i ∈ ω.

Theorem 3.2.10. There is a degree a >_{s} 0′ such that a ≱_{s} deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) whenever deg_{s}(D_{i}) >_{s} 0′ and is join-irreducible for each i ∈ ω.

Proof.Let D_{i} be such that deg_{s}(D_{i}) >_{s} 0′ and is join-irreducible for each i ∈ ω. By Corollary 3.2.4, for every i ∈ ω there is an f_{i} >_{T} 0 such that D_{i} ≥_{s} B_{f_{i}}. The mass problem B_{f_{i}} is Turing upward-closed for each i ∈ ω, so D_{i} ⊆ B_{f_{i}} for each i ∈ ω. Thus ⋃_{i∈ω} i^{⌢}D_{i} ⊆ ⋃_{i∈ω} i^{⌢}B_{f_{i}}. Hence it suffices to find a degree a >_{s} 0′ that does not bound any degree of the form deg_{s}(⋃_{i∈ω} i^{⌢}B_{f_{i}}), where f_{i} >_{T} 0 for each i ∈ ω.

We use the same construction used in [75] to prove Theorem 3.2.3. Build mass problems A_{s} ⊆ {g | g >_{T} 0} such that {g | g >_{T} 0} \ A_{s} is finite for each s ∈ ω. Set A_{0} = {g | g >_{T} 0}. At stage s + 1, choose h_{s} >_{T} 0 such that h_{s} does not compute any of the (finitely many) functions in {g | g >_{T} 0} \ A_{s}. If Φ_{s}(h_{s}) is total and >_{T} 0, let g_{s} = Φ_{s}(h_{s}) and set A_{s+1} = A_{s} \ {g_{s}}. Otherwise set A_{s+1} = A_{s}. Put A = ⋂_{s∈ω} A_{s} and put a = deg_{s}(A).

To see a >_{s} 0′, observe that by construction Φ_{s}(h_{s}) ∉ A for each s ∈ ω. Now let f_{i} >_{T} 0 for each i ∈ ω. We need to show that Φ_{e}(A) ⊈ ⋃_{i∈ω} i^{⌢}B_{f_{i}} for every index e. To do this, we first show that the functions in {g | g >_{T} 0} \ A have distinct Turing degree. Suppose that g_{i} leaves A at stage i + 1 and g_{j} leaves A at stage j +1 for i+1 < j +1 (i.e., at stage i+1 we had Φ_{i}(h_{i}) = g_{i} >_{T} 0, and at stage j + 1 we had Φ_{j}(h_{j}) = g_{j} >_{T} 0). Then g_{i} ≰_{T} g_{j} because otherwise g_{i} ≤_{T} g_{j} ≤_{T} h_{j}, contradicting that h_{j} was chosen ≱_{T} g_{i} at stage j + 1. Now suppose Φ_{e}(g) is total for all g ∈ A. Fix any σ ∈ ω^{<ω} such that Φ_{e}(σ)(0)↓, and let n be such that Φ_{e}(σ)(0) = n. A is missing at most one function ≡_{T} f_{n}, so let g ∈ A be such that σ ⊂ g and g ≡_{T} f_{n}. Then Φ_{e}(g)(0) = n, but Φ_{e}(g) ∉ n^{⌢}B_{f_{n}}. Hence Φ_{e}(A) ⊈ ⋃_{i∈ω} i^{⌢}B_{f_{i}}.

Question 3.2.11. Let a be the degree constructed in Theorem 3.2.10. Does a → deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) = deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) whenever deg_{s}(D_{i}) >_{s} 0′ and is joinirreducible for each i ∈ ω? Is PTh(D_{s} /a) ⊆ JAN?

Finally, we note that the answer to Question 3.0.10 is “no” for D_{w} in place of D_{s}. Let f >_{T} 0 have minimal Turing degree, and let a = deg_{w}(B_{f}). Then, in D_{w}, [0, a] = {0, 0′, a} and JAN ⊊ PTh(D_{w} /a) ⊊ CPC.

3.3 New degrees whose corresponding logic is contained in JAN

We extend Theorem 3.2.1 by proving that PTh(D_{s} /a) ⊆ JAN for degrees a such that a ≥_{s} deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) for some collection of join-irreducible degrees deg_{s}(D_{i}) >_{s} 0′, i ∈ ω.

A propositional formula is called positive if the connective ‘¬’ does not appear in it. For a logic L let L^{+} denote the positive formulas in L, and for a Brouwer algebra B let PTh^{+}(B) denote the set of positive formulas valid in B. JAN is the maximum intermediate logic L for which L^{+} = IPC^{+} [30]. This means that L^{+} = IPC^{+} implies L ⊆ JAN for any intermediate logic L. Thus PTh^{+}(B) = IPC^{+} implies PTh(B) ⊆ JAN for any Brouwer algebra B.

Let B_{1} and B_{2} be Brouwer algebras. An injection f: B_{1} → B_{2} is called a Bembedding if it preserves 0, 1, +, ×, and → (and therefore also ¬). An injection f: B_{1} → B_{2} is called a B^{+}-embedding if it preserves 0, +, ×, and → (but not necessarily 1 or ¬). If B_{1} B-embeds into B_{2}, then PTh(B_{2}) ⊆ PTh(B_{1}), and if B_{1} B^{+}-embeds into B_{2}, then PTh^{+}(B_{2}) ⊆ PTh^{+}(B_{1}). Both of these facts are easily checked in light of [49] Theorem VI.2.4. If a ≤ b are in a Brouwer algebra B, then B /a B^{+}-embeds into B /b by the identity. This implies that PTh^{+}(B /b) ⊆ PTh^{+}(B /a), and it follows that the a for which PTh(B /a) ⊆ JAN is upward-closed in any Brouwer algebra B.

Our goal is to B^{+}-embed a certain class of Brouwer algebras into D_{s} /a. For any set X, let Fr(X) denote the free distributive lattice generated by X and let 0 ⊕ Fr(X) denote Fr(X) with a new bottom element 0. The elements of Fr(X) are all of the form ∑_{v∈V} ∏_{u∈U_{v}} x_{u}^{v} where V and the U_{v} are finite sets of indices and the x_{u}^{v} are all in X (see for example Balbes and Dwinger [7] Section V.3). For such representations, ∑_{v∈V} ∏_{u∈U_{v}} x_{u}^{v} ≤ ∑_{j∈J} ∏_{i∈I_{j}} y_{i}^{j} if and only if

(∀v ∈ V )(∃j ∈ J)(∀i ∈ I_{j})(∃u ∈ U_{v})(x_{u}^{v} = y_{i}^{j}).

If a, b ∈ Fr(X) are such that a ≱ b, then a → b exists. To see this, let a = ∑_{v∈V} ∏_{u∈U_{v}} x_{u}^{v} and b = ∑_{j∈J} ∏_{i∈I_{j}} y_{i}^{j} be representations for a and b. Then check

a → b = ∑{∏_{i∈I_{j}} y_{i}^{j} | (∀v ∈ V )(∏_{i∈I_{j}} y_{i}^{j} ≰ ∏_{u∈U_{v}} x_{u}^{v})}.

If a ≥ b are in Fr(X) for an infinite X, then a → b fails to exist because in this case Fr(X) has no least element. We see then that a → b exists for every a, b ∈ 0 ⊕ Fr(X).

If X is finite, then so are Fr(X) and 0 ⊕ Fr(X). Hence both are Brouwer algebras. Let Fr(n) denote the free distributive lattice with n generators. The logic LM = ⋂_{n∈ω} PTh(0 ⊕ Fr(n)) is called the Medvedev logic of finite problems. (LM is usually defined in terms of Brouwer algebras isomorphic to the 0⊕Fr(n). See [75] for details.) We take advantage of the fact that LM^{+} = IPC^{+} [42].

If X is infinite, then 0 ⊕ Fr(X) fails to be a Brouwer algebra only because it lacks a top element. Therefore the notion of a B^{+}-embedding makes sense when we allow B_{1} or B_{2} to be 0 ⊕ Fr(X). If we let 0 ⊕ Fr(X) ⊕ 1 denote Fr(X) with a new bottom element 0 and a new top element 1, then 0 ⊕ Fr(X) ⊕ 1 is always a Brouwer algebra.

For any partial order (P, ≤_{P}), let Fr(P, ≤_{P}) denote the free distributive lattice generated by (P, ≤_{P}). Fr(P, ≤_{P}) is the quotient Fr(P)/≡ where, for a = ∑_{v∈V} ∏_{u∈U_{v}} x_{u}^{v} and b = ∑_{j∈J} ∏_{i∈I_{j}} y_{i}^{j} in Fr(P), a ≡ b if and only if (a ⪯ b) ∧(b ⪯ a), and a ⪯ b if and only if

(∀v ∈ V )(∃j ∈ J)(∀i ∈ I_{j})(∃u ∈ U_{v})(x_{u}^{v} ≤_{P} y_{i}^{j}).

Fr(P, ≤_{P}) is always a distributive lattice, and 0 ⊕ Fr(P, ≤_{P}) ⊕ 1 is always a Brouwer algebra; also see [72].

The following lemmas facilitate our embeddings. Lemma 3.3.3 is a slight generalization of the claim in the proof of [72] Lemma 2.3 and of [68] Lemma 6. The embedding is done in Theorem 3.3.4 which is nearly identical to [73] Theorem 2.11. Part of the reason for reproducing the proof here is to (hopefully) correct the notational inconsistencies in the proof in [73]. We restate [73] Theorem 2.11 for reference. Theorem 3.3.1 ([73] Theorem 2.11). Let d = ∏_{i=0}^{n} d_{i} where d_{i} >_{s} 0′ and d_{i} is joinirreducible for each i ≤ n. Then 0 ⊕ Fr(P, ≤_{P}) ⊕ 1 B-embeds into D_{s} /d for every countable partial order (P, ≤_{P}).

(The above theorem is stated more generally in [73]. Each degree d_{i} for i ≤ n is allowed to be either join-irreducible or De-irreducible. A degree a is dense if it is of the form deg_{s}(A) where A is dense in ω^{ω}, and a degree d is De-irreducible if a → d = d for all dense degrees a. We do not consider De-irreducible degrees in our version of [73] Theorem 2.11, which is Theorem 3.3.4, because in Theorem 3.3.4 we require that the mass problems D_{i} (which play the role of the degrees d_{i} in [73] Theorem 2.11) are Turing upward-closed. Mass problems that are Turing upward-closed are dense and hence their degrees are not Deirreducible.)

Lemma 3.3.2 ([22]). If X ≱_{s} Y are mass problems, then there is a W ⊆ X with |W| ≤ ω such that W ≱_{s} Y.

Proof.X ≱_{s} Y means that there is no Turing functional Φ such that Φ(X) ⊆ Y. Thus for each functional Φ_{e} there must be some function f_{e} ∈ X such that Φ_{e}(f_{e}) ∉ Y. Let W consist of a choice of one such f_{e} ∈ X for each functional Φ_{e}.

Lemma 3.3.3. Let U, V, and F_{i} for i ∈ ω be mass problems such that ⋃_{i∈ω} i^{⌢}F_{i} ≤_{s} U + V and σ^{⌢}U ⊆ U for all σ ∈ ω^{<ω}. Then there are mass problems V_{i} for i ∈ ω such that ⋃_{i∈ω} i^{⌢}V_{i} ≡_{s} V and F_{i} ≤_{s} U + V_{i} for each i ∈ ω.

Proof.Let U, V, and F_{i} for i ∈ ω be as in the statement of the lemma. Let Φ be such that Φ(U + V) ⊆ ⋃_{i∈ω} i^{⌢}F_{i}. For each i ∈ ω, define V_{i} = {g ∈ V | (∃σ ∈ ω^{<ω})(Φ(σ ⊕ g)(0) = i)}. V ≤_{s} ⋃_{i∈ω} i^{⌢}V_{i} is clear. ⋃_{i∈ω} i^{⌢}V_{i} ≤_{s} V by the reduction which, given g, searches for a σ ∈ ω^{<ω} such that Φ(σ ⊕ g)(0)↓ and outputs Φ(σ⊕g)(0)^{⌢}g. To see i^{⌢}F_{i} ≤_{s} U + V_{i}, consider the reduction which, given f ⊕ g, searches for a σ ∈ ω^{<ω} such that Φ(σ ⊕ g)(0) = i and outputs Φ((σ^{⌢}f) ⊕ g). If f ⊕ g ∈ U + V_{i}, then such a σ is found, σ^{⌢}f is in U, and Φ((σ^{⌢}f) ⊕ g) is in i^{⌢}F_{i}.

Theorem 3.3.4. Let d = deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) where deg_{s}(D_{i}) >_{s} 0′, deg_{s}(D_{i}) is joinirreducible, and D_{i} is Turing upward-closed for each i ∈ ω. Then 0⊕Fr(2^{ω}) B^{+}-embeds into D_{s} /d.

Proof.Let D_{i} for i ∈ ω be as in the statement of the theorem, let D = ⋃_{i∈ω} i^{⌢}D_{i}, and let d = deg_{s}(D). Lemma 3.2.7 proves that d >_{s} 0′. By Lemma 3.3.2, let A ⊆ {f | f >_{T} 0} be a countable mass problem such that A ≱_{s} D. Let {f_{x} | x ∈ 2^{ω}} be a collection of functions such that f_{x} |_{T} f_{y} for all x, y ∈ 2^{ω} with x ≠ y and such that f ≰_{T} f_{x} for all f ∈ A and x ∈ 2^{ω}. Such a sequence can be constructed via standard recursion-theoretic techniques: build a perfect tree whose paths are Turing incomparable and do not compute any functions in A. See for example [37] Section II.4. Notice that B_{f_{x}} ≤_{s} A (because A ⊆ B_{f_{x}}) for each x ∈ 2^{ω}.

Define G: 0 ⊕ Fr(2^{ω}) → D_{s} as follows. Let G(0) = 0 and let G(x) = deg_{s}(B_{f_{x}}) on the generators x ∈ 2^{ω} of Fr(2^{ω}). Then extend G to all of 0 ⊕ Fr(2^{ω}) so that G(∑_{v∈V} ∏_{u∈U_{v}} x_{u}^{v}) = ∑_{v∈V} ∏_{u∈U_{v}} G(x_{u}^{v}). G preserves 0, +, and × by definition, and G is injective and preserves → by Lemma 3.2.2 items (iii) and (iv). Hence G is a B^{+}-embedding (this is essentially [72] Corollary 2.5). Now define H: 0 ⊕ Fr(2^{ω}) → D_{s} /d by H(a) = G(a) × d for all a ∈ 0 ⊕ Fr(2^{ω}). This H is the desired B^{+}-embedding. By definition, H preserves 0, +, and ×. We must show that H is injective and that H preserves →.

Clearly H(a) = 0 if and only if a = 0, so to show that H is injective we let a, b ∈ Fr(2^{ω}) be such that H(a) ≤_{s} H(b) and show that a ≤ b. Let a = ∑_{v∈V} ∏_{u∈U_{v}} x_{u}^{v} be a representation for a and let b = ∑_{j∈J} ∏_{i∈I_{j}} y_{i}^{j} be a representation for b. H(a) ≤_{s} H(b) means that

∑_{v∈V} ∏_{u∈U_{v}} G(x_{u}^{v}) × d ≤_{s} ∑_{j∈J} ∏_{i∈I_{j}} G(y_{i}^{j}) × d.

Therefore

∑_{v∈V} ∏_{u∈U_{v}} G(x_{u}^{v}) × d ≤_{s} ∑_{j∈J} ∏_{i∈I_{j}} G(y_{i}^{j}) = ∏{∑_{j∈J} G(y_{α(j)}^{j}) | α ∈ ∏_{j∈J} I_{j}}

where the equality is by distributivity (∏_{j∈J} I_{j} denotes the Cartesian product of the I_{j}’s). Hence

∑_{v∈V} ∏_{u∈U_{v}} G(x_{u}^{v}) × d ≤_{s} ∑_{j∈J} G(y_{α(j)}^{j}) for each α ∈ ∏_{j∈J} I_{j}.

Each ∑_{j∈J} G(y_{α(j)}^{j}) is meet-irreducible by Lemma 3.2.2 item (ii). Also, d ≰_{s} ∑_{j∈J} G(y_{α(j)}^{j}) for each α ∈ ∏_{j∈J} I_{j} because ∑_{j∈J} G(y_{α(j)}^{j}) ≤_{s} deg_{s}(A) but d ≰_{s} deg_{s}(A). Thus

∑_{v∈V} ∏_{u∈U_{v}} G(x_{u}^{v}) ≤_{s} ∑_{j∈J} G(y_{α(j)}^{j}) for each α ∈ ∏_{j∈J} I_{j},

and this implies that

∑_{v∈V} ∏_{u∈U_{v}} G(x_{u}^{v}) ≤_{s} ∏{∑_{j∈J} G(y_{α(j)}^{j}) | α ∈ ∏_{j∈J} I_{j}}.

The left-hand side of the above inequality is G(a) and the right-hand side is G(b). G is a B^{+}-embedding, so we conclude a ≤ b.

If either of a, b ∈ 0 ⊕ Fr(2^{ω}) is 0, then clearly H(a → b) = H(a) → H(b). So as before, let a = ∑_{v∈V} ∏_{u∈U_{v}} x_{u}^{v} and let b = ∑_{j∈J} ∏_{i∈I_{j}} y_{i}^{j} be in Fr(2^{ω}). We see H(a → b) ≥_{s} H(a) → H(b) because

H(a → b) + H(a) = H((a → b) + a) ≥_{s} H(b).

To show that H(a → b) ≤_{s} H(a) → H(b), we show that if z ∈ D_{s} is such that H(b) ≤_{s} H(a) + z, then H(a → b) ≤_{s} z. Suppose H(b) ≤_{s} H(a) + z. That is,

∑_{j∈J} ∏_{i∈I_{j}} G(y_{i}^{j}) × d ≤_{s}(∑_{v∈V} ∏_{u∈U_{v}} G(x_{u}^{v}) × d) + z.  (3.1)

Since a → b = ∑{∏_{i∈I_{j}} y_{i}^{j} | (∀v ∈ V )(∏_{i∈I_{j}} y_{i}^{j} ≰ ∏_{u∈U_{v}} x_{u}^{v})}, we have

H(a → b) = G(a → b) × d

= ∑{∏_{i∈I_{j}} G(y_{i}^{j}) | (∀v ∈ V )(∏_{i∈I_{j}} G(y_{i}^{j}) ≰_{s} ∏_{u∈U_{v}} G(x_{u}^{v}))} × d.

It suffices to show that, given j ∈ J, if ∏_{i∈I_{j}} G(y_{i}^{j}) satisfies

(∀v ∈ V )(∏_{i∈I_{j}} G(y_{i}^{j}) ≰_{s} ∏_{u∈U_{v}} G(x_{u}^{v})),

then ∏_{i∈I_{j}} G(y_{i}^{j}) × d ≤_{s} z. Suppose ∏_{i∈I_{j}} G(y_{i}^{j}) is such a meet. Then we know

(∀v ∈ V )(∃u ∈ U_{v})(∏_{i∈I_{j}} G(y_{i}^{j}) ≰_{s} G(x_{u}^{v})).

By choosing such a u ∈ U_{v} for every v ∈ V and by appealing to Lemma 3.2.2 items (i) and (ii), we see that there is an α ∈ ∏_{v∈V} U_{v} such that

∏_{i∈I_{j}} G(y_{i}^{j}) ≰_{s} ∑_{v∈V} G(x_{α(v)}^{v}).  (3.2)

Distributing ∑_{v∈V} ∏_{u∈U_{v}} G(x_{u}^{v}) in the right-hand side of (3.1) yields

∏_{i∈I_{j}} G(y_{i}^{j}) × d ≤_{s} ∑_{v∈V} G(x_{α(v)}^{v}) + z.

The degree ∑_{v∈V} G(x_{α(v)}^{v}) is a finite join of degrees of the form deg_{s}(B_{f}) and thus has a representative U such that σ^{⌢}U ⊆ U for all σ ∈ ω^{<ω}. So by Lemma 3.3.3 there are mass problems Z_{i} for i ∈ I_{j} and Ẑ_{i} for i ∈ ω such that

z = (∏_{i∈I_{j}} deg_{s}(Z_{i})) × deg_{s}(⋃_{i∈ω} i^{⌢}Ẑ_{i}),

G(y_{i}^{j}) ≤_{s} ∑_{v∈V} G(x_{α(v)}^{v}) + deg_{s}(Z_{i}) for each i ∈ I_{j}, and
deg_{s}(D_{i}) ≤_{s} ∑_{v∈V} G(x_{α(v)}^{v}) + deg_{s}(Ẑ_{i}) for each i ∈ ω.

Each G(y_{i}^{j}) is join-irreducible, and G(y_{i}^{j}) ≰_{s} ∑_{v∈V} G(x_{α(v)}^{v}) by (3.2). Thus G(y_{i}^{j}) ≤_{s} deg_{s}(Z_{i}) for each i ∈ ω, so ∏_{i∈I_{j}} G(y_{i}^{j}) ≤_{s} ∏_{i∈I_{j}} deg_{s}(Z_{i}). Each deg_{s}(D_{i}) is join-irreducible by assumption, and deg_{s}(D_{i}) ≰_{s} ∑_{v∈V} G(x_{α(v)}^{v}) because the right-hand side is ≤_{s} deg_{s}(A) but the left-hand side is not. It follows that deg_{s}(D_{i}) ≤_{s} deg_{s}(Ẑ_{i}) for each i ∈ ω, and so Ẑ_{i} ⊆ D_{i} for each i ∈ ω because each D_{i} is Turing upward-closed. Thus ⋃_{i∈ω} i^{⌢}Ẑ_{i} ⊆ D, so d ≤_{s} deg_{s}(⋃_{i∈ω} i^{⌢}Ẑ_{i}). Therefore ∏_{i∈I_{j}} G(y_{i}^{j}) × d ≤_{s}(∏_{i∈I_{j}} deg_{s}(Z_{i})) × deg_{s}(⋃_{i∈ω} i^{⌢}Ẑ_{i}) = z as desired.

Corollary 3.3.5. If a ≥_{s} d are degrees such that d = deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) where deg_{s}(D_{i}) >_{s} 0′ and is join-irreducible for each i ∈ ω, then 0 ⊕ Fr(2^{ω}) B^{+}-embeds into D_{s} /a.

Proof.Let a, d, and D_{i} for i ∈ ω be as in the statement of the corollary. Let d_{0} = deg_{s}(⋃_{i∈ω} i^{⌢}C(D_{i})) and notice that d ≥_{s} d_{0}. D_{i} ≡_{s} C(D_{i}) for each i ∈ ω by Lemma 3.1.2, so d_{0} satisfies the hypotheses of Theorem 3.3.4. Thus 0 ⊕ Fr(2^{ω}) B^{+}-embeds into D_{s} /d_{0} which B^{+}-embeds into D_{s} /a.

Corollary 3.3.6. If a ≥_{s} d are degrees such that d = deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) where deg_{s}(D_{i}) >_{s} 0′ and is join-irreducible for each i ∈ ω, then PTh(D_{s} /a) ⊆ JAN.

Proof.The Brouwer algebra 0 ⊕ Fr(n) B^{+}-embeds into 0 ⊕ Fr(2^{ω}) for each n, and 0 ⊕ Fr(2^{ω}) B^{+}-embeds into D_{s} /a by Corollary 3.3.5. Thus PTh^{+}(D_{s} /a) ⊆ ⋂_{n∈ω} PTh^{+}(0 ⊕ Fr(n)) = LM^{+} = IPC^{+}. So PTh(D_{s} /a) ⊆ JAN.

Theorem 3.3.4 can be modified to B-embed 0⊕Fr(2^{ω})⊕1 into D_{s} /d for degrees d as in the statement of Theorem 3.3.4. However, if a ≤ b in a Brouwer algebra B, it is not in general the case that B /a B-embeds into B /b. So the proof of Corollary 3.3.5 fails for B-embedding 0 ⊕ Fr(2^{ω}) ⊕ 1. Theorem 3.3.4 can also be modified to prove a more precise analogue of [73] Theorem 2.11 (restated as Theorem 3.3.1). Let d = deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) where deg_{s}(D_{i}) >_{s} 0′, deg_{s}(D_{i}) is joinirreducible, and D_{i} is Turing upward-closed for each i ∈ ω. Then 0 ⊕ Fr(P, ≤_{P} ) ⊕ 1 B-embeds into D_{s} /d for every countable partial order (P, ≤_{P}).

3.4 𝔉_{cl} is not prime

Recall that a filter 𝔉 in a lattice is called prime if a + b ∈ 𝔉 → a ∈ 𝔉 ∨ b ∈ 𝔉 for all a and b in the lattice. Theorem 3.1.3 can be phrased as a characterization of the prime principal filters in D_{s}: a degree a generates a prime filter if and only if a = deg_{s}(ω^{ω} \ I) for some Turing ideal I. There is little general theory of the non-principal filters in D_{s}, but several specific cases have been studied in Dyment [22], Sorbi [70], Bianchini and Sorbi [9], and Lewis, Shore, and Sorbi [39]. See also [74] for a summary of many of the results appearing in these works. We consider the filters 𝔉 and 𝔉_{cl}.

Definition 3.4.1.

• A degree a is called dense (closed) if a = deg_{s}(A) for an A that is dense (closed) in ω^{ω}.

• ℑ denotes the ideal generated by {a | a is dense }.

• 𝔉 denotes D_{s} \ ℑ.

• 𝔉_{cl} denotes the filter generated by {a | a >_{s} 0 and is closed }.

The join and meet of two dense degrees is dense [22], and the join and meet of two closed degrees is closed [9]. It follows that ℑ = {b | (∃a ≥_{s} b)(a is dense )} and 𝔉_{cl} = {b | (∃a ≤_{s} b)(a >_{s} 0 and is closed )}. The basic properties of ℑ, 𝔉, and 𝔉_{cl} are as follows. ℑ is a prime ideal [70], 𝔉 is a prime filter [9], ℑ is not principal [22], 𝔉 and 𝔉_{cl} are not principal [9], and 𝔉_{cl} ⊊ 𝔉 [9]. Both [9] and [74] ask for a proof that 𝔉_{cl} is not prime. We provide a proof of this fact now.

Lemma 3.4.2. For any f ∈ ω^{ω} there are A, B ⊆ ω^{ω} such that A + B ≥_{s}{f} and, for any closed C ⊆ ω^{ω}, if A ≥_{s} C or B ≥_{s} C, then C contains a recursive function.

Proof.Fix a recursive bijection ω ↔ ω^{<ω}. For e, n ∈ ω, if

∀m∀σ(∃τ ⊇ σ)(Φ_{e}(n^{⌢}τ)(m)↓),

then define η(e, n, i) ∈ ω^{<ω} by induction on i ∈ ω as follows. Let η(e, n, 0) = n^{⌢}σ, where σ is the least string such that Φ_{e}(n^{⌢}σ)(0)↓. Given η(e, n, i), let η(e, n, i + 1) = η(e, n, i)^{⌢}0^{⌢}σ, where σ is the least string such that Φ_{e}(η(e, n, i)^{⌢}0^{⌢}σ)(i+1)↓.

Let f ∈ ω^{ω}. We construct A and B such that the following conditions hold.

• If g ∈ A, then g(0) has the form g(0) = ⟨ ℓ, ⟨ n_{0}, x_{0}, y_{0} ⟩, …, ⟨ n_{ℓ−1}, x_{ℓ−1}, y_{ℓ−1} ⟩ ⟩,

where ℓ ∈ ω and n_{i} ∈ ω, x_{i} ∈ {0, 1}, and y_{i} ∈ ω for each i < ℓ.

• If g ∈ A and ⟨ n_{e}, 0, y_{e} ⟩ is in the e^{th} position of g(0), then – ∃m∃σ(∀τ ⊇ σ)(Φ_{e}(n_{e}^{⌢}τ)(m)↑), and

– any h ∈ B with h(0) = n_{e} is of the form h = n_{e}^{⌢}σ^{⌢}f, where |σ| = y_{e}.

• If g ∈ A and ⟨ n_{e}, 1, y_{e} ⟩ is the e^{th} position of g(0), then – ∀m∀σ(∃τ ⊇ σ)(Φ_{e}(n_{e}^{⌢}τ)(m)↓), and

– any h ∈ B with h(0) = n_{e} is of the form h = η(e, n_{e}, i)^{⌢}1^{⌢}f for some

i ∈ ω.

• The above properties hold with the roles of A and B reversed.

We construct A and B in stages. The construction is similar to the construction in Lemma 3.1.2 in that if g goes into A before h goes into B, then h(0) codes how to recover f from g, and similarly with the roles of A and B reversed. Start at stage 0 with A = ∅, B = ∅, s = ⟨ ⟩, and t = ⟨ ⟩.

Stage e + 1: Set n_{e} = e^{⌢}t.

Case 1: ∃m∃σ(∀τ ⊇ σ)(Φ_{e}(n_{e}^{⌢}τ)(m)↑). Choose such a σ and put n_{e}^{⌢}σ^{⌢}f in A. Update s = s^{⌢} ⟨ n_{e}, 0, |σ| ⟩.

Case 2: ∀m∀σ(∃τ ⊇ σ)(Φ_{e}(n_{e}^{⌢}τ)(m)↓). Put the functions η(e, n_{e}, i)^{⌢}1^{⌢}f in A for each i ∈ ω. Update s = s^{⌢} ⟨ n_{e}, 1, 0 ⟩.

Repeat the above procedure with the roles of A and B reversed and the roles of s and t reversed. This completes stage e + 1. Then go on to stage e + 2. This completes the construction.

Suppose A ≥_{s} C where C is closed. We show that C contains a recursive function. The proof with B in place of A is the same. Let Φ_{e}(A) ⊆ C. Consider stage e + 1 of the above construction. Case 1 must not have occurred because otherwise A would contain a function n_{e}^{⌢}σ^{⌢}f such that Φ_{e}(n_{e}^{⌢}σ^{⌢}f) is not total. Thus case 2 occurred, and so A contains the function η(e, n_{e}, i)^{⌢}1^{⌢}f for each i ∈ ω. Let k be the recursive function k = n_{e}^{⌢}σ_{0}^{⌢}0^{⌢}σ_{1}^{⌢}0^{⌢}σ_{2}^{⌢}0^{⌢} ⋯, where η(e, n_{e}, i) = n_{e}^{⌢}σ_{0}^{⌢}0^{⌢} ⋯^{⌢} 0^{⌢}σ_{i} for each i ∈ ω (think of k as the “limit” of the strings η(e, n_{e}, i) as i → ∞). Then Φ_{e}(η(e, n_{e}, i)^{⌢}1^{⌢}f) ∈ C and Φ_{e}(η(e, n_{e}, i)^{⌢}1^{⌢}f) ↾ i = Φ_{e}(k) ↾ i for each i ∈ ω. Thus C contains the recursive function Φ_{e}(k) because C is closed.

We now describe a uniform procedure for producing f from g ⊕ h ∈ A + B. First decode h(0) as h(0) = ⟨ ℓ, ⟨ n_{0}, x_{0}, y_{0} ⟩, …, ⟨ n_{ℓ−1}, x_{ℓ−1}, y_{ℓ−1} ⟩ ⟩ and look for g(0) among the n_{e}. If ⟨ g(0), 0, y_{e} ⟩ appears in h(0) at position e, then output g from position y_{e} + 1 onward as in this case g = σ^{⌢}f for some string σ of length y_{e} + 1. If ⟨ g(0), 1, 0 ⟩ appears in h(0) at position e, then g = η(e, g(0), i)^{⌢}1^{⌢}f for some i ∈ ω. Compute which i by successively computing the η(e, g(0), j), matching them against g, and checking if the next bit of g is 0 (in which case compute η(e, g(0), j + 1)) or 1 (in which case j = i). Output f once i is found.

The number g(0) appears among the n_{e} coded into h(0) if g went into A before h went into B. Otherwise h went into B before g went into A, so h(0) appears among the n_{e} coded in g(0). In this case, switch the roles of g and h and apply the above procedure to compute f.

Theorem 3.4.3. 𝔉_{cl} is not prime. In fact, if 𝔊 ⊆ 𝔉_{cl}, 𝔊 ≠ {1} is a filter, then 𝔊 is not prime.

Proof.Suppose 𝔊 ⊆ 𝔉_{cl} is a filter such that 𝔊 ≠ {1}. Let f >_{T} 0 be such that deg_{s}({f}) ∈ 𝔊. Let A, B ⊆ ω^{ω} be as in Lemma 3.4.2 for this f. Let a = deg_{s}(A) and b = deg_{s}(B). Then a, b ∉ 𝔊 because a, b ∉ 𝔉_{cl}, but a + b ∈ 𝔊 because a + b ≥_{s} deg_{s}({f}).

If x and y are degrees such that y is closed and y ≰_{s} x, then there is no dense degree z such that y ≤_{s} x + z [39]. Therefore, if 𝔊 ⊆ 𝔉_{cl}, 𝔊 ≠ {1} is a filter, then any degrees a and b witnessing that 𝔊 is not prime must both be in 𝔉 \ 𝔊.

The results of the previous section suggest two new filters to study.

Definition 3.4.4.

• 𝔊 denotes the filter generated by

{d | d >_{s} 0′ and is join-irreducible }.

• ℌ denotes the filter generated by

{deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) | (∀i ∈ ω)(deg_{s}(D_{i}) >_{s} 0′ and is join-irreducible )}.

𝔊 is exactly the set of all degrees b for which b ≥_{s} ∏_{i=0}^{n} d_{i} for some joinirreducible degrees d_{i} >_{s} 0′, i ≤ n, and ℌ is exactly the set of all degrees b for which b ≥_{s} deg_{s}(⋃_{i∈ω} i^{⌢}D_{i}) for some join-irreducible degrees deg_{s}(D_{i}) >_{s} 0′, i ∈ ω.

Theorem 3.4.5. 𝔉_{cl} ⊊ 𝔊 ⊊ ℌ ⊊ {a | a >_{s} 0′}. 𝔊 ⊈ 𝔉 (hence also ℌ ⊈ 𝔉). Neither 𝔊 nor ℌ is principal.

Proof.Every closed degree >_{s} 0 bounds a join-irreducible degree >_{s} 0′ [75]. Hence 𝔉_{cl} ⊆ 𝔊. 𝔊 ⊆ ℌ is clear. To see 𝔊 ⊈ 𝔉, observe that every B_{f} is dense, so if f >_{T} 0, then deg_{s}(B_{f}) ∈ 𝔊 \ 𝔉. This also shows 𝔊 ⊈ 𝔉_{cl}. The degree constructed in Theorem 3.2.8 witnesses ℌ ⊈ 𝔊. The degree constructed in Theorem 3.2.10 witnesses {a | a >_{s} 0′} ⊈ ℌ. We show that 𝔊 is not principal. The proof for ℌ is the same. First, if A is countable and contains no recursive functions, then there is a function f >_{T} 0 such that g ≰_{T} f for all g ∈ A. Thus B_{f} ≤_{s} A (as A ⊆ B_{f}) for this f. Every deg_{s}(B_{f}) for f >_{T} 0 is in 𝔊, so every deg_{s}(A) where A is countable and contains no recursive function is in 𝔊. If 𝔊 were principal, it would be generated by a degree x such that x ≤_{s} deg_{s}(A) for all countable A not containing a recursive function. By Lemma 3.3.2, the only such x are 0 and 0′. We know 0 and 0′ are not in 𝔊, so 𝔊 cannot be principal.

We end with a question.

Question 3.4.6.

• Is 𝔉 ⊆ 𝔊? Is 𝔉 ⊆ ℌ?

• Is 𝔊 prime? Is ℌ prime?

• Is {a | PTh(D_{s} /a) ⊆ JAN} a filter?

To prove that {a | PTh(D_{s} /a) ⊆ JAN} is a filter, it suffices to prove that PTh(D_{s} /(a × b)) ⊆ JAN whenever both PTh(D_{s} /a) and PTh(D_{s} /b) are ⊆ JAN because {a | PTh(D_{s} /a) ⊆ JAN} is upward-closed in D_{s}.

CHAPTER 4 FORCING NO BIG CHAINS IN THE POWER SET OF THE REALS

Let κ ≤ λ be infinite cardinals. Independence phenomena concerning the question of whether or not there is a chain (i.e. linearly ordered subset) in (2^{κ}, ⊆) of cardinality λ were studied extensively by Baumgartner [8] and Mitchell [44]. However, the particular case of κ = 2^{ω} and λ = 2^{2^{ω}} was not discussed explicitly. In the positive direction, the following results appear in Baumgartner [8]. Baumgartner attributes his techniques to Sierpinski.

Theorem 4.0.7 (see [8]). If 2^{<κ} = κ, then there is a chain in (2^{κ}, ⊆) of cardinality 2^{κ}.

Corollary 4.0.8 (see [8]). ZFC + CH proves that there is a chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}. Thus if ZFC is consistent, then so is ZFC + “there is a chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}.”

Proof.CH implies 2^{<ω_{1}} = ω_{1}. So apply Theorem 4.0.7 with κ = ω_{1}.

In the negative direction, Comfort and Remus noticed in [20] that the consistency of ZFC + “there is no chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}” indeed follows from Baumgartner’s results in [8]. The main purpose of this chapter is to give a more explicit proof of the consistency of ZFC + “there is no chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}” (Corollary 4.2.6). We thank both Todorcevic and Mitchell for independently suggesting the proof that we give.

The author was introduced to the foregoing problem via the Medvedev degrees. In [77], Terwijn uses Corollary 4.0.8 to prove that if ZFC is consistent, then so is ZFC + “there is a chain in the Medvedev degrees of cardinality 2^{2^{ω}}.” We prove the following theorem in ZFC. For a cardinal λ, (2^{2^{ω}}, ⊆) has a chain of cardinality λ if and only if the Medvedev degrees do (Theorem 4.3.1). Thus the statement “there is a chain in the Medvedev degrees of cardinality 2^{2^{ω}}” is independent of ZFC.

4.1 Forcing prerequisites

This section contains the basic facts about forcing and about chain conditions that we need in the next section. We follow [35] when possible. The reader familiar with this material may skip ahead.

First, a partial order may be replaced by an isomorphic partial order without changing the forcing extension.

Lemma 4.1.1 (see [35] section VII.7). Let ℙ, ℚ ∈ M and let i: ℙ → ℚ be an isomorphism in M. Let G ⊆ ℙ. Then G is ℙ-generic over M if and only if i′′G is ℚ-generic over M. In this case M[G] = M[i′′G].

Forcing with a partial order ℙ × ℚ is equivalent to forcing with the partial orders ℙ and ℚ one-at-a-time in either order.

Lemma 4.1.2 (see [35] section VIII.1). Let ℙ, ℚ ∈ M and let G_{0} ⊆ ℙ, G_{1} ⊆ ℚ. Then the following are equivalent:

(i) G_{0} × G_{1} is ℙ × ℚ-generic over M, (ii) G_{0} is ℙ-generic over M and G_{1} is ℚ-generic over M[G_{0}], and (iii) G_{1} is ℚ-generic over M and G_{0} is ℙ-generic over M[G_{1}].

If (i)–(iii) hold, then M[G_{0} ×G_{1}] = M[G_{0}][G_{1}] = M[G_{1}][G_{0}]. Moreover, if G is ℙ × ℚgeneric over M, then G = H_{0} × H_{1} for some H_{0} ⊆ ℙ and H_{1} ⊆ ℚ which thus satisfy (i)–(iii).

Let κ be a cardinal. A partial order ℙ is κ-closed if every descending chain of conditions having order-type < κ has a lower bound. A partial order ℙ has the κ-chain condition (abbreviated κ-cc) if every X ⊆ ℙ of cardinality κ has two compatible elements. Notice that ℙ is κ-cc if and only if for every sequence of conditions (p_{ξ} | ξ ∈ κ) there are distinct ξ, ζ ∈ κ such that p_{ξ} and p_{ζ} are compatible. A partial order ℙ has property K_{κ} if every X ⊆ ℙ of cardinality κ has a pairwise compatible subset of cardinality κ. Notice that if κ is regular, then ℙ has property K_{κ} if and only if for every sequence of conditions (p_{ξ} | ξ ∈ κ) there is a set X ⊆ κ of cardinality κ such that if ξ, ζ ∈ X then p_{ξ} and p_{ζ} are compatible. Clearly property K_{κ} implies κ-cc. We consider κ = ω_{1} and κ = ω_{2}. Traditionally ω_{1}-cc is called ccc and property K_{ω_{1}} is called property K.

Partial orders that are κ-closed preserve cardinals ≤ κ, and partial orders that are κ-cc preserve cardinals ≥ κ. It follows that a partial order that is both κ-closed and κ^{+}-cc preserves all cardinals.

The following lemmas easily generalize to other cardinals, but we state the versions that we use which are in terms of ω_{2}. Lemma 4.1.3. If ℙ and ℚ have property K_{ω_{2}}, then so does ℙ × ℚ.

Proof.Let ((p_{ξ}, q_{ξ}) | ξ ∈ ω_{2}) be a sequence of conditions from ℙ × ℚ. Applying ℙ’s property K_{ω_{2}} to the sequence (p_{ξ} | ξ ∈ ω_{2}) yields a set Y ⊆ ω_{2} such that |Y | = ω_{2} and p_{ξ} and p_{ζ} are compatible for every ξ, ζ ∈ Y. Applying ℚ’s property K_{ω_{2}} to the sequence (q_{ξ} | ξ ∈ Y) yields a set X ⊆ Y such that |X| = ω_{2} and q_{ξ} and q_{ζ} are compatible for every ξ, ζ ∈ X. The conditions (p_{ξ}, q_{ξ}) and (p_{ζ}, q_{ζ}) are compatible for every ξ, ζ in X, so X witnesses property K_{ω_{2}} for the sequence ((p_{ξ}, q_{ξ}) | ξ ∈ ω_{2}).

Lemma 4.1.4. If ℙ is ω_{2}-cc and X ⊆ ℙ has cardinality ω_{2}, then there is a p ∈ ℙ such that p ⊩ |Ġ ∩ X̌| = |ω̌_{2}|.

Proof.Fix a bijection f: ω_{2} ↔ X. If there is no such p, then there is a name α̇ such that

1_{ℙ} ⊩ (α̇ ∈ ω̌_{2}) ∧(∀ξ ∈ ω̌_{2})(f̌(ξ) ∈ Ġ ∩ X̌ → ξ ∈ α̇).

Now let A ⊆ ℙ be a maximal antichain of conditions with the property (∀p ∈ A)(∃β ∈ ω_{2})(p ⊩ α̇ = β̌). We have that |A| < ω_{2} because ℙ is ω_{2}-cc. Thus we are able to choose γ ∈ ω_{2} such that if p ∈ A and p ⊩ α̇ = β̌ then β ∈ γ. Clearly f(γ) ⊩ f̌(γ̌) ∈ Ġ ∩ X̌; therefore f(γ) ⊩ γ̌ ∈ α̇. Choose p ≤ f(γ) such that p ⊩ α̇ = β̌ for some β ∈ ω_{2}. This β must be ≥ γ. Hence p is incompatible with everything in A, thereby contradicting A’s maximality.

Lemma 4.1.4 is stated in its most natural way, but we make greater use of the following reformulation.

Lemma 4.1.5. Let ℙ be ω_{2}-cc, let X ⊆ ω_{2} have cardinality ω_{2}, and let (p_{ξ} | ξ ∈ X) be a sequence of conditions from ℙ. Let Ż = {⟨ ξ̌, p_{ξ} ⟩ | ξ ∈ X} be a name for the set {ξ ∈ X | p_{ξ} ∈ G}. Then there is a p ∈ ℙ such that p ⊩ |Ż| = |ω̌_{2}|.

Proof.If |{p_{ξ} | ξ ∈ X}| < ω_{2}, choose p such that |{ξ ∈ X | p_{ξ} = p}| = ω_{2}. Otherwise choose p by applying Lemma 4.1.4 to the set {p_{ξ} | ξ ∈ X}.

A partial order with property K_{ω_{2}} retains property K_{ω_{2}} in an extension created by forcing with an ω_{2}-cc partial order.

Lemma 4.1.6. If ℙ is ω_{2}-cc and ℚ has property K_{ω_{2}}, then 1_{ℙ} ⊩ ℚ̌ has property K_{ω_{2}}.

Proof.Suppose not. Then there is a ℙ-name ḟ and a condition p ∈ ℙ such that

p ⊩ (ḟ: ω̌_{2} → ℚ̌) ∧(∀X ⊆ ω̌_{2})(|X| = |ω̌_{2}| →(∃ξ, ζ ∈ X)(ḟ(ξ) ⊥ ḟ(ζ))).

That is, p forces that the sequence (f(ξ) | ξ ∈ ω_{2}) of conditions from ℚ is a counterexample to ℚ having property K_{ω_{2}} (as ℙ preserves ω_{2}). For each ξ ∈ ω_{2}, choose a condition p_{ξ} ∈ ℙ and a condition q_{ξ} ∈ ℚ such that p_{ξ} ≤ p and p_{ξ} ⊩ ḟ(ξ̌) = q̌_{ξ}. This defines a sequence of conditions (p_{ξ} | ξ ∈ ω_{2}) from ℙ and a sequence of conditions (q_{ξ} | ξ ∈ ω_{2}) from ℚ.

We show that the sequence (q_{ξ} | ξ ∈ ω_{2}) is a counterexample to ℚ having property K_{ω_{2}}. To this end, let X ⊆ ω_{2} have cardinality ω_{2}, let Ż be a ℙ-name for the set {ξ ∈ X | p_{ξ} ∈ G}, and apply Lemma 4.1.5 to the sequence (p_{ξ} | ξ ∈ X) to get a condition p′ ≤ p such that p′ ⊩ |Ż| = |ω̌_{2}|. We now have that p′ ⊩ (Ż ⊆ ω̌_{2}) ∧(|Ż| = |ω̌_{2}|). Thus, as p′ ≤ p, we also have that p′ ⊩ (∃ξ, ζ ∈ Ż)(ḟ(ξ) ⊥ ḟ(ζ)). Let ξ̇ and ζ̇ be ℙ-names such that p′ ⊩ (ξ̇, ζ̇ ∈ Ż) ∧(ḟ(ξ̇) ⊥ ḟ(ζ̇)). Finally, choose p′′ ≤ p′ and choose ξ, ζ ∈ X such that p′′ ⊩ (ξ̇ = ξ̌) ∧(ζ̇ = ζ̌). Then p′′ ⊩ ξ̌, ζ̌ ∈ Ż, and so, by the definition of Ż, p′′ ⊩ p̌_{ξ}, p̌_{ζ} ∈ Ġ. Hence p′′ ⊩ (ḟ(ξ̌) ⊥ ḟ(ζ̌)) ∧(ḟ(ξ̌) = q̌_{ξ}) ∧(ḟ(ζ̌) = q̌_{ζ}). So it must be that q_{ξ} ⊥ q_{ζ}.

We work with partial orders of the form Fn(I, 2, κ), which is the set of partial functions p: I → 2 of cardinality < κ ordered by p ≤ q if and only if p extends q. If |I| ≥ κ, forcing with Fn(I, 2, κ) adds |I| subsets of κ, thereby making 2^{κ} ≥ |I| in the extension (see [35] section VII.6).

In order to show that our partial orders have the desired chain conditions, we recall the notion of a Δ-system and the corresponding Δ-system lemma. A set A is called a Δ-system if there is a set r such that a ∩ b = r whenever a and b are distinct members of A.

Lemma 4.1.7 (Δ-system lemma; see [35] section II.1). Let κ be an infinite cardinal. Let θ > κ be regular and satisfy (∀α < θ)(|α^{<κ}| < θ). Assume |A| ≥ θ and (∀x ∈ A)(|x| < κ). Then there is a Δ-system B ⊆ A with |B| = θ.

Finally, in order to show that the cardinal arithmetic works out as desired in our extensions, we recall that sets in an extension M[G] have nice names in M. If ℙ ∈ M is a partial order and σ is a name, then a nice name for a subset of σ is a name τ of the form ⋃{{π} × A_{π} | π ∈ dom(σ)} where each A_{π} is an antichain in ℙ.

Lemma 4.1.8 (see [35] section VII.5). If ℙ ∈ M is a partial order and σ, μ are names, then there is a nice name τ for a subset of σ such that 1_{ℙ} ⊩ (μ ⊆ σ) →(μ = τ).

A nice name for a subset of an existing set x ∈ M is particularly nice. A nice name τ for a subset of the canonical name x̌ = {(y̌, 1_{ℙ}) | y ∈ x} can be taken to be of the form ⋃{{y̌} × A_{y} | y ∈ x}.

4.2 Forcing no chains in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}

Recall that a (set-theoretic) tree is a partial order (T, ≤_{T}) such that, for each t ∈ T, the set of predecessors {s ∈ T | s <_{T} t} is well-ordered by <_{T}. We usually write T for (T, ≤_{T}). The level ℓ(t) of a t ∈ T is the order-type of {s ∈ T | s <_{T} t}, the ξ^{th} level of T is Lev_{ξ} = {t ∈ T | ℓ(t) = ξ}, and the height of T is the least ξ such that Lev_{ξ} = ∅. A branch through T is a maximal linearly ordered subset of T. The length of a branch is its order-type. We rephrase our question about chains as a question about trees via the following theorem.

Theorem 4.2.1 (Baumgartner, Mitchell; see [8]). Let κ ≤ λ be infinite cardinals. The following are equivalent:

• There is a tree of height ≤ κ and cardinality ≤ κ with ≥ λ branches.

• There is a chain in (2^{κ}, ⊆) of cardinality λ.

In light of Theorem 4.2.1 (with κ = 2^{ω} and λ = 2^{2^{ω}}), we force there to be no chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}} by forcing there to be no tree of height ≤ 2^{ω} and cardinality ≤ 2^{ω} with ≥ 2^{2^{ω}} branches. Our proof is similar to Silver’s proof that one can force there to be no Kurepa trees given that there is an inaccessible cardinal (see [35] section VIII.3).

Our strategy is as follows.

(i) Let M ⊧ GCH, and, in M, let ℙ = Fn(ω_{2}, 2, ω) and ℚ = Fn(ω_{ω_{2}}, 2, ω_{1}). Force with ℙ × ℚ.

(ii) ℙ × ℚ preserves cardinals. If G is ℙ × ℚ-generic over M, then, in M[G], 2^{ω} = ω_{2}, 2^{ω_{1}} = ω_{ω_{2}}, and 2^{ω_{2}} = ω_{ω_{2}}^{+}.

(iii) Suppose T ⊆ ω_{2}×ω_{2} is a tree in M[G] of height ≤ ω_{2} and of cardinality ≤ ω_{2}. T has ≤ |ω_{2}^{<ω_{2}}| = ω_{ω_{2}} branches of cardinality < ω_{2}. Thus we need only show that T has < ω_{ω_{2}}^{+} branches of cardinality ω_{2}. To do this, factor ℙ × ℚ into an isomorphic partial order ℝ_{0} × ℝ_{1} with the following properties.

– M[G] = M[G_{0}][G_{1}] where G_{0} is ℝ_{0}-generic over M and G_{1} is ℝ_{1}-generic

over M[G_{0}],

– T ∈ M[G_{0}], and, in M[G_{0}], T has ≤ ω_{3} branches, and

– any branch of T in M[G_{0}][G_{1}] of cardinality ω_{2} is already in M[G_{0}].

Thus T has ≤ ω_{3} < ω_{ω_{2}}^{+} branches of cardinality ω_{2} in M[G].

Let M, ℙ, and ℚ be as in step (i) of the strategy.

Lemma 4.2.2. In M, the partial order ℙ has property K, and the partial orders ℙ, ℚ, and ℙ × ℚ all have property K_{ω_{2}}.

Proof.To see that ℙ has property K, let X ⊆ ℙ have cardinality ω_{1} and let A be the set of domains of the functions in X. Then |A| = ω_{1} and (∀x ∈ A)(|x| < ω). As (∀α < ω_{1})(|α^{<ω}| < ω_{1}), Lemma 4.1.7 gives us a Δ-system B ⊆ A of cardinality ω_{1} with finite root r. There are only a finite number of functions f: r → 2, so there must be one such f for which the set {g ∈ X | (dom g ∈ B) ∧(g ⊇ f)} has cardinality ω_{1}. This is the desired pairwise compatible subset of X. The proof that ℙ has property K_{ω_{2}} is the same as the above, but with ω_{2} in place of ω_{1}.

For ℚ, let X ⊆ ℚ have cardinality ω_{2}, and let A be the set of domains of the functions in X. |A| = ω_{2} because for a fixed countable x ⊆ ω_{ω_{2}} there are only 2^{ω} = ω_{1} possible functions x → 2. We have (∀x ∈ A)(|x| < ω_{1}) and, as |ω_{1}^{<ω_{1}}| = ω_{1}, we have (∀α < ω_{2})(|α^{<ω_{1}}| < ω_{2}). Thus Lemma 4.1.7 gives us a Δ-system B ⊆ A of cardinality ω_{2} with countable root r. There are only ω_{1} functions f: r → 2, so there must be one such f for which the set {g ∈ X | (dom g ∈ B) ∧ (g ⊇ f)} has cardinality ω_{2}. This is the desired pairwise compatible subset of X.

Finally, ℙ × ℚ has property K_{ω_{2}} by Lemma 4.1.3.

The next lemma completes step (ii).

Lemma 4.2.3. ℙ × ℚ preserves cardinals. If G is ℙ × ℚ-generic over M, then, in M[G], 2^{ω} = ω_{2}, 2^{ω_{1}} = ω_{ω_{2}}, and 2^{ω_{2}} = ω_{ω_{2}}^{+}.

Proof.We use Lemma 4.1.2 to equate forcing with ℙ × ℚ to forcing with ℚ then with ℙ. Let G be ℙ × ℚ-generic over M. Factor G as G = G_{0} × G_{1} where G_{0} ⊆ ℙ and G_{1} ⊆ ℚ so that M[G] = M[G_{1}][G_{0}]. Forcing with ℚ adds ω_{ω_{2}} subsets of ω_{1}, and ℚ is both ω_{1}-closed and ω_{2}-cc. Thus, in M[G_{1}], 2^{ω_{1}} ≥ ω_{ω_{2}} and M[G_{1}] has the same cardinals as M. Forcing cannot add any new finite functions, so M[G_{1}]’s version of Fn(ω_{2}, 2, ω) is exactly ℙ because forcing with ℚ preserves ω_{2}. Below we need that (ω_{2})^{ω} = ω_{2} in M[G_{1}]. This equality is true in M by GCH and remains true in M[G_{1}] because of the fact that ℚ is ω_{1}-closed implies that forcing with ℚ adds no new functions ω → ω_{2}.

ℙ is ccc, and forcing with ℙ adds ω_{2} subsets of ω. Thus M[G_{1}][G_{0}] has the same cardinals as M[G_{1}], and, in M[G_{1}][G_{0}], 2^{ω} ≥ ω_{2}. So we have that M[G] = M[G_{1}][G_{0}] has the same cardinals as M (i.e. ℙ × ℚ preserves cardinals) and that, in M[G], 2^{ω} ≥ ω_{2} and 2^{ω_{1}} ≥ ω_{ω_{2}}. Furthermore, in M[G], 2^{ω_{2}} ≥ ω_{ω_{2}}^{+} because 2^{ω_{2}} ≥ 2^{ω_{1}} ≥ ω_{ω_{2}} and 2^{ω_{2}} cannot have cofinality ω_{2}. We show the desired reverse inequalities by counting nice names.

To see 2^{ω} ≤ ω_{2}, count the nice ℙ-names for subsets of ω in M[G_{1}]. Such a name has the form ⋃{{ň} × A_{n} | n ∈ ω} where each A_{n} is an antichain in ℙ. ℙ is ccc and has cardinality ω_{2}, hence ℙ has (ω_{2})^{ω} = ω_{2} antichains. Therefore there are ≤ (ω_{2})^{ω} = ω_{2} nice names for subsets of ω in M[G_{1}]. Forcing with ℙ preserves cardinals, so there are ≤ ω_{2} subsets of ω in M[G].

To see 2^{ω_{1}} ≤ ω_{ω_{2}}, count the nice ℙ × ℚ-names for subsets of ω_{1} in M. Such a name has the form ⋃{{α̌} × A_{α} | α ∈ ω_{1}} where each A_{α} is an antichain in ℙ × ℚ. ℙ × ℚ has property K_{ω_{2}} by Lemma 4.2.2, so its antichains have cardinality ≤ ω_{1}. The fact that GCH holds in M implies that ℙ × ℚ has cardinality ω_{ω_{2}} and consequently that ℙ × ℚ has ≤ (ω_{ω_{2}})^{ω_{1}} = ω_{ω_{2}} antichains. Therefore there are ≤ (ω_{ω_{2}})^{ω_{1}} = ω_{ω_{2}} nice names for subsets of ω_{1} in M. Forcing with ℙ × ℚ preserves cardinals, so there are ≤ ω_{ω_{2}} subsets of ω_{1} in M[G].

Finally, to see 2^{ω_{2}} ≤ ω_{ω_{2}}^{+}, count the nice ℙ × ℚ-names for subsets of ω_{2} in M. Such a name has the form ⋃{{α̌} × A_{α} | α ∈ ω_{2}} where each A_{α} is an antichain in ℙ × ℚ. There are ≤ (ω_{ω_{2}})^{ω_{2}} = ω_{ω_{2}}^{+} nice names for subsets of ω_{1} in M. Forcing with ℙ × ℚ preserves cardinals, so there are ≤ ω_{ω_{2}}^{+} subsets of ω_{2} in M[G].

The next lemma is the crux of step (iii).

Lemma 4.2.4. Suppose, in some model N, that T ⊆ ω_{2} × ω_{2} is a tree of height ω_{2} and that ℝ is a partial order with property K_{ω_{2}}. If G is ℝ-generic over N, then there are no new branches of T in N[G] of cardinality ω_{2}.

Proof.In N, let B ⊆ 2^{ω_{2}} be the set of branches through T and let ḃ be a name for a branch through T. Suppose for a contradiction that there is a condition p ∈ ℝ forcing that ḃ is a new branch of cardinality ω_{2}. That is, p ⊩ (ḃ is a branch) ∧(|ḃ| = |ω̌_{2}|) ∧(ḃ ∉ B̌) (note ℝ preserves ω_{2}). For each ξ < ω_{2}, we can find an x_{ξ} ∈ Lev_{ξ} and a p_{ξ} ≤ p such that p_{ξ} ⊩ x̌_{ξ} ∈ ḃ. Apply ℝ’s property K_{ω_{2}} to the sequence (p_{ξ} | ξ ∈ ω_{2}) to get a set X ⊆ ω_{2} such that |X| = ω_{2} and such that p_{ξ} and p_{ζ} are compatible for every ξ, ζ ∈ X. Notice that if ξ < ζ are in X, then the compatibility of p_{ξ} and p_{ζ} implies that x_{ξ} <_{T} x_{ζ}. Therefore the set c = {x ∈ ω_{2} | (∃ξ ∈ X)(x ≤_{T} x_{ξ})} is a branch in B. Let Ż be a name for the set {ξ ∈ X | p_{ξ} ∈ G}, and, by Lemma 4.1.5, choose p′ ≤ p such that p′ ⊩ |Ż| = |ω̌_{2}|. We now show that p′ ⊩ ḃ = č, which contradicts that p ⊩ ḃ ∉ B̌. To this end, suppose that H is ℝ-generic with p′ ∈ H, and now work in N[H]. Suppose that x ∈ b ∩ Lev_{ξ}. There is some ζ > ξ with p_{ζ} ∈ H because |Z| = |ω_{2}|. Therefore x_{ζ} ∈ b because p_{ζ} ⊩ x̌_{ζ} ∈ ḃ. Hence x <_{T} x_{ζ} and so x ∈ c. This proves b ⊆ c, and equality follows from the fact that both b and c are branches of cardinality ω_{2} in a tree of height ω_{2}.

We can now carry out step (iii).

Theorem 4.2.5. If ZFC is consistent, then so is ZFC + “there is no tree of height ≤ 2^{ω} and cardinality ≤ 2^{ω} with ≥ 2^{2^{ω}} branches.”

Proof.Let M ⊧ GCH. In M, let ℙ = Fn(ω_{2}, 2, ω) and let ℚ = Fn(ω_{ω_{2}}, 2, ω_{1}). Let G be ℙ × ℚ-generic over M. By Lemma 4.2.3,

M[G] ⊧ (2^{ω} = ω_{2}) ∧(2^{ω_{1}} = ω_{ω_{2}}) ∧(2^{ω_{2}} = ω_{ω_{2}}^{+}).

We show that in M[G] there is no tree of height ≤ ω_{2} and cardinality ≤ ω_{2} with ≥ ω_{ω_{2}}^{+} branches. A tree of cardinality ω_{2} is isomorphic to a tree ⊆ ω_{2} × ω_{2}, so it suffices to show that in M[G] there is no tree ⊆ ω_{2} ×ω_{2} of height ≤ ω_{2} with ≥ ω_{ω_{2}}^{+} branches.

Suppose T ⊆ ω_{2} × ω_{2} is a tree of height ≤ ω_{2} in the extension M[G]. Then, in M, T has a nice name of the form Ṫ = ⋃{{š} × A_{s} | s ∈ ω_{2} × ω_{2}} where each A_{s} is an antichain in ℙ × ℚ, a partial order with property K_{ω_{2}} by Lemma 4.2.2. Thus each A_{s} has cardinality ≤ ω_{1}, and hence the set X = {α ∈ ω_{ω_{2}} | (∃(p, q) ∈ ⋃_{s∈ω_{2}×ω_{2}} A_{s})(α ∈ dom q)} has cardinality ≤ ω_{2}. Add elements of ω_{ω_{2}} to X so that it has cardinality exactly ω_{2}. Let ℚ_{X} = Fn(X, 2, ω_{1}) and let ℚ_{ω_{ω_{2}}\X} = Fn(ω_{ω_{2}} \ X, 2, ω_{1}). It is easy to check that ℚ_{X} × ℚ_{ω_{ω_{2}}\X} ≅ ℚ, so factor ℙ × ℚ as (ℙ × ℚ_{X}) × ℚ_{ω_{ω_{2}}\X}. In terms of step (iii) as described above, ℙ × ℚ_{X} is ℝ_{0} and ℚ_{ω_{ω_{2}}\X} is ℝ_{1}. By Lemma 4.1.1, there is a (ℙ × ℚ_{X}) × ℚ_{ω_{ω_{2}}\X}-generic set H such that M[H] = M[G], and, by Lemma 4.1.2, H = H_{0} × H_{1} where H_{0} is ℙ × ℚ_{X}-generic over M and H_{1} is ℚ_{ω_{ω_{2}}\X}-generic over M[H_{0}]. By our choice of X, the ℙ × ℚ-name Ṫ is also a ℙ × ℚ_{X}-name. So our tree T is in the intermediate extension M[H_{0}]. The models M, M[H_{0}], and M[H_{0}][H_{1}] = M[H] = M[G] all have the same cardinals because M and M[G] do by Lemma 4.2.3. Also, M[H_{0}] ⊧ (2^{ω} = ω_{2}) ∧(2^{ω_{1}} = ω_{2}) ∧(2^{ω_{2}} = ω_{3}) by a proof similar to that of Lemma 4.2.3. Furthermore, ℚ_{ω_{ω_{2}}\X} has property K_{ω_{2}} in M[H_{0}] by Lemma 4.1.6 because ℚ_{ω_{ω_{2}}\X} and ℙ × ℚ_{X} have property K_{ω_{2}} in M by a proof similar to that of Lemma 4.2.2. Now, T has ≤ 2^{ω_{2}} = ω_{3} branches in M[H_{0}]. Lemma 4.2.4 tells us that T does not have any more branches of cardinality ω_{2} in M[H_{0}][H_{1}] than it does in M[H_{0}]. So, in M[G], T has ≤ ω_{3} branches of cardinality ω_{2} and ≤ |ω_{2}^{<ω_{2}}| = ω_{ω_{2}} branches of cardinality < ω_{2} for a grand total of ≤ ω_{ω_{2}} branches.

In light of Theorem 4.2.1, our main goal is a corollary of Theorem 4.2.5.

Corollary 4.2.6. If ZFC is consistent, then so is ZFC + “there is no chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}.”

Corollary 4.2.7. The statement “there is a chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}” is independent of ZFC.

Proof.Corollary 4.0.8 gives the consistency of the statement, and Corollary 4.2.6 gives the consistency of its negation.

4.3 Big chains in the Medvedev degrees

We show, in ZFC, that the cardinalities of chains in (2^{2^{ω}}, ⊆) are the same as the cardinalities of chains in D_{s}. The forward direction of Theorem 4.3.1 is due to Terwijn [77].

Theorem 4.3.1. For any cardinal κ, there is a chain in (2^{2^{ω}}, ⊆) of cardinality κ if and only if there is a chain in D_{s} of cardinality κ.

Proof.Let ℭ ⊆ 2^{2^{ω}} be a chain of cardinality κ. Let (f_{ξ} | ξ ∈ 2^{ω}) be a ≤_{T}-antichain (in the pairwise incomparable sense) of functions in ω^{ω}. For each C ∈ ℭ, let X_{C} = {f_{ξ} | ξ ∈ C}. Then, for C, D ∈ ℭ, C ⊊ D implies X_{C} >_{s} X_{D}. The inequality X_{C} ≥_{s} X_{D} holds because X_{C} ⊆ X_{D}. To see that the inequality is strict, let ζ ∈ D \ C so that f_{ζ} ∈ X_{D} \ X_{C}. The function f_{ζ} does not compute any function in X_{C} because the f_{ξ}’s were chosen to be a ≤_{T}-antichain. Thus there is no Turing functional Φ for which Φ(X_{D}) ⊆ X_{C}. The set {deg_{s}(X_{C}) | C ∈ ℭ} is therefore a chain in D_{s} of cardinality κ.

Conversely, let 𝔛 ⊆ D_{s} be a chain of cardinality κ. We produce a chain in (2^{([ω^{ω}]^{≤ω})}, ⊆) of cardinality κ, where [ω^{ω}]^{≤ω} = {X ⊆ ω^{ω} | |X | ≤ ω}. This suffices because |[ω^{ω}]^{≤ω}| = 2^{ω}. Let 𝔛̂ ⊆ 2^{ω^{ω}} consist of a choice of one representative X ⊆ ω^{ω} from each degree in 𝔛 so that 𝔛̂ is a chain of cardinality κ under ≤_{s}. For each X ∈ 𝔛̂, let D_{X} = {W ⊆ X | |W| ≤ ω} and let C_{X} = ⋃{D_{Y} | (Y ∈ 𝔛̂) ∧(Y ≥_{s} X)}. Then, for X , Y ∈ 𝔛̂, X <_{s} Y implies C_{X} ⊋ C_{Y}. The inequality C_{X} ⊇ C_{Y} is clear. To see that the inequality is strict, apply Lemma 3.3.2 (as X ≱_{s} Y) to get a W ⊆ X such that |W| ≤ ω and W ≱_{s} Y. W is in D_{X} and hence in C_{X}, but W cannot be in C_{Y} as this would imply W ≥_{s} Y. Thus {C_{X} | X ∈ 𝔛̂} is the desired chain.

Theorem 4.3.1 is also valid for D_{w} in place of D_{s}. In the forward direction of Theorem 4.3.1, we actually proved that, for C, D ∈ ℭ, C ⊊ D implies X_{C} >_{w} X_{D}. The proof given for the reverse direction is also valid for D_{w}, but a simpler one is possible. Notice that a mass problem X ⊆ ω^{ω} is Muchnik-equivalent to its Turing upward-closure {f ∈ ω^{ω} | (∃g ∈ X)(f ≥_{T} g)}. Furthermore, if X and Y are Turing upward-closed then X ≤_{w} Y if and only if X ⊇ Y. So if 𝔛 ⊆ D_{w} is a chain of cardinality κ, choose a set of representatives 𝔛̂ ⊆ 2^{ω^{ω}} that are Turing upward-closed. Then 𝔛̂ is a chain in (2^{ω^{ω}}, ⊆) of cardinality κ.

Corollary 4.3.2. The following statements are both independent of ZFC. “There is a chain in the Medvedev degrees of cardinality 2^{2^{ω}}” and “there is a chain in the Muchnik degrees of cardinality 2^{2^{ω}}.”

Proof.By Theorem 4.3.1, both statements are equivalent under ZFC to the statement “there is a chain in (2^{2^{ω}}, ⊆) of cardinality 2^{2^{ω}}.” The result then follows from Corollary 4.2.7.

In contrast, it is a ZFC theorem that (2^{2^{ω}}, ⊆), D_{s}, and D_{w} all have antichains (in the pairwise incomparable sense) of cardinality 2^{2^{ω}} [47].

CHAPTER 5 MENGER’S THEOREM IN Π_{1}^{1}-CA_{0}

The results of this chapter also appear in [55] by the author.

König’s duality theorem for finite bipartite graphs is a classic theorem in graph theory and one of the pillars of matching theory. It expresses a duality between matchings and covers in bipartite graphs. Let (X, Y, E) be a bipartite graph. A matching is a set of edges M ⊆ E such that no two edges in M share a vertex. A cover is a set of vertices C ⊆ X ∪Y such that every edge in E has a vertex in C. Finite König’s duality theorem says that the cardinalities of matchings and the cardinalities of covers meet in the middle. Finite König’s Duality Theorem. In every finite bipartite graph, the maximum cardinality of a matching equals the minimum cardinality of a cover.

Finite Menger’s theorem generalizes finite König’s duality theorem from bipartite graphs to arbitrary graphs. Let G be a graph with vertices V (G) and edges E(G). A web is a triple (G, A, B) where G is a graph and A and B are distinguished sets of vertices A, B ⊆ V (G). The notion of a matching in a bipartite graph is generalized by the notion of a set of disjoint A-B paths^{1} in a web. An A-B path in a web (G, A, B) is a path that starts in A and ends in B. Two paths are disjoint if they have no vertices in common. The notion of a cover in a bipartite graph is generalized by the notion of an A-B separator in a web. An A-B separator in a web (G, A, B) is a set of vertices C ⊆ V (G) such that every A-B path in G contains a vertex of C (so that removing C from the graph separates A from B).

Finite Menger’s Theorem. In every finite web (G, A, B), the maximum cardinality of a set of disjoint A-B paths equals the minimum cardinality of an A-B separator.

Finite Menger’s theorem is itself a special case of the famous max-flow mincut theorem for network flows. See [21] Section 2.1 for a full treatment of finite König’s duality theorem, [21] Section 3.3 for finite Menger’s theorem, and

[21] Section 6.2 for the max-flow min-cut theorem.

The conclusions of finite König’s duality theorem and finite Menger’s theorem remain true for infinite bipartite graphs and infinite webs, but they are more an exercise in cardinal arithmetic than they are in combinatorics. To deepen the combinatorial content of these theorems, Erdős conjectured that there always exist a matching and a cover that simultaneously witness each other’s optimality. His reformulations are what we now call König’s duality theorem and Menger’s theorem.

König’s Duality Theorem. In every bipartite graph (X, Y, E), there is a matching M and a cover C such that C consists of exactly one vertex from each edge in M.

Menger’s Theorem. In every web (G, A, B), there is a set of disjoint A-B paths M and an A-B separator C such that C consists of exactly one vertex from each path in M.

1For us, “path” always means “simple path,” that is, no repeated vertices.

The most general case, Menger’s theorem for webs of arbitrary cardinality, is now known to be true. The proof took more than forty years to discover. The first progress was by Podewski and Steffens, who proved König’s duality theorem for countable bipartite graphs [48]. Aharoni next proved König’s duality theorem for arbitrary bipartite graphs [1]. He then proved Menger’s theorem for countable webs [2]. Finally, Aharoni and Berger proved Menger’s theorem for arbitrary webs [3].

The question motivating our work is the following.

Question 5.0.3. What is the axiomatic strength of Menger’s theorem for countable webs in the context of second-order arithmetic?

Aharoni, Magidor, and Shore [4] and Simpson [63] answered Question 5.0.3 for König’s duality theorem for countable bipartite graphs. Aharoni, Magidor, and Shore noticed that Aharoni’s proof of König’s duality theorem in [1] actually proves a stronger statement, which they call extended König’s duality theorem. They proved that extended König’s duality theorem is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0}, and they proved that König’s duality theorem implies ATR_{0} over RCA_{0} [4]. Simpson produced a new proof of König’s duality theorem in ATR_{0} by exploiting the fact that ATR_{0} proves the existence of models of Σ_{1}^{1}-AC_{0} [63]. Therefore König’s duality theorem for countable bipartite graphs is equivalent to ATR_{0} over RCA_{0}.

A priori, Menger’s theorem for countable webs implies ATR_{0} over RCA_{0} because it implies König’s duality theorem for countable bipartite graphs over RCA_{0}. Here we provide a proof of Menger’s theorem for countable webs in Π_{1}^{1}-CA_{0} (Theorem 5.2.4). The general plan for our proof is inspired by Aharoni’s proof in [2] and Diestel’s presentation of it in [21] Section 8.4. As with König’s duality theorem, we notice that this proof in fact proves a stronger statement, which we call extended Menger’s theorem, that is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0}. By general considerations, Menger’s theorem cannot imply Π_{1}^{1}-CA_{0} over RCA_{0}. Menger’s theorem can be written as a Π_{2}^{1} sentence in the language of secondorder arithmetic, and no true Π_{2}^{1} sentence implies Π_{1}^{1}-CA_{0}, even over ATR_{0} (see [4] Proposition 4.17). Question 5.0.3 now becomes more specific.

Question 5.0.4. Is Menger’s theorem for countable webs provable in ATR_{0}?

5.1 Warps, waves, and alternating walks

In this section we use ACA_{0} to develop the basic tools we need to prove Menger’s theorem in Π_{1}^{1}-CA_{0}. Our notation and terminology mostly follows [2] with some ideas borrowed from [21] Section 8.4.

All the graphs that we consider are countable because we are working in second-order arithmetic. All the graphs that we consider are directed. Menger’s theorem for undirected graphs follows from Menger’s theorem for directed graphs by the usual trick of replacing an undirected edge by two directed edges. Henceforth a “graph” is a countable directed graph. If G is a graph and x ∈ V (G), then in-deg_{G}(x) denotes the in-degree of x, and out-deg_{G}(x) denotes the out-degree of x.

As defined above, a web is a triple (G, A, B) where G is a graph and A and B are distinguished sets of vertices A, B ⊆ V (G). We often abuse this notation by writing G for (G, A, B). For convenience, we always assume that there are no edges directed into A, that there are no edges directed out of B, and that A ∩ B = ∅.

If H and H′ are subgraphs of a graph G, then H ∪ H′ is the subgraph of G induced by V (H) ∪ V (H′), and G \ H is the subgraph induced by V (G) \ V (H).

Let G be a graph. If P is a path in G, we write in(P) for the first vertex of P (if it exists) and ter(P) for the last vertex of P (if it exists). If P is a path with in(P) ∈ A and ter(P) ∈ B for some A, B ⊆ V (G), then we call P an A-B path. If P is a path and x ∈ V (P), then P x denotes the subpath of P consisting of all the vertices up to and including x, and \overline{P x} denotes the subpath of P consisting of all the vertices up to and not including x. Similarly, xP denotes the subpath of P consisting of all the vertices following x and including x, and \underline{xP} denotes the subpath of P consisting of all the all the vertices following x and not including x. If P and Q are paths with V (P) ∩ V (Q) = {x}, then P xQ is the path obtained by concatenating the paths P x and xQ. If V (P) ∩ V (Q) = {ter(P)} = {in(Q)}, then P Q denotes P ter(P)Q, the concatenation of the paths P and Q.

For the purposes of this chapter, a (graph-theoretic) tree is a directed acyclic graph T that has a distinguished root r ∈ V (T) such that for any x ∈ V (T) there is a unique path in T from r to x. The path in a tree T from its root to an x ∈ V (T) is denoted T x. If P is a finite path, a tree with trunk P is a tree T of the form P ∪ T′ where T′ is a tree rooted at ter(P). A tree with trunk P has root in(P). If G = (G, A, B) is a web, an A-B tree in G is a subgraph of G that is a tree with root in A and exactly one vertex in B.

Definition 5.1.1. A warp in a web G = (G, A, B) is a subgraph W of G such that

• A ⊆ V (W),

• every x ∈ V (W) has in-deg_{W} (x) ≤ 1 and out-deg_{W} (x) ≤ 1, and

• every x ∈ V (W) is reachable from some a ∈ A by a path in W.

A warp is thus a collection of disjoint paths in G with each path starting at a distinct vertex in A and such that for every a ∈ A there is a path in the warp starting at a. Such paths may be one-way infinite. It is often convenient to think of a warp W as the collection of its component paths {P_{a} | a ∈ A ∧ in(P_{a}) = a} with the understanding that this collection is coded by the set {⟨ a, ⟨ n, x ⟩ ⟩ | x is the n^{th} vertex of P_{a}}. “P is a path in W ” always means that P is one of these component paths.

If W is a warp, then let ter(W) = {x ∈ V (W) | out-deg_{W} (x) = 0}. That is, ter(W) is the set of terminal vertices of the paths in W. The statement “if W is a warp then ter(W) exists” is equivalent to ACA_{0} over RCA_{0}, hence our assumption of ACA_{0} throughout this section.

Definition 5.1.2. A wave in a web G = (G, A, B) is a warp W such that ter(W) is an A-B separator.

The warp {P_{a} | a ∈ A} in which each path P_{a} is the trivial path (a) is always a wave, and we call it the trivial wave.

Definition 5.1.3. For warps W and Y in a web G = (G, A, B), Y is an extension of W (written W ≤ Y) if and only if W is a subgraph of Y.

Definition 5.1.4. If (W_{i} | i ∈ ℕ) is a sequence of warps such that W_{i} ≤ W_{i+1} for each i ∈ ℕ, then ⋃_{i∈ℕ} W_{i} denotes the limit warp defined by V (⋃_{i∈ℕ} W_{i}) = ⋃_{i∈ℕ} V (W_{i}) and E(⋃_{i∈ℕ} W_{i}) = ⋃_{i∈ℕ} E(W_{i}).

It is easy to check in RCA_{0} that a limit warp, if it exists, is indeed a warp. However, the statement “if (W_{i} | i ∈ ℕ) is a sequence of warps such that W_{i} ≤ W_{i+1} for each i ∈ ℕ, then ⋃_{i∈ℕ} W_{i} exists” is equivalent to ACA_{0} over RCA_{0}.

Definition 5.1.5. Let W = {P_{a} | a ∈ A} be a wave in a web G = (G, A, B). Then

• P_{a} is W -essential in G if and only if P_{a} is finite and there is a ter(P_{a})-B path in G disjoint from V (W) \ {ter(P_{a})},

• a ∈ A is W -essential in G if and only if P_{a} is W -essential in G, and

• ess_{G}(W) = {a ∈ A | a is W -essential in G }.

The motivation behind the definition of W -essential in G is that if P is a path in a wave W that is W -essential in G, then ter(W) needs ter(P) to separate A from B. If Q is a ter(P)-B path disjoint from V (W) \ {ter(P)}, then P Q is an A-B path disjoint from ter(W) \ {ter(P)}. One readily checks that {ter(P) | P is a W -essential path in G } is an A-B separator.

Definition 5.1.6. If W and Y are waves in a web G = (G, A, B) with W ≤ Y, then Y is a good extension of W if and only if ess_{G}(W) = ess_{G}(Y), and Y is a bad extension of W if and only if ess_{G}(W) ≠ ess_{G}(Y).

If W and Y are waves in a web G = (G, A, B) with W ≤ Y, then it is always the case that ess_{G}(Y) ⊆ ess_{G}(W). Thus Y is a good extension of W if and only if ess_{G}(W) ⊆ ess_{G}(Y).

Lemma 5.1.7 (in ACA_{0}; see [2] Lemma 2.5). If (W_{i} | i ∈ ℕ) is a sequence of waves in a web G = (G, A, B) such that W_{i} ≤ W_{i+1} for each i ∈ ℕ, then ⋃_{i∈ℕ} W_{i} is a wave in G.

Proof.Let W = ⋃_{i∈ℕ} W_{i}. It is easy to check that W is a warp. We need to show that ter(W) is an A-B separator. Let P be an A-B path, and let X = {⟨ x, i ⟩ | x ∈ V (P) ∩ ter(W_{i})}, which exists by arithmetical comprehension. Each W_{i} is a wave, hence X is infinite. As V (P) is finite, there must be an x ∈ V (P) such that {i | x ∈ ter(W_{i})} is infinite. Then x = ter(Q) for the path Q in W containing x. If not, then there is a vertex following x on Q, the corresponding edge must appear in W_{n} for some n, and so x ∉ ter(W_{i}) for all i ≥ n.

Definition 5.1.8. Let W and Y be warps in a web G = (G, A, B) with W ≤ Y. Let Q be a finite path in both Y and W (i.e., the path Q is in W and is not properly extended in Y). A (Y \ W)-alternating walk from ter(Q) is a walk R = x_{0}e_{0}x_{1}e_{1} ⋯ e_{n−1}x_{n} such that

(i) x_{0} = ter(Q), (ii) for all i ≤ n, x_{i} ∈ (V (G) \ V (W)) ∪ ter(W), (iii) for all i < n, if e_{i} ∉ E(Y), then e_{i} = (x_{i}, x_{i+1}), (iv) for all i < n, if e_{i} ∈ E(Y), then e_{i} = (x_{i+1}, x_{i}) (i.e., R traverses e_{i} backwards),

(v) for all i, j ≤ n with i ≠ j, if x_{i} = x_{j}, then x_{i} ∈ V (Y), and (vi) for all i, j ≤ n with i ≠ j, e_{i} ≠ e_{j}. (vii) for all 0 < i ≤ n, if x_{i} ∈ V (Y), then either e_{i−1} or e_{i} is in E(Y).

Note that if x_{n} is the last vertex on a (Y \ W)-alternating walk from ter(Q) and x_{n} ∈ V (Y), then item (vii) implies that e_{n−1} ∈ E(Y). A (Y \ W)-alternating walk from ter(Q) is similar to a Y -walk as defined in [2] and to a walk which alternates with respect to Y as defined in [21] Section 3.3. The difference is that a (Y \ W)-alternating walk from ter(Q) is not allowed to use the vertices in V (W) \ ter(W), hence the notation “Y \ W.”

Definition 5.1.9. Let W = {P_{a} | a ∈ A} and Y = {Q_{a} | a ∈ A} be warps in a web G = (G, A, B) with W ≤ Y. Let Q_{a_{0}} be a finite path in both Y and W. Then alt_{G}(Y \ W, ter(Q_{a_{0}})) denotes the warp {Q′_{a} | a ∈ A} where Q′_{a} = Q_{a}x if x is the last vertex on Q which lies on a (Y \ W)-alternating walk from ter(Q_{a_{0}}) and Q′_{a} = P_{a} if no such x exists.

Our definition of alt_{G}(Y \ W, ter(Q_{a_{0}})) is analogous to the definition of M(a_{0}, W) in [2]. Also, note that W ≤ alt_{G}(Y \ W, ter(Q_{a_{0}})) ≤ Y. The first inequality is by Definition 5.1.8 item (ii) and the second inequality is clear.

The crucial lemma from this section is Lemma 5.1.12. Lemma 5.1.10 and Lemma 5.1.11 are used to prove Lemma 5.1.12.

Lemma 5.1.10 (in ACA_{0}). Let W and Y be warps in a web G = (G, A, B) with W ≤ Y. Let Q be a finite path in both Y and W. Let R be a (Y \ W)-alternating walk from ter(Q). Then there is a warp Z ≥ W in G with ter(Z) = (ter(Y) \ {ter(Q)}) ∪ {ter(R)}.

Proof.Let R = x_{0}e_{0}x_{1}e_{1} ⋯ e_{n−1}x_{n} where x_{0} = ter(Q) and x_{n} = ter(R). Assume n > 0, for otherwise we may take Z = Y. Let Z′ be the subgraph of G with E(Z′) = E(Y) △ E(R) and V (Z′) = A ∪ {x | (∃e ∈ E(Z′))(x is a vertex of e)}. One readily checks the following equalities.

• If x ∈ V (Y) \ V (R), then in-deg_{Z′}(x) = in-deg_{Y} (x) and out-deg_{Z′}(x) = out-deg_{Y} (x),

• for 0 < i < n, if x_{i} ∈ V (R) \ V (Y), then in-deg_{Z′}(x_{i}) = 1 and out-deg_{Z′}(x_{i}) = 1, • for 0 < i < n, if x_{i} ∈ V (R)∩V (Y) is in V (Z′), then in-deg_{Z′}(x_{i}) = in-deg_{Y} (x_{i})

and out-deg_{Z′}(x_{i}) = out-deg_{Y} (x_{i}),

• in-deg_{Z′}(x_{0}) = in-deg_{Y} (x_{0}) and out-deg_{Z′}(x_{0}) = 1, and

• in-deg_{Z′}(x_{n}) = 1 and out-deg(x_{n}) = 0.

It follows that in-deg_{Z′}(x) ≤ 1 and out-deg_{Z′}(x) ≤ 1 for all x ∈ V (Z′), which means that every component of Z′ is either a path or a cycle. Let Z be the subgraph of Z′ consisting of the component paths of Z′ (i.e., Z is the subgraph of Z′ induced by {x ∈ V (Z′) | x is not on a cycle in Z′ }). Z contains every vertex x ∈ V (Z′) with in-deg_{Z′}(x) = 0 or out-deg_{Z′}(x) = 0. In particular, A ⊆ V (Z) and ter(Z) = (ter(Y) \ {ter(Q)}) ∪ {ter(R)}. To show that Z is a warp, we need only show that in(P) exists and is in A for every path P in Z. The above equations imply that if x ∈ V (P) \ A, then in-deg_{Z}(x) ≠ 0 and hence that x has an immediate predecessor on P. This fact together with the fact that R is finite implies that there is an x ∈ V (P) such that (V (P x) ∩ V (R)) \ A = ∅. Thus the edges of P x must all be edges of Y, which means that P x must be an initial segment of some path in Y. Hence in(P) exists and is in A. Finally, Z ≥ W by Definition 5.1.8 item (ii).

Lemma 5.1.11 (in ACA_{0}; see [2] Lemma 2.7). Let W and Y be waves in a web G = (G, A, B) with W ≤ Y. Let Q be a finite path in both Y and W. Then alt_{G}(Y \ W, ter(Q)) is a wave.

Proof.Let U = alt_{G}(Y \ W, ter(Q)). Suppose for a contradiction that P is an A-B path disjoint from ter(U). W is a wave, so the last vertex on P that is in V (W) must be in ter(W). Let w be this vertex, and let S the path in W with ter(S) = w. The path SwP is an A-B path disjoint from ter(U). Y is a wave, so wP intersects ter(Y), which must happen at a vertex in V (Y) \ V (U). Let y be the first vertex on wP in V (Y) \ V (U). Let z be the last vertex on wP y in V (U), which exists because w ∈ V (U). Claim. There is a (Y \ W)-alternating walk from ter(Q) ending at z.

Proof of claim.Let Q′ be the path in Y containing z. We show that there is a (Y \ W)-alternating walk R from ter(Q) that meets Q′ at a vertex r which is past z on Q′. If r is the first such vertex on R, then RrQ′z (following the edges of Q′ backwards) is the desired walk. If z = w, then Q′ extends S, so if there is no such walk R then by Definition 5.1.9 S is a path in Z which contradicts that P is disjoint from ter(U). On the other hand, if z ≠ w, then z ∉ V (W) by choice of w. As z ∈ V (U) \ V (W) and z ∉ ter(U), again by Definition 5.1.9 it must be the case that some (Y \ W)-alternating walk R from ter(Q) meets Q′ at a vertex past z.

Now let R be the walk provided by the claim, let r be the last vertex of zP y on R, and let y′ be the vertex immediately preceding y on the path in Y containing y. Then RrP y(y′, y)y′ is a (Y \ W)-alternating walk from ter(Q) on which y lies which contradicts y ∉ V (U).

Lemma 5.1.12 (in ACA_{0}; see [2] Lemma 2.8). Let W be a wave in a web G = (G, A, B) that has no bad extensions in G. Let x ∈ V (G)\V (W) be such that there is a wave Y ≥ W in G \ {x} = (G \ {x}, A \ {x}, B \ {x}) with ess_{G\{x}}(Y) ⊊ ess_{G}(W). Then there is a wave Z ≥ W in G with x ∈ ter(Z).

Proof.Let G = (G, A, B), W = {P_{a} | a ∈ A}, x, and Y = {Q_{a} | a ∈ A} be as in the statement of the lemma. Let a_{0} ∈ ess_{G}(W) \ ess_{G\{x}}(Y). If we replace Q_{a_{0}} with P_{a_{0}} in Y, then we retain that this path is not Y -essential in G\{x}. Thus we may assume Q_{a_{0}} = P_{a_{0}}. In particular, Q_{a_{0}} is a finite path that is not Y -essential in G \ {x}. Claim. In G, there is an alternating (Y \ W)-walk from ter(Q_{a_{0}}) ending at x.

Proof of claim.If there is a ter(Q_{a_{0}})-x path disjoint from V (Y) \ {ter(Q_{a_{0}})}, then this path is the desired walk. So suppose instead there is no such path. Let U = alt_{G\{x}}(Y \ W, ter(Q_{a_{0}})). U is a wave in G \ {x} by Lemma 5.1.11. Furthermore, a_{0} ∉ ess_{G\{x}}(Y) implies that a_{0} ∉ ess_{G\{x}}(U) because if P is a ter(Q_{a_{0}})-B path in G \ {x}, then the first vertex on P in V (Y) \ {ter(Q_{a_{0}})} is also in V (U). We prove that U is not a wave in G. To do this, it suffices to show that every ter(Q_{a_{0}})-B path in G intersects V (U)\{ter(Q_{a_{0}})}. Therefore if U were a wave in G, it would be a bad extension of W in G because a_{0} would be in ess_{G}(W) \ ess_{G}(U). This is a contradiction. Consider a ter(Q_{a_{0}})-B path P. If x ∉ V (P), then P is a path in G \ {x} and hence P intersects V (U) \ {ter(Q_{a_{0}})} because a_{0} ∉ ess_{G\{x}}(U). If x ∈ V (P), then by assumption P x intersects V (Y) \ {ter(Q_{a_{0}})}. Again, the first vertex on P in V (Y) \ {ter(Q_{a_{0}})} is also in V (U).

We now know that U is a wave in G \ {x} but not in G. Thus there is an A-B path S in G avoiding ter(U), and x must lie on S. Let z be the last vertex of Sx that is in V (U). It must be that z ∈ ((V (U) \ V (W)) ∪ ter(W)) \ ter(U). Hence there must be an alternating (Y \ W)-walk R from ter(Q_{a_{0}}) to z. Let y be the last vertex of zSx which lies on R. Then RyP x is the desired alternating (Y \ W)-walk from ter(Q_{a_{0}}) to x.

By the claim, let R be an alternating (Y \ W)-walk from ter(Q_{a_{0}}) ending at x. Apply Lemma 5.1.10 to get a warp Z ≥ W in G with ter(Z) = (ter(Y) \ {ter(Q_{a_{0}})})∪{x}. Z is a wave because ter(Y)\{ter(Q_{a_{0}})} is an (A\{x})-(B \{x}) separator in G\{x}, thus (ter(Y)\{ter(Q_{a_{0}})})∪{x} is an A-B separator in G.

5.2 Menger’s theorem in Π_{1}^{1}-CA_{0}

We plan to prove Menger’s theorem as follows. Given a web G = (G, A, B), start with W a ≤-maximal wave in G. Let C be the terminal vertices of the paths in W that are W -essential. Then extend these W -essential paths to be the collection of disjoint A-B paths M. Lemma 5.2.1 provides the ≤-maximal wave W, and Lemma 5.2.2 is the tool we use to extend the W -essential paths to a collection of disjoint A-B paths. The proof of Lemma 5.2.1 is the only argument in which we employ the full strength of Π_{1}^{1}-CA_{0}.

Lemma 5.2.1 (in Π_{1}^{1}-CA_{0}; see [2] Corollary 2.5a). In every web there is a ≤-maximal wave.

Proof.Let G = (G, A, B) be a web, let (g_{n} | n ∈ ℕ) be an enumeration of V (G), and by Theorem 1.6.3 let M be a countable coded β-model with (G, A, B) ∈ M. Using ACA_{0} outside M, we construct a sequence of integers (i_{n} | n ∈ ℕ) such that (M)_{i_{n}} is a wave for each n ∈ ℕ and (M)_{i_{n}} ≤ (M)_{i_{n+1}} for each n ∈ ℕ. Let i_{0} be an index such that (M)_{i_{0}} is the trivial wave {(a) | a ∈ A}. Suppose we have i_{0}, …, i_{n}. If there is an i ∈ ℕ such that (M)_{i} is a wave with (M)_{i} ≥ (M)_{i_{n}} and g_{n} ∈ V ((M)_{i}), then let i_{n+1} be such an i. Otherwise let i_{n+1} = i_{n}. With the desired sequence (i_{n} | n ∈ ℕ) in hand, let W be the limit W = ⋃_{n∈ℕ}(M)_{i_{n}}, which is a wave by Lemma 5.1.7. This W is ≤-maximal in (G, A, B). If not, there is a wave Y ≥ W with some g_{n} ∈ V (Y) \ V (W). As (M)_{i_{n}} ≤ W ≤ Y, at stage n + 1 in the construction the Σ_{1}^{1} formula (∃Y)(Y is a wave ∧ Y ≥ (M)_{i_{n}} ∧ g_{n} ∈ V (Y)) is true and hence is true in M because M is a β-model. Therefore we chose i_{n+1} so that g_{n} ∈ V ((M)_{i_{n+1}}), contradicting g_{n} ∉ V (W).

In [2] and [21], Lemma 5.2.1 is obtained by a simple application of Zorn’s lemma. Our proof above is the most effective proof possible, in the sense that Lemma 5.2.1 is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0} (see Corollary 5.3.3).

The following Lemma 5.2.2 is the key tool used to complete the proof of Menger’s theorem. We first give a proof of Lemma 5.2.2 in the style of ordinary mathematics for the sake of clarity. We then explain how to formalize Lemma 5.2.2 in a way that will allow us to complete the proof of Menger’s theorem in Π_{1}^{1}-CA_{0}.

Lemma 5.2.2 (see [2] Theorem 3.4). Let W be a wave in G = (G, A, B) that has no bad extensions in G, let a ∈ ess_{G}(W), and let P_{a} be the component path of W starting at a. Then there is a finite a-B tree T with trunk P_{a} such that V (T)∩(V (W)\V (P_{a})) = ∅, and there is a wave Y in G \ T such that Y ≥ W \ P_{a}, ess_{G\T} (Y) = ess_{G}(W) \ {a}, and Y has no bad extensions in G \ T.

Proof.We assume that the conclusion of the lemma is false and construct a bad extension of W in G, which is a contradiction.

Let (Q_{n} | n ∈ ℕ) list all the ter(P_{a})-B paths with each path occurring on the list infinitely often. We construct sequences (T_{n} | n ∈ ℕ) and (Y_{n} | n ∈ ℕ) such that for each n ∈ ℕ

(i) T_{n} is a finite tree in G with trunk P_{a}, (ii) Y_{n} is a wave in G \ T_{n}, (iii) T_{n−1} ⊆ T_{n} (if n > 0), (iv) Y_{n−1} ≤ Y_{n} (if n > 0),

(v) ess_{G\T_{n}}(Y_{n}) = ess_{G}(W) \ {a}, and

(vi) Y_{n} has no bad extensions in G \ T_{n}.

Start with T_{0} = P_{a} and Y_{0} = W \ P_{a}. Items (i), (ii), and (v) are easily checked for n = 0. Furthermore, if Y were a bad extension of Y_{0} in G \ T_{0}, then Y ∪ {P_{a}} would be a bad extension of W in G. Hence we have item (vi) for n = 0 as well.

Suppose we have constructed T_{n} and Y_{n}. If V (Q_{n})∩V (Y_{n}) ≠ ∅ or T = T_{n} ∪Q_{n} is not a tree with trunk P_{a}, set T_{n+1} = T_{n} and Y_{n+1} = Y_{n}. Otherwise V (Q_{n}) ∩ V (Y_{n}) = ∅ and T = T_{n} ∪ Q_{n} is a tree with trunk P_{a}. The situation now is that

• T is a finite a-B tree with trunk P_{a} such that V (T) ∩ (V (W) \ V (P_{a})) = ∅ and

• Y_{n} is a wave in G \ T such that Y_{n} ≥ W \ P_{a}.

We are assuming that the lemma is false, so either ess_{G\T} (Y_{n}) ⊊ ess_{G}(W) \ {a} or Y_{n} has a bad extension in G\T. Both cases imply the existence of a wave Y ≥ Y_{n} in G \ T with ess_{G\T} (Y) ⊊ ess_{G}(W) \ {a} = ess_{G\T_{n}}(Y_{n}) (where the equality is by item (v)). Let x be the first vertex on Q_{n} such that there exists a wave Y ≥ Y_{n} in G\(T_{n} ∪Q_{n}x) with ess_{G\(T_{n}∪Q_{n}x)}(Y) ⊊ ess_{G\T_{n}}(Y_{n}) (note x ≠ ter(P_{a}) by item (vi)). Let T_{n+1} = T_{n} ∪ \overline{Q_{n}x}. By the choice of x, Y_{n} has no bad extensions in G \ T_{n+1}, but there is an extension Y ≥ Y_{n} in (G \ T_{n+1}) \ {x} with ess_{(G\T_{n+1})\{x}}(Y) ⊊ ess_{G\T_{n+1}}(Y_{n}). Thus by Lemma 5.1.12 there is a wave Y_{n+1} ≥ Y_{n} in G \ T_{n+1} with x ∈ ter(Y_{n+1}). With this T_{n+1} and Y_{n+1}, items (i)–(iv) are clear for n+1. Item (v) is by the choice of x, which implies that ess_{G\T_{n+1}}(Y_{n+1}) = ess_{G\T_{n}}(Y_{n}) = ess_{G}(W) \ {a}. Item (vi) is again by the choice of x because a bad extension Y ≥ Y_{n+1} in G \ T_{n+1} would be a Y ≥ Y_{n} in G \ T_{n+1} with ess_{G\T_{n+1}}(Y) ⊊ ess_{G\T_{n}}(Y_{n}).

Let T = ⋃_{n∈ℕ} T_{n}, and let Y = ⋃_{n∈ℕ} Y_{n}. By construction, V (Y_{n}) ∩ V (T_{m}) = ∅ for all n, m ∈ ℕ, which means that each Y_{n} is a wave in G \ T. Therefore Y is a wave in G \ T by Lemma 5.1.7. It remains to show that ter(Y) is an A-B separator in G. Our desired contradiction follows because then Y ∪ {P_{a}} would be a bad extension of W in G because a ∈ ess_{G}(W) \ ess_{G}(Y ∪ {P_{a}}).

Let P be an A-B path in G. It suffices to show that there is a final segment S of P that lies in G \ T and intersects V (Y). This is because if x is the last vertex of S in V (Y) and Q is the component path of Y containing x, then QxS is an A-B path in G \ T which means that xS (and hence P) must intersect ter(Y). Thus let x be the last vertex of P on T (if there is no such x, then P is a path in G \ T and thus intersects ter(Y)), let n be such that x ∈ T_{n}, and let m > n be such that Q_{m} = ter(P_{a})T_{n}xP. Consider stage m + 1 of the construction. If V (Q_{m})∩V (Y_{m}) ≠ ∅, then it must be that \underline{xP} intersects Y_{m} and hence intersects Y as desired. Otherwise V (Q_{m}) ∩ V (Y_{m}) = ∅ and T_{m} ∪ Q_{m} is a tree with trunk P_{a}. Thus we choose Y_{m+1} to contain a vertex of \underline{xP}, so Y intersects \underline{xP} as desired.

Lemma 5.2.3 (in ACA_{0}). If M is a countable coded ω-model of Σ_{1}^{1}-DC_{0}, then Lemma 5.2.2 holds in M.

Proof.Consider the formula φ(G, Y, P, x) which says there exists a number z such that

(i) z codes a finite subset of V (P),

∀y(∀Y′(Y′ is a wave ≥ Y in G \ \overline{P y} → ess_{G\\overline{P y}}(Y′) = ess_{G}(Y)) → y ∈ z),

(iii)

∀s(s codes a finite set
∧ ∀y(∃Y′(Y′ is a wave ≥ Y in G \ \overline{P y} ∧ ess_{G\\overline{P y}}(Y′) ⊊ ess_{G}(Y)) → y ∈ s)

→ V (P) \ z ⊆ s), and

(iv) x is the first vertex on P not in z.

Claim. In M, suppose that G = (G, A, B) is a web, Y is a wave in G, and P is a finite path in G disjoint from V (Y) such that Y has no bad extensions in G but there exists a wave Y′ ≥ Y in G \ P with ess_{G\P} (Y′) ⊊ ess_{G}(Y). Then M ⊧ φ(G, Y, P, x) if and only if, in M, x is the first vertex on P such that there exists a wave Y′ ≥ Y in G \ P x with ess_{G\P x}(Y′) ⊊ ess_{G}(Y).

Proof of claim.For the forward direction, using ACA_{0} outside M, let

Z = \{y ∈ V (P) | ess_{G\\overline{P y}}(Y′) = ess_{G}(Y)
for all waves Y′ ≥ Y in G \ \overline{P y} that are in M \}.

Let z be a number coding Z and let s be a number coding V (P) \ Z. In M, z and s code the same sets that they do outside of M, and M interprets that

z codes \{y ∈ V (P) | ess_{G\\overline{P y}}(Y′) = ess_{G}(Y) for all waves Y′ ≥ Y in G \ \overline{P y} \} and
s codes \{y ∈ V (P) | ess_{G\\overline{P y}}(Y′) ⊊ ess_{G}(Y) for some wave Y′ ≥ Y in G \ \overline{P y} \}.

Hence in M, this z is the only z which satisfies items (i)–(iii). Thus if φ(G, Y, P, x) holds in M, x must be the first vertex on P not in z for this z. Thus x must be the first vertex on P such that, in M, there exists a wave Y′ ≥ Y in G \ P x with ess_{G\P x}(Y′) ⊊ ess_{G}(Y).

For the converse, by using ACA_{0} outside of M, let x be the first vertex on P such that, in M, there exists a wave Y′ ≥ Y in G\P x with ess_{G\P x}(Y′) ⊊ ess_{G}(Y). Let z be a number coding V (\overline{P x}). In M, z also codes V (\overline{P x}), and this z witnesses M ⊧ φ(G, Y, P, x).

The reason for the somewhat convoluted definition of φ is that prenexing this φ yields a Σ_{1}^{1} formula.

Suppose for a contradiction that Lemma 5.2.2 is false in M and, in M, let W, G = (G, A, B), and P_{a} be a counterexample to Lemma 5.2.2. We use Σ_{1}^{1}-DC_{0} in M to run the construction from Lemma 5.2.2. This produces in M a bad extension of W in G, which is a contradiction.

We apply Σ_{1}^{1}-DC_{0} to the formula η(n, X, Y) below. Our η has fixed parameters G, W, P_{a}, and (Q_{n} | n ∈ ℕ) (a list of all ter(P_{a})-B paths with each occurring infinitely often). We think of a set Y ⊆ ℕ as coding a pair Y = (tY, wY) where wY is a wave in G \ tY. Formally, tY = (Y)_{0} and wY = (Y)_{1}. Our formula η(n, X, Y) says that if t(X)_{n−1} is the tree and w(X)_{n−1} is the wave constructed at stage n − 1 in Lemma 5.2.2, then tY is the tree and wY is the wave constructed at stage n in Lemma 5.2.2. Formally, η(n, X, Y) says the following.

• If n = 0, then tY = P_{a} and wY = W \ P_{a}.

• If n > 0, if t(X)_{n−1} is a finite tree with trunk P_{a} such that V (t(X)_{n−1}) ∩ (V (W) \ V (P_{a})) = ∅, if w(X)_{n−1} ≥ W \ P_{a} is a wave in G \ t(X)_{n−1}, if ess_{G\t(X)_{n−1}}(w(X)_{n−1}) = ess_{G}(W) \ {a}, and if w(X)_{n−1} has no bad extensions in G \ t(X)_{n−1}, then

– if V (Q_{n−1}) ∩ V (w(X)_{n−1}) ≠ ∅ or if t(X)_{n−1} ∪ Q_{n−1} is not a tree, then tY = t(X)_{n−1} and wY = w(X)_{n−1}, and

– if V (Q_{n−1}) ∩ V (w(X)_{n−1}) = ∅ and if t(X)_{n−1} ∪ Q_{n−1} is a tree, then there is an x such that tY = t(X)_{n−1} ∪ \overline{Q_{n−1}x}, wY is a wave in G \ tY, x ∈ ter(wY), wY ≥ w(X)_{n−1}, and φ(G \ t(X)_{n−1}, w(X)_{n−1}, Q_{n−1}, x).

Prenexing η yields a Σ_{1}^{1} formula. To see this, observe that all the subformulas of η are arithmetic, with the exception of “w(X)_{n−1} has no bad extensions in G \ t(X)_{n−1},” which is Π_{1}^{1} and appears in the antecedent of η, and φ, which is Σ_{1}^{1} and appears in the consequent of η.

We show that M ⊧ ∀n∀X∃Y η(n, X, Y). The interesting case is when n > 0 and X ∈ M is such that, in M,

• t(X)_{n−1} is a finite tree in G with trunk P_{a} such that V (t(X)_{n−1}) ∩ (V (W) \ V (P_{a})) = ∅,

• w(X)_{n−1} ≥ W \ P_{a} is a wave in G \ t(X)_{n−1},

• ess_{G\t(X)_{n−1}}(w(X)_{n−1}) = ess_{G}(W) \ {a},

• w(X)_{n−1} has no bad extensions in G \ t(X)_{n−1},

• V (Q_{n−1}) ∩ V (w(X)_{n−1}) = ∅, and

• t(X)_{n−1} ∪ Q_{n−1} is a tree in G with trunk P_{a}.

By applying ACA_{0} outside M, let x be the first vertex on Q_{n−1} such that, in M, there is a wave Z ≥ w(X)_{n−1} in G \ (t(X)_{n−1} ∪ Q_{n−1}x) with ess_{G\(t(X)_{n−1}∪Q_{n−1}x)}(Z) ⊊ ess_{G\t(X)_{n−1}}(w(X)_{n−1}). Such an x exists by the assumption that G, W, and P_{a} are a counterexample to Lemma 5.2.2 in M. By the claim, φ(G \ t(X)_{n−1}, w(X)_{n−1}, Q_{n−1}, x) holds in M. As M ⊧ ACA_{0}, apply Lemma 5.1.12 inside M to get a wave Z ≥ w(X)_{n−1} in G\(t(X)_{n−1}∪\overline{Q_{n−1}x}) with x ∈ ter(Z). Set tY = t(X)_{n−1} ∪ \overline{Q_{n−1}x} and wY = Z to get a Y ∈ M witnessing ∃Y η(n, X, Y).

Now apply Σ_{1}^{1}-DC_{0} inside M to conclude that M ⊧ ∃Z∀nη(n, (Z)^{n}, (Z)_{n}), and let Z ∈ M be such a Z. By induction (on an arithmetic formula) outside M, verify that, for all n ∈ ℕ, • t(Z)_{n} is a finite tree in G with trunk P_{a} such that V (t(Z)_{n})∩(V (W)\V (P_{a})) =

∅,

• w(Z)_{n} ≥ W \ P_{a} is a wave in G \ t(Z)_{n},

• t(Z)_{n−1} ⊆ t(Z)_{n} (if n > 0),

• w(Z)_{n−1} ≤ w(Z)_{n} (if n > 0),

• ess_{G\t(Z)_{n}}(w(Z)_{n}) = ess_{G}(W) \ {a_{0}},

• w(Z)_{n} has no bad extensions in G \ t(Z)_{n},

and additionally that if V (Q_{n}) ∩ V (w(Z)_{n}) = ∅ and t(Z)_{n} ∪ Q_{n} is a finite tree in G with trunk P_{a}, then ter(w(Z)_{n+1}) contains a vertex of Q_{n}. Inside M, let T = ⋃_{n∈ℕ} t(Z)_{n} and Y = ⋃_{n∈ℕ} w(Z)_{n}. Just as in the proof of Lemma 5.2.2, Y is a wave in G \ T and ter(Y) is an A-B separator in G. Thus Y ∪ {P_{a}} ∈ M is the desired bad extension of W in G, which gives the contradiction.

Theorem 5.2.4. Menger’s theorem for countable webs is provable in Π_{1}^{1}-CA_{0}.

Proof.Let G = (G, A, B) be a countable web. By Lemma 5.2.1, let W = {P_{a} | a ∈ A} be a ≤-maximal wave in G. Let C = {ter(P_{a}) | a ∈ ess_{G}(W)}. We extend the paths in {P_{a} | a ∈ ess_{G}(W)} to be a collection of disjoint A-B paths M. M and C then witness Menger’s theorem for G.

By Theorem 1.6.4, let M be a countable coded ω-model of Σ_{1}^{1}-DC_{0} containing G and W. By Lemma 5.2.3, Lemma 5.2.2 holds in M. Also, M ⊧ “W is a ≤-maximal wave” , therefore M ⊧ “W has no bad extensions in G” because W has no proper extensions in G whatsoever. Let (a_{n} | n ∈ ℕ) enumerate ess_{G}(W). Outside M, we construct sequences (X_{n} | n ∈ ℕ), (Y_{n} | n ∈ ℕ), and (Q_{n} | n ∈ ℕ) such that, for all n ∈ ℕ,

• X_{n} ∈ M, Y_{n} ∈ M, and Q_{n} ∈ M,

• X_{n} ⊆ V (G) is a finite set, X_{n} ∩ A = {a_{i} | i ≤ n}, and X_{n} ⊆ X_{n+1},

• Y_{n} is a wave in G\X_{n} such that Y_{n} ≥ W \⋃_{i≤n} P_{a_{i}}, Y_{n} has no bad extensions in G \ X_{n}, and ess_{G\X_{n}}(Y_{n}) = {a_{i} | i > n}, and

• Q_{n} is an A-B path extending P_{a_{n}}.

To get started, by Lemma 5.2.3 let T ∈ M be a finite a_{0}-B tree in G with trunk P_{a_{0}} such that V (T) ∩ (V (W) \ V (P_{a_{0}})) = ∅, and let Y_{0} ∈ M be a wave in G \ T such that Y_{0} ≥ W \ P_{a_{0}}, ess_{G\T} (Y_{0}) = {a_{i} | i > 0}, and Y_{0} has no bad extensions in G \ T. Let X_{0} = T, and let Q_{0} be the a_{0}-B path in T. Suppose we have X_{n}, Y_{n} and Q_{n}. Let P′_{a_{n+1}} be the path in Y_{n} starting at a_{n+1}, and note that P′_{a_{n+1}} extends P_{a_{n+1}}. By Lemma 5.2.3, let T ∈ M be a finite a_{n+1}-B tree in G \ X_{n} with trunk P′_{a_{n+1}} such that V (T) ∩ (V (Y_{n}) \ V (P′_{a_{n+1}})) = ∅, and let Y_{n+1} ∈ M be a wave in G \ (X_{n} ∪ T) such that Y_{n+1} ≥ Y_{n} \ P′_{a_{n+1}}, ess_{G\(X_{n}∪T)}(Y_{n+1}) = {a_{i} | i > n + 1}, and Y_{n+1} has no bad extensions in G \ (X_{n} ∪ T). Let X_{n+1} = X_{n} ∪ T and let Q_{n+1} be the a_{n+1}-B path in T. In the end, the collection M = {Q_{n} | n ∈ ℕ} consists of disjoint A-B paths in G, and C = {ter(P_{a}) | a ∈ ess_{G}(W)} is an A-B separator containing exactly one vertex from each path in M.

5.3 Extended Menger’s theorem

Although Menger’s theorem for countable webs cannot be equivalent to Π_{1}^{1}-CA_{0} over RCA_{0} as discussed in the introduction, the proof given in Theorem 5.2.4 is equivalent to Π_{1}^{1}-CA_{0} in the sense that it proves a stronger statement, called extended Menger’s theorem, that is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0}. This additional strength comes from our application of Lemma 5.2.1.

Extended Menger’s Theorem. Let (G, A, B) be a countable web. Then there is a set of disjoint A-B paths M and an A-B separator C such that C consists of exactly one vertex from each path in M. Furthermore, C is the set of terminal vertices of the essential paths in a ≤-maximal wave.

Let G be a graph. For x ∈ V (G), N(x) = {y ∈ V (G) | (x, y) ∈ E(G)} denotes the set of neighbors of x. For X ⊆ V (G), D(X) = {y ∈ V (G) | N(y) ⊆ X} denotes the demand of X. In the proof of König’s duality theorem for countable bipartite graphs in [4], the following lemma plays the role that Lemma 5.2.1 plays in the proof of Menger’s theorem for countable webs given in Theorem 5.2.4.

Lemma 5.3.1 ([4] Lemma 3.2). Let (X, Y, E) be a countable bipartite graph. Then there is a ⊆-maximum Y_{0} ⊆ Y for which there is a matching of Y_{0} into D(Y_{0}).

The application of Lemma 5.3.1 yields a stronger form of König’s duality theorem, called extended König’s duality theorem.

Extended König’s Duality Theorem. Let (X, Y, E) be a countable bipartite graph. Then there is a matching M and a cover C such that C consists of exactly one vertex from each edge in E. Furthermore, for every y ∈ Y, y ∈ C if and only if there is a Y_{0} ⊆ Y containing y and a matching of Y_{0} into D(Y_{0}).

Extended König’s duality theorem is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0} by

[4] Theorem 4.18. In fact, Lemma 5.3.1 itself is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0} by [4] Corollary 4.20. In contrast, recall from the introduction that König’s duality theorem is equivalent to ATR_{0} over RCA_{0}. We show that the existence of a ≤-maximal wave, that is, Lemma 5.2.1, implies Lemma 5.3.1 over RCA_{0}. It follows that both Lemma 5.2.1 and extended Menger’s theorem are equivalent to Π_{1}^{1}-CA_{0} over RCA_{0}.

Lemma 5.3.2. Lemma 5.2.1 implies Lemma 5.3.1 over RCA_{0}.

Proof.We prove the lemma in two steps. First, we prove that Lemma 5.2.1 implies ACA_{0} over RCA_{0}. Second, we prove that Lemma 5.2.1 implies Lemma 5.3.1 over ACA_{0}.

First work in RCA_{0}. We use the fact that ACA_{0} is equivalent to the statement “for every injection f: ℕ → ℕ there is a Z ⊆ ℕ such that ∀n(n ∈ Z ↔ ∃m(f(m) = n))” (see [67] Lemma III.1.3). So let f: ℕ → ℕ be an injection. Let (X, Y, E) be the bipartite graph with sides X = {x_{n} | n ∈ ℕ} and Y = {y_{n} | n ∈ ℕ} and edges E = {(x_{m}, y_{n}) | f(m) = n}. Let G be the web G = ((X, Y, E), X, Y), and by Lemma 5.2.1 let W be a ≤-maximal wave in G. Let Z = {n | y_{n} ∈ V (W)}.

We show that ∀n(n ∈ Z ↔ ∃m(f(m) = n)). If f(m) = n, then (x_{m}, y_{n}) is the only edge incident to either x_{n} or y_{n} because f is an injection. Thus the path in W starting at x_{m} is either the trivial path (x_{m}) or the path (x_{m}, y_{n}). If the path is (x_{m}), then the path could be extended to (x_{m}, y_{n}), giving a proper extension of the wave W and contradicting maximality. Thus the path is (x_{m}, y_{n}), hence y_{n} ∈ V (W) and n ∈ Z. Conversely, if n ∈ Z, then y_{n} ∈ V (W) so (x_{m}, y_{n}) must be an edge for some m ∈ ℕ. This can only happen if f(m) = n.

Now work in ACA_{0}. Let (X, Y, E) be a countable bipartite graph. By Lemma 5.2.1, let W be a ≤-maximal wave in the web G = ((X, Y, E), X, Y). Let Y_{0} = Y ∩ ter(W). We show that Y_{0} witnesses Lemma 5.3.1 for (X, Y, E). Let M be the matching consisting of the paths in W of length 1. If y ∈ Y_{0}, then by choice of Y_{0} and M there is an x ∈ X such that (x, y) in M. If x ∉ D(Y_{0}), then there is a y′ ∈ Y \ Y_{0} such that (x, y′) ∈ E. Clearly y′ ∉ ter(W), and x ∉ ter(W) as well because (x, y) is a path in W. Thus (x, y′) is an X-Y path in G avoiding ter(W), contradicting that W is a wave. Therefore M is a matching of Y_{0} into D(Y_{0}).

To see that Y_{0} is ⊆-maximum, suppose for a contradiction that there is a Y′ ⊆ Y and a matching M′ of Y′ into D(Y′) such that Y′ ⊈ Y_{0}. Let W′ be the subgraph of (X, Y, E) with vertices V (W)∪Y′ and edges E(W)∪{(x, y) ∈ M′ | y ∉ Y_{0}}. W is a proper subgraph of W′, so if we can show that W′ is a wave, then we have that W < W′, contradicting the maximality of W. Consider an edge (x, y) ∈ M′ with y ∉ Y_{0}. It must be that x ∈ ter(W) because otherwise (x, y) would be an X-Y path in G avoiding ter(W). It follows that W′ is a warp. To see that ter(W′) is an X-Y separator, consider an edge (x, y) ∈ E. We know ter(W) is an X-Y separator, so either x ∈ ter(W) or y ∈ ter(W). If y ∈ ter(W) then y ∈ ter(W′), so assume x ∈ ter(W). If x ∉ ter(W′), then there must have been an edge (x, y′) ∈ M′ for some y′ ∈ Y′ \ Y_{0}. By assumption, M′ is a matching from Y′ into D(Y′), so x ∈ D(Y′) because M′ matches y′ and x. Therefore y ∈ Y′ because x ∈ D(Y′) and (x, y) is an edge. Clearly Y′ ⊆ ter(W′), so y ∈ ter(W′) as desired.

Corollary 5.3.3. Lemma 5.2.1 is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0}.

Proof.The given proof of Lemma 5.2.1 is in Π_{1}^{1}-CA_{0}. By Lemma 5.3.2, Lemma 5.2.1 implies Lemma 5.3.1 over RCA_{0}. By [4] Corollary 4.20, Lemma 5.3.1 is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0}.

Corollary 5.3.4. Extended Menger’s theorem is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0}.

Proof.Theorem 5.2.4 proves extended Menger’s theorem in Π_{1}^{1}-CA_{0}. Extended Menger’s theorem asserts the existence of a ≤-maximal wave, which is equivalent to Π_{1}^{1}-CA_{0} over RCA_{0} by Corollary 5.3.3.

CHAPTER 6 PARTIAL ANALYSES OF BIRKHOFF’S THEOREM AND OF UNFRIENDLY PARTITIONS

In this chapter, we present a few partial results concerning the reverse mathematics of Birkhoff’s theorem and of unfriendly partitions.

6.1 Countable Birkhoff’s theorem in WKL_{0}

Three classic theorems in finite matching theory are Hall’s theorem, König’s duality theorem, and Birkhoff’s theorem. The reverse mathematics of König’s duality theorem was begun in [4] and was completed in [63]. The theorem is that König’s duality theorem (for countable graphs) is equivalent to ATR_{0} over RCA_{0}. The reverse mathematics of Menger’s theorem, a generalization of König’s duality theorem, was analyzed in the previous chapter.

If G is a graph and S ⊆ V (G), let N(S) = {y ∈ V (G) | (∃s ∈ S)((x, y) ∈ E)} denote the neighbors of S. A bipartite graph (X, Y, E) satisfies Hall’s condition if and only if, for all finite S ⊆ X, |S| ≤ |N(S)|. A bipartite graph (X, Y, E) satisfies symmetric Hall’s condition if and only if, for all finite S ⊆ X and all finite S ⊆ Y, |S| ≤ |N(S)|. The classical versions of Hall’s theorem are the following.

Finite Hall’s Theorem. In every finite bipartite graph (X, Y, E), there is a complete matching of X into Y (i.e., a matching in which every vertex of X is matched) if and only if (X, Y, E) satisfies Hall’s condition.

Finite Symmetric Hall’s Theorem. In every finite bipartite graph (X, Y, E), there is a perfect matching if and only if (X, Y, E) satisfies symmetric Hall’s condition.

Call a graph locally finite if every vertex is has finitely many neighbors. By easy compactness arguments, a locally finite bipartite graph (X, Y, E) has a complete matching of X into Y if and only if it satisfies Hall’s condition, and a locally finite bipartite graph has a perfect matching if and only if it satisfies symmetric Hall’s condition.

Hirst analyzed the reverse mathematics of Hall’s theorem for countable locally finite graphs in [27]. For reference, we summarize Hirst’s results in the following theorems.

Theorem 6.1.1 ([27] Theorem 2.2 and [27] Theorem 3.1). The following are equivalent over RCA_{0}.

(i) ACA_{0}. (ii) Every countable locally finite bipartite graph (X, Y, E) satisfying Hall’s condition

has a complete matching of X into Y. (iii) Every countable locally finite bipartite graph satisfying symmetric Hall’s condition

has a perfect matching.

Call a countable graph (with V (G) = ℕ) bounded if there is a function f: ℕ → ℕ such that ∀x∀y((x, y) ∈ E(G) → y ≤ f(x)). That is, f(x) is a bound on the neighbors of x. Of course a countable graph is locally finite if and only if it is bounded, but it takes ACA_{0} to prove this. In WKL_{0}, Hall’s theorem is provable for bounded graphs, but locally finite graphs are not in general bounded. For k ∈ ℕ, call a graph k-regular if and only if every vertex has exactly k neighbors. Under RCA_{0}, a countable k-regular graph is bounded.

Theorem 6.1.2 ([27] Theorem 2.3 and [27] Theorem 3.3). The following are equivalent over RCA_{0}.

(i) WKL_{0}. (ii) Every countable bounded bipartite graph (X, Y, E) satisfying Hall’s condition has

a complete matching of X into Y. (iii) Every countable bounded bipartite graph satisfying symmetric Hall’s condition

has a perfect matching. (iv) For every k ≥ 1, every countable k-regular bipartite graph has a perfect matching.

(v) Every countable 2-regular bipartite graph has a perfect matching.

We make use of the implication (v) ⇒ (i), and we now provide a proof that is simpler than the original proof in [27].

Proof of Theorem 6.1.2 (v) ⇒ (i). By [67] Lemma IV.4.4, it suffices to show that if f, g: ℕ → ℕ are injections such that ∀m∀n(f(m) ≠ g(n)) then ∃Z∀m(f(m) ∈ Z ∧ g(m) ∉ Z). Thus let f and g be injections such that ∀m∀n(f(m) ≠ g(n)). We build a 2-regular bipartite graph G = (X, Y, E) such that any matching in G determines a set Z such that ∀m(f(m) ∈ Z ∧ g(m) ∉ Z). Let

X = {x_{n} | n ∈ ℕ} ∪ {x_{n,i}^{j} | n, i ∈ ℕ ∧ j < 4}
Y = {y_{n} | n ∈ ℕ} ∪ {y_{n,0}^{0}, y_{n,0}^{1} | n ∈ ℕ} ∪ {y_{n,i}^{j} | n ∈ ℕ ∧ i ≥ 1 ∧ j < 4}
E = {(x_{n}, y_{n,0}^{0}) | n ∈ ℕ} ∪ {(x_{n}, y_{n,0}^{1}) | n ∈ ℕ}
∪ {(x_{n,0}^{2}, y_{n}) | n ∈ ℕ} ∪ {(x_{n,0}^{3}, y_{n}) | n ∈ ℕ}
∪ {(x_{n,i}^{0}, y_{n,i+1}^{0}) | n, i ∈ ℕ} ∪ {(x_{n,i}^{1}, y_{n,i+1}^{1}) | n, i ∈ ℕ}
∪ {(x_{n,i}^{0}, y_{n,i}^{0}) | f(i) ≠ n ∧ g(i) ≠ n} ∪ {(x_{n,i}^{1}, y_{n,i}^{1}) | f(i) ≠ n ∧ g(i) ≠ n}
∪ {(x_{n,i}^{2}, y_{n,i}^{2}) | n ∈ ℕ ∧ i ≥ 1} ∪ {(x_{n,i}^{3}, y_{n,i}^{3}) | n ∈ ℕ ∧ i ≥ 1}
∪ {(x_{n,i}^{2}, y_{n,i+1}^{2}) | f(i) ≠ n ∧ g(i) ≠ n} ∪ {(x_{n,i}^{3}, y_{n,i+1}^{3}) | f(i) ≠ n ∧ g(i) ≠ n}
∪ {(x_{n,i}^{2}, y_{n,i}^{0}) | f(i) = n} ∪ {(x_{n,i}^{3}, y_{n,i}^{1}) | f(i) = n}
∪ {(x_{n,i}^{2}, y_{n,i}^{1}) | g(i) = n} ∪ {(x_{n,i}^{3}, y_{n,i}^{0}) | g(i) = n}
∪ {(x_{n,i}^{0}, y_{n,i+1}^{2}) | f(i) = n ∨ g(i) = n} ∪ {(x_{n,i}^{1}, y_{n,i+1}^{3}) | f(i) = n ∨ g(i) = n}

Clearly G is bipartite. To see that G is 2-regular, we check that every vertex has exactly two neighbors.

• x_{n} is adjacent to y_{n,0}^{0} and y_{n,0}^{1}, and y_{n} is adjacent to x_{n,0}^{2} and x_{n,0}^{3}.

• x_{n,i}^{0} is adjacent to y_{n,i+1}^{0}, and it is adjacent to y_{n,i}^{0} (if f(i) ≠ n ∧ g(i) ≠ n) or to y_{n,i+1}^{2} (otherwise).

• x_{n,i}^{1} is adjacent to y_{n,i+1}^{1}, and it is adjacent to y_{n,i}^{1} (if f(i) ≠ n ∧ g(i) ≠ n) or to y_{n,i+1}^{3} (otherwise).

• x_{n,i}^{2} is adjacent to y_{n} (if i = 0) or y_{n,i}^{2} (if i ≥ 1), and it is adjacent to y_{n,i+1}^{2} (if f(i) ≠ n ∧ g(i) ≠ n) or y_{n,i}^{0} (if f(i) = n) or y_{n,i}^{1} (if g(i) = n).

• x_{n,i}^{3} is adjacent to y_{n} (if i = 0) or y_{n,i}^{3} (if i ≥ 1), and it is adjacent to y_{n,i+1}^{3} (if f(i) ≠ n ∧ g(i) ≠ n) or y_{n,i}^{1} (if f(i) = n) or y_{n,i}^{0} (if g(i) = n).

• y_{n,i}^{0} is adjacent to x_{n} (if i = 0) or x_{n,i−1}^{0} (if i ≥ 1), and it is adjacent to x_{n,i}^{0} (if f(i) ≠ n ∧ g(i) ≠ n) or x_{n,i}^{2} (if f(i) = n) or x_{n,i}^{3} (if g(i) = n).

• y_{n,i}^{1} is adjacent to x_{n} (if i = 0) or x_{n,i−1}^{1} (if i ≥ 1), and it is adjacent to x_{n,i}^{1} (if f(i) ≠ n ∧ g(i) ≠ n) or x_{n,i}^{3} (if f(i) = n) or x_{n,i}^{2} (if g(i) = n).

• y_{n,i}^{2} is adjacent to x_{n,i}^{2}, and it is adjacent to x_{n,i−1}^{2} (if f(i) ≠ n ∧ g(i) ≠ n) or x_{n,i−1}^{0} (otherwise).

• y_{n,i}^{3} is adjacent to x_{n,i}^{3}, and it is adjacent to x_{n,i−1}^{3} (if f(i) ≠ n ∧ g(i) ≠ n) or x_{n,i−1}^{1} (otherwise).

Thus G is a 2-regular bipartite graph. Let M be a perfect matching, and by Δ_{1}^{0} comprehension, let

Z = {n | ((x_{n}, y_{n,0}^{0}) ∈ M ∧(x_{n,0}^{2}, y_{n}) ∈ M) ∨((x_{n}, y_{n,0}^{1}) ∈ M ∧(x_{n,0}^{3}, y_{n}) ∈ M)}.

Suppose f(m) = n. Exactly one of (x_{n}, y_{n,0}^{0}) and (x_{n}, y_{n,0}^{1}) is in M. If (x_{n}, y_{n,0}^{0}) is in M, then so are (x_{n,i}^{0}, y_{n,i+1}^{0}) for 0 ≤ i < m, (x_{n,i}^{2}, y_{n,i}^{2}) for 0 < i ≤ m, and (x_{n,0}^{2}, y_{n}). If (x_{n}, y_{n,0}^{1}) is in M, then so are (x_{n,i}^{1}, y_{n,i+1}^{1}) for 0 ≤ i < m, (x_{n,i}^{3}, y_{n,i}^{3}) for 0 < i ≤ m, and (x_{n,0}^{3}, y_{n}). Hence ((x_{n}, y_{n,0}^{0}) ∈ M ∧(x_{n,0}^{2}, y_{n}) ∈ M) ∨((x_{n}, y_{n,0}^{1}) ∈ M ∧(x_{n,0}^{3}, y_{n}) ∈ M), thus n ∈ Z.

Conversely, suppose g(m) = n. Exactly one of (x_{n}, y_{n,0}^{0}) and (x_{n}, y_{n,0}^{1}) is in M. If (x_{n}, y_{n,0}^{0}) is in M, then so are (x_{n,i}^{0}, y_{n,i+1}^{0}) for 0 ≤ i < m, (x_{n,i}^{3}, y_{n,i}^{3}) for 0 < i ≤ m, and (x_{n,0}^{3}, y_{n}). If (x_{n}, y_{n,0}^{1}) is in M, then so are (x_{n,i}^{1}, y_{n,i+1}^{1}) for 0 ≤ i < m, (x_{n,i}^{2}, y_{n,i}^{2}) for 0 < i ≤ m, and (x_{n,0}^{2}, y_{n}). Hence ((x_{n}, y_{n,0}^{0}) ∈ M ∧(x_{n,0}^{3}, y_{n}) ∈ M) ∨((x_{n}, y_{n,0}^{1}) ∈ M ∧(x_{n,0}^{2}, y_{n}) ∈ M), thus n ∉ Z.

Thus ∀m(f(m) ∈ Z ∧ g(m) ∉ Z) as desired.

We are now ready to discuss Birkhoff’s theorem. For n ∈ ℕ, an n×n matrix of non-negative reals is called doubly stochastic if every row sums to one and every column sums to one. A permutation matrix is a matrix whose entries are either 0 or 1 and such that every row and every column contains exactly one 1. A linear combination λ_{0}P_{0} + λ_{1}P_{1} + ⋯ + λ_{n}P_{n} of permutation matrices P_{0}, P_{1}, …, P_{n} is convex if and only if λ_{i} ≥ 0 for each 0 ≤ i ≤ n and λ_{0} + λ_{1} + ⋯ + λ_{n} = 1.

Finite Birkhoff’s Theorem. Every n × n doubly stochastic matrix is a convex linear combination of n × n permutation matrices.

Birkhoff asked in problem 111 of [12] for an ℕ × ℕ version of this theorem. The solution we consider was provided by Isbell in [29]. Isbell makes the following definitions.

Definition 6.1.3. For an ℕ × ℕ matrix A, let

∥A∥ = max(sup_{i∈ℕ} ∑_{j∈ℕ} |A(i, j)|, sup_{j∈ℕ} ∑_{i∈ℕ} |A(i, j)|).

Definition 6.1.4. A doubly stochastic ℕ × ℕ matrix A satisfies (∗) if and only if for every ϵ > 0 there is an n such that, in any row or column, the sum of the n largest entries is ≥ 1 − ϵ.

Isbell’s solution to Birkhoff’s problem 111, what we call countable Birkhoff’s theorem, is the following.

Countable Birkhoff’s Theorem ([29] Theorem 3). If A is an ℕ × ℕ doubly stochastic matrix satisfying (∗), then for every ϵ > 0 there is a convex linear combination of ℕ × ℕ permutation matrices B such that ∥A − B∥ ≤ ϵ.

Countable Birkhoff’s theorem may be rephrased as follows. In the Banach space of ℕ × ℕ matrices A with ∥A∥ finite and norm ∥·∥, the convex closure of the set of permutation matrices is the set of doubly stochastic matrices satisfying (∗).

The main result of this section is a proof of countable Birkhoff’s theorem in WKL_{0}. In order to give this proof, we first need a version of Theorem 6.1.2 for bipartite multigraphs. The key to approximating a doubly stochastic matrix by a convex combination of permutation matrices is the following fact. Any doubly stochastic matrix consisting of rational entries with a fixed denominator m is a convex linear combination of permutation matrices. In the language of matchings, this says that every m-regular bipartite multigraph is the union of m edge-disjoint perfect matchings. We use multigraphs because p/m at entry (i, j) in the matrix corresponds to p edges (i, j) in the graph. Classically, the fact is proved by applying Hall’s theorem for regular graphs m times. However, iterating Hall’s theorem this way is not formalizable in WKL_{0}. Instead, we use a compactness argument that produces the m matchings simultaneously. To get started, we need the following finite cases. The arguments are standard and are formalizable in RCA_{0}.

Lemma 6.1.5 (in RCA_{0}). Let G = (A, B, E) be a finite bipartite multigraph graph with A_{0} ⊆ A and B_{0} ⊆ B such that (∀S ⊆ A_{0})(|S| ≤ |N(S)|) and (∀S ⊆ B_{0})(|S| ≤ |N(S)|). Then there is a matching (not necessarily perfect) covering A_{0} ∪ B_{0}.

Proof.Induct on ⟨ |A_{0}|, |B_{0}| ⟩. If there are no edges between A_{0} and B_{0}, then we can apply finite Hall’s theorem to A_{0} and B_{0} independently to get a matching of A_{0} into B\B_{0} and a matching of B_{0} into A\A_{0}. The union of these two matchings is a matching covering A_{0}∪B_{0}. This handles the cases where |A_{0}| = 0 or |B_{0}| = 0.

Suppose that |N(S)| ≥ |S| + 1 for all non-empty S ⊆ A_{0} and all non-empty S ⊆ B_{0}. We may assume that there are a ∈ A_{0} and b ∈ B_{0} such that (a, b) is an edge. Let A′ = A \ {a}, A′_{0} = A_{0} \ {a}, B′ = B \ {b}, and B′_{0} = B_{0} \ {b}. Then (∀S ⊆ A′_{0})(|S| ≤ |N(S)|) and (∀S ⊆ B′_{0})(|S| ≤ |N(S)|). By induction, there is a matching covering A′_{0} ∪ B′_{0}. Adding the edge (a, b) gives a matching covering A_{0} ∪ B_{0}.

Suppose instead that |N(S)| = |S| for some non-empty S ⊆ A_{0} (the case where S ⊆ B_{0} is symmetric). Finite Hall’s theorem gives a matching M_{1} of S into N(S), and this matching is perfect because |N(S)| = |S|. Put A′ = A \ S, A′_{0} = A_{0} \ S, B′ = B \ N(S), and B′_{0} = B_{0} \ N(S). Let H = (A′, B′, E ∩ (A′ × B′)). We need to show that |T | ≤ |N_{H}(T)| for all T ⊆ A′_{0} and all T ⊆ B′_{0}. Let T ⊆ A′_{0}. If |N_{H}(T)| < |T | then |N_{G}(T ∪ S)| = |N_{H}(T)| + |N_{G}(S)| < |T | + |S| = |T ∪ S|, which is a contradiction because T ∪ S ⊆ A_{0}. Let T ⊆ B′_{0}. N_{H}(T) = N_{G}(T) because if b ∈ T had a neighbor in S, then b would be in N(S) and hence not in T. Since B′_{0} ⊆ B_{0}, it follows that |T | ≤ |N_{G}(T)| = |N_{H}(T)|. Thus by induction there is a matching M_{2} in H covering A′_{0} ∪ B′_{0}. The matching M_{1} ∪ M_{2} covers A_{0} ∪ B_{0}.

The formula φ(⟨ n, m ⟩) on which we induct says “for every finite bipartite multigraph (A, B, E) and for every A_{0} ⊆ A and B_{0} ⊆ B with |A_{0}| = n and |B_{0}| = m such that (∀S ⊆ A_{0})(|S| ≤ |N(S)|) and (∀S ⊆ B_{0})(|S| ≤ |N(S)|), there is a matching covering A_{0} ∪ B_{0}.” Given a sensible coding of finite multigraphs G = (A, B, E) over ℕ, bounds on the possible codes for subsets of A, B, and E can be calculated from the code for G. Thus the only unbounded quantifier in φ is “for every bipartite multigraph,” hence φ can be written as a Π_{1}^{0} formula. RCA_{0} proves the Π_{1}^{0} induction scheme (see [67] Corollary II.3.10), so the foregoing argument is formalizable in RCA_{0}.

Lemma 6.1.6 (in RCA_{0}). Let G = (A, B, E) be a finite bipartite multigraph with A_{0} ⊆ A and B_{0} ⊆ B such that every vertex of A∪B has multidegree ≤ m and every vertex of A_{0}∪B_{0} has multidegree = m. Then there are m edge-disjoint matchings (not necessarily perfect), each covering A_{0} ∪ B_{0}.

Proof.Induct on m. Suppose m = 1. Let S ⊆ A_{0}. The number of edges incident to S is |S|, so the number of edges incident to N(S) is ≥ |S|. Hence the average multidegree of a vertex in N(S) is ≥ |S|/|N(S)|. If |N(S)| < |S|, then this average is > 1, which means some vertex of B has multidegree > 1, a contradiction. Hence |S| ≤ |N(S)| for all S ⊆ A_{0}, and similarly this holds for all S ⊆ B_{0}. By Lemma 6.1.5, there is a matching covering A_{0} ∪ B_{0}.

Now consider the case m + 1. Remove all edges without an endpoint in A_{0} ∪ B_{0}. If a ∈ A \ A_{0} has m + 1 edges incident to B_{0}, then put a in A_{0}. Similarly, if b ∈ B \ B_{0} has m + 1 edges incident to A_{0}, then put b in B_{0}. In the resulting multigraph, all vertices of A_{0} ∪ B_{0} have multidegree = m + 1, and all vertices of (A \ A_{0}) ∪ (B \ B_{0}) have multidegree ≤ m. Let S ⊆ A_{0}. The number of edges incident to S is (m + 1)|S|, so the number of edges incident to N(S) is ≥ (m + 1)|S|. Hence the average multidegree of a vertex in N(S) is ≥ (m + 1)|S|/|N(S)|. If |N(S)| < |S|, then this average is > m + 1, which means that some vertex of B has multidegree > m + 1. Hence |S| ≤ |N(S)| for all S ⊆ A_{0}, and similarly this holds for all S ⊆ B_{0}. By Lemma 6.1.5, there is a matching M covering A_{0} ∪ B_{0}. Remove the edges of this matching from the multigraph. In the resulting multigraph, all vertices of A_{0} ∪ B_{0} have multidegree = m, and all vertices of A∪B have multidegree ≤ m. By induction there exist m edge-disjoint matchings, each covering A_{0} ∪ B_{0}. Add M to this list and we have m + 1 such matchings.

As in Lemma 6.1.5, the property on which we induct can be written as a Π_{1}^{0} formula, and hence the foregoing argument is formalizable in RCA_{0}.

We can now decompose a countable m-regular bipartite multigraph into m edge-disjoint perfect matchings in WKL_{0}.

Lemma 6.1.7 (in WKL_{0}). Let (X, Y, E) be a countable m-regular bipartite multigraph. Then (X, Y, E) is the union of m edge-disjoint perfect matchings.

Proof.Assume X and Y are copies of ℕ. We will define a tree T ⊆ ((ℕ^{2})^{m})^{<ℕ} ordered by extension. An element of T looks like

⟨ ⟨ ⟨ y_{0}^{0}, x_{0}^{0} ⟩, …, ⟨ y_{0}^{m−1}, x_{0}^{m−1} ⟩ ⟩, …, ⟨ ⟨ y_{n}^{0}, x_{n}^{0} ⟩, …, ⟨ y_{n}^{m−1}, x_{n}^{m−1} ⟩ ⟩ ⟩ .

The idea is that ⟨ y_{0}^{i}, x_{0}^{i} ⟩, …, ⟨ y_{n}^{i}, x_{n}^{i} ⟩ codes the i^{th} partial matching: for k ≤ n, k ∈ X is matched to y_{k}^{i} ∈ Y and k ∈ Y is matched to x_{k}^{i} ∈ X. Thus we define T to be the set of all such sequences with the following properties.

(i) If k ≤ n and i < m then (k, y_{k}^{i}) and (x_{k}^{i}, k) are edges. (ii) If k, l ≤ n and i < m then y_{k}^{i} = l if and only if x_{l}^{i} = k (this guarantees that

the i^{th} partial matching is a matching). (iii) For each k ≤ n, the number of times each y ∈ Y occurs as some y_{k}^{i} for i < m

is ≤ the multiplicity of the edge (k, y), and the number of times each x ∈ X occurs as some x_{k}^{i} for i < m is ≤ the multiplicity of the edge (x, k) (this guarantees that the matchings are disjoint).

One readily proves that T is bounded using the fact that (X, Y, E) is mregular (hence bounded). Thus to apply weak König’s lemma, we need only to show that for each n ∈ ℕ there is a sequence in T of length n. Let A_{0} = {0, …, n − 1} ⊆ X be the first n elements of X and B_{0} = {0, …, n − 1} ⊆ Y be the first n elements of Y. Let N_{1} be the largest neighbor of a vertex in B_{0} and let N_{2} be the largest neighbor of a vertex in A_{0}. Let A = {0, …, N_{1}} ⊆ X, and let B = {0, …, N_{2}} ⊆ Y. By Lemma 6.1.6, there are m edge-disjoint matchings in the finite bipartite multigraph (A, B, E ∩(A×B)), each covering A_{0} ∪B_{0}. Such a sequence of matchings corresponds to an element of the tree of length n. Apply weak König’s lemma to T. Any infinite path through T gives a decomposition of (X, Y, E) into m edge-disjoint perfect matchings.

Lemma 6.1.7 is in fact equivalent to WKL_{0} over RCA_{0}.

Theorem 6.1.8. The following are equivalent over RCA_{0}.

(i) WKL_{0}. (ii) Every countable m-regular bipartite multigraph is the union of m edge-disjoint

perfect matchings.

Proof.Lemma 6.1.7 proves (i) ⇒ (ii). For (ii) ⇒ (i), observe that (ii) implies that every 2-regular bipartite graph has a perfect matching which implies (i) by Theorem 6.1.2.

Theorem 6.1.9. Countable Birkhoff’s theorem is provable in WKL_{0}.

Proof.Let A be an ℕ × ℕ doubly stochastic matrix of reals satisfying (∗). Let ϵ > 0 be rational. By (∗), let n be such that, in any row or column, the sum of the n largest entries is ≥ 1 − ϵ/4. We first do some pre-processing of A in RCA_{0}. Let m = ⌈\frac{n}{ϵ/4}⌉. Define an ℕ × ℕ matrix of rationals C by letting C(i, j) be the largest p/m < A(i, j). Then ∥A − C∥ ≤ ϵ/2 as follows. Fix a row i, and let S be a set of n indices such that ∑_{j∈S} A(i, j) ≥ 1 − ϵ/4. Then

∑_{j∈ℕ}(|A(i, j) − C(i, j)|) = ∑_{j∈S}(|A(i, j) − C(i, j)|) + ∑_{j∉S}(|A(i, j) − C(i, j)|).

By choice of C, C(i, j) < A(i, j) and |A(i, j)−C(i, j)| ≤ 1/m for all i and j. Therefore ∑_{j∈S}(|A(i, j) − C(i, j)|) ≤ n(1/m) ≤ ϵ/4 and ∑_{j∉S}(|A(i, j) − C(i, j)|) ≤ ∑_{j∉S} A(i, j) ≤ ϵ/4 (because ∑_{j∈S} A(i, j) ≥ 1 − ϵ/4). Thus ∑_{j∈ℕ}(|A(i, j) − C(i, j)|) ≤ ϵ/2. Similarly, each of the absolute column sums of A − C is ≤ ϵ/2, so ∥A − C∥ ≤ ϵ/2.

Since each entry of C is of the form p/m, and since C(i, j) < A(i, j) for all i and j, it follows that each row and column of C sums to a rational of the form p/m < 1. Therefore we can recursively define an ℕ × ℕ doubly stochastic matrix B by alternating adding entries of the form 1/m in each row and column of C. It must be that every row and column of C sums to at least 1 − ϵ/2 because ∥A − C∥ ≤ ϵ/2. Therefore at most ϵ/2 is added to each row and column of C to get B, hence ∥C − B∥ ≤ ϵ/2. Thus ∥A − B∥ ≤ ϵ. We use WKL_{0} to show that B is a convex linear combination of permutation matrices.

The matrix B is an ℕ × ℕ doubly stochastic matrix in which every entry is of the form p/m. View B as a bipartite multigraph (X, Y, E) where X = {x_{i} | i ∈ ℕ}, Y = {y_{i} | i ∈ ℕ}, and E contains p copies of the edge (x_{i}, y_{j}) if B(i, j) = p/m. (X, Y, E) is m-regular because B is doubly stochastic. By Lemma 6.1.7, let M_{0}, M_{1}, …, M_{m−1} be m edge-disjoint perfect matchings such that E = ⋃_{k<m} M_{k}. For each k < m, let P_{k} be the permutation matrix corresponding to M_{k} (i.e., P_{k}(i, j) = 1 if and only if (x_{i}, y_{j}) ∈ M_{k}). Then B = (1/m)P_{0} + (1/m)P_{1} + ⋯ + (1/m)P_{m−1} as desired.

Unfortunately we do not have a reversal of Theorem 6.1.9.

Isbell’s solution to Birkhoff’s problem 111 is not the only one. Solutions by Kendall [34] and Révész [50], for example, also deserve reverse mathematical attention.

6.2 Variations on countable Birkhoff’s theorem

In the phrasing of countable Birkhoff’s theorem above, we are given a doubly stochastic ℕ × ℕ matrix A satisfying (∗) and an ϵ > 0, and we ask for a ℕ × ℕ matrix B that is a convex linear combination of permutation matrices such that ∥A−B∥ ≤ ϵ. Instead, we could ask for a sequence of convex linear combinations of permutation matrices converging to A. That is, we could ask for a sequence ⟨ λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n} | n ∈ ℕ ⟩ such that ∥A − (λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n})∥ ≤ 1/n for each n.

Another variation on countable Birkhoff’s theorem is to strengthen (∗).

Definition 6.2.1. If A is an ℕ × ℕ doubly stochastic matrix, a function f: ℕ → ℕ witnesses (∗) for A if and only if, for every n ∈ ℕ, in any row or column the sum of the h(n) largest entries is ≥ 1 − 1/n.

Theorem 6.2.2. The following statement is provable in WKL_{0}. If A is an ℕ × ℕ doubly stochastic matrix and h witnesses (∗) for A, then there is a sequence ⟨ λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n} | n ∈ ℕ ⟩ such that ∥A − (λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n})∥ ≤ 1/n for each n.

Proof.In the proof of Theorem 6.1.9, given ϵ > 0, we found a ℕ × ℕ doubly stochastic matrix B such that ∥A − B∥ ≤ ϵ and every entry of B is a rational of the form p/m for fixed m. Then, using Lemma 6.1.7, we define from B an infinite bounded tree T such that any infinite path of T codes a decomposition of B into a convex linear combination of permutation matrices.

Now, the function h witnessing (∗) for A lets us make the argument from Theorem 6.1.9 uniformly in 1/n. In RCA_{0} we can find a sequence ⟨ B_{n} | n ∈ ℕ ⟩ of ℕ × ℕ doubly stochastic matrices such that every entry of B_{n} is a rational of the form p/m_{n} and ∥A − B_{n}∥ ≤ 1/n for each n. From the sequence ⟨ B_{n} | n ∈ ℕ ⟩, produce a sequence of infinite bounded trees ⟨ T_{n} | n ∈ ℕ ⟩ such that every infinite path of T_{n} codes a decomposition of B_{n} into a convex linear combination of permutation matrices. Then code the sequence of trees ⟨ T_{n} | n ∈ ℕ ⟩ into a single infinite bounded tree T:

(∀σ ∈ ℕ^{<ℕ})(σ ∈ T ↔(∀ ⟨ i, j ⟩ < |σ|)(σ(⟨ i, 0 ⟩)σ(⟨ i, 1 ⟩) ⋯ σ(⟨ i, j ⟩) ∈ T_{i})).

Any infinite path through T codes infinite paths through every T_{n} and hence codes a sequence ⟨ λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n} | n ∈ ℕ ⟩ such that ∥A − (λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n})∥ ≤ 1/n for each n.

Unfortunately we do not have a reversal of Theorem 6.2.2.

If A is an ℕ × ℕ doubly stochastic matrix satisfying (∗), then finding an h witnessing (∗) for A requires ACA_{0}.

Lemma 6.2.3. The following are equivalent over RCA_{0}.

(i) ACA_{0}. (ii) If A is an ℕ × ℕ doubly stochastic matrix satisfying (∗), then there is an h wit-

nessing (∗) for A.

Proof.(i) ⇒ (ii): Let A be an ℕ × ℕ doubly stochastic matrix satisfying (∗). In ACA_{0}, we may define a function f by

f(⟨ n, i ⟩) = the least m such that, for all j ≤ i,

there are m entries in row j that sum to ≥ 1 − 1/n, and there are m entries in column j that sum to ≥ 1 − 1/n .

For each n, lim_{i} f(⟨ n, i ⟩) exists because A satisfies (∗). Again in ACA_{0}, define h(n) = lim_{i} f(⟨ n, i ⟩).

(ii) ⇒ (i): By [67] Lemma III.1.3, it suffices to show that if f: ℕ → ℕ is an injection, then there is a Z ⊆ ℕ such that ∀n(n ∈ Z ↔ ∃m(f(m) = n)). Let f: ℕ → ℕ be an injection. Define a ℕ × ℕ block diagonal matrix

A = (\begin{matrix}C_{0} \\ & C_{1} \\ & & C_{2} \\ & & & . & . & . \end{matrix})

as follows. Each C_{m} will be a finite doubly stochastic matrix, which means that A will be doubly stochastic. If f(m) = n, then C_{m} will witness h(n) ≥ m. Define the matrix C_{m} as follows. Suppose f(m) = n. Let N = max(m, n). Then let

C_{m} = (\begin{matrix}a_{1} & ⋯ & a_{n−1} & b_{1} & ⋯ & b_{ℓ_{m}} & c_{1} & ⋯ & c_{ℓ′_{m}} \\ ⋮ & ... \\ a_{n−1} & & 1 − a_{n−1} \\ b_{1} & & & 1 − b_{1} \\ ⋮ & & & & ... \\ b_{ℓ_{m}} & & & & & 1 − b_{ℓ_{m}} \\ c_{1} & & & & & & 1 − c_{1} \\ ⋮ & & & & & & & ... \\ c_{ℓ′_{m}} & & & & & & & & 1 − c_{ℓ′_{m}} \end{matrix} )

where

• a_{1} = a_{2} = 1/4,

• 1 − (a_{1} + ⋯ + a_{k}) = 1/k for 2 ≤ k ≤ n − 1,

• b_{1} = ⋯ = b_{ℓ_{m}} = 1/ℓ_{m}(n^{2}−n), where ℓ_{m} is least ≥ N such that 1/ℓ_{m}(n^{2}−n) < a_{n−1} (this makes 1 − (a_{1} + ⋯ + a_{n−1} + b_{1} + ⋯ + b_{ℓ_{m}}) = 1/n), and

• c_{1} = ⋯ = c_{ℓ′_{m}} = 1/ℓ′_{m}n, where ℓ′_{m} is least such that 1/ℓ′_{m}n < 1/ℓ_{m}(n^{2} − n).

We show that A satisfies (∗). Let ϵ > 0, and let 1/n < ϵ. The function f is an injection, so there must be a greatest k for which f(k) ≤ n. Let N be the maximum of 2, n, and the maximum size (i.e., number of rows) of C_{0}, C_{1}, …, C_{k}. Then in any row or column of A, the largest N entries sum to at least 1 − 1/n. This is true for any row or column which goes through a block C_{i} for i ≤ k because such rows and columns have at most N non-zero entries. Consider a row or column going through a block C_{i} with i > k. If the row or column contains two non-zero entries which sum to one, then the N largest entries sum to 1. Otherwise, the row or column contains a_{1}, …, a_{f(i)−1}, f(i) > n by choice of k, and a_{1} + ⋯ + a_{n} = 1 − 1/n by construction.

By (ii), let h be a function witnessing (∗) for A. By Δ_{1}^{0} comprehension, let Z = {n | (∃m ≤ h(n))(f(m) = n)}. Clearly if n ∈ Z, then ∃m(f(m) = n). Conversely, suppose that ∃m(f(m) = n). In the first row through block C_{m}, the m largest entries are among a_{1}, …, a_{n−1}, b_{1}, …, b_{ℓ_{m}}, and a_{1} + ⋯ + a_{n−1} + b_{1} + ⋯ + b_{ℓ_{m}} = 1 − 1/n. Hence h(n) ≥ m, so (∃m ≤ h(n))(f(m) = n). Thus ∀n(n ∈ Z ↔ ∃m(f(m) = n)).

The strongest version of countable Birkhoff’s theorem we consider is equivalent to ACA_{0} over RCA_{0}.

Theorem 6.2.4. The following are equivalent over RCA_{0}.

(i) ACA_{0}. (ii) If A is an ℕ × ℕ doubly stochastic matrix satisfying (∗), then there is a sequence

⟨ λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n} | n ∈ ℕ ⟩ such that ∥A − (λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n})∥ ≤ 1/n for each n.

Proof.(i) ⇒ (ii): By Lemma 6.2.3, let h witness (∗) for A. Then apply Theorem 6.2.2.

(ii) ⇒ (i): Let A be a ℕ × ℕ doubly stochastic matrix satisfying (∗). By (ii), let ⟨ λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n} | n ∈ ℕ ⟩ be such that ∥A − (λ_{0}^{n}P_{0}^{n} + ⋯ + λ_{m_{n}}^{n}P_{m_{n}}^{n})∥ ≤ 1/n for each n. Let h(n) = m_{n}. Then h witnesses (∗) for A, so (i) follows from Lemma 6.2.3.

6.3 Reverse mathematics and unfriendly partitions

A partition V (G) = X_{0}∪X_{1} of the vertices of a graph G is unfriendly if and only if (∀i < 2)(∀v ∈ X_{i})(|N(v)∩X_{i}| ≤ |N(v)∩X_{1−i}|). That is, every vertex in the graph has at least as many neighbors outside its side of the partition as inside its side of the partition. Every finite graph has an unfriendly partition: simply choose any partition that maximizes the number of edges that have an endpoint in each side of the partition. It follows by compactness that every locally finite graph has an unfriendly partition. Shelah and Milner give an example of a graph that has no unfriendly partition [58]. The cardinality of their example is the ω^{th} successor of the continuum. Amazingly, it is not known whether every countable graph has an unfriendly partition. The assertion that every countable graph has an unfriendly partition is now known as the unfriendly partition conjecture.

Aharoni, Milner, and Prikry show that a graph with only finitely many vertices of infinite degree has an unfriendly partition [5]. They also show that if G is a graph for which there exist a finite number of infinite cardinals κ_{0} < κ_{1} < ⋯ < κ_{k} with κ_{i} regular for 1 ≤ i ≤ k such that < κ_{0} vertices have finite degree and every vertex with infinite degree has degree κ_{i} for some 0 ≤ i ≤ k, then G has an unfriendly partition. More recently, Bruhn, Diestel, Georgakopoulos, and Sprüssel show that every graph with no infinite path has an unfriendly partition [13]. Although an infinite graph may not have an unfriendly partition, Shelah and Milner also show that every graph, regardless of cardinality, has an unfriendly 3-partition, that is, a partition V (G) = X_{0}∪X_{1}∪X_{2} such that (∀i < 3)(∀v ∈ X_{i})(|{u ∈ N(v) | u ∈ X_{i}}| ≤ |{u ∈ N(v) | u ∉ X_{i}}|) [58].

The unfriendly partition conjecture motivates our study of the reverse mathematics of unfriendly partitions. We present some initial results.

Theorem 6.3.1. The following are equivalent over RCA_{0}.

(i) ACA_{0}. (ii) Every locally finite graph has an unfriendly partition.

Proof.(i) ⇒ (ii): Let G = (V, E) be a locally finite graph and enumerate V as {v_{i} | i ∈ ℕ}. By arithmetical comprehension, let f: ℕ → ℕ be such that (∀i)((v_{i}, v_{j}) ∈ E → j < f(i)). Define a tree T ⊆ 2^{<ℕ} by

σ ∈ T ⇔(∀i < |σ|)(|σ| ≤ f(i) →

|{j < |σ| | v_{j} ∈ N(v_{i}) ∧ σ(j) = σ(i)}| ≤ |{j < |σ| | v_{j} ∈ N(v_{i}) ∧ σ(j) ≠ σ(i)}|).

The fact that every finite graph has an unfriendly partition implies that T is infinite. Apply König’s lemma to T. Any infinite path through T codes an unfriendly partition of G.

(ii) ⇒ (i): By [67] Lemma III.1.3, it suffices to show that if f: ℕ → ℕ is an injection, then there is a Z ⊆ ℕ such that ∀n(n ∈ Z ↔ ∃m(f(m) = n)). Let f: ℕ → ℕ be an injection. By Δ_{1}^{0} comprehension, let G = (V, E) be the graph where

V = {a_{n}, b_{n}, c_{n}, d_{n}, e_{n} | n ∈ ℕ}
E = {(a_{n}, b_{n}), (c_{n}, d_{n}), (d_{n}, e_{n}) | n ∈ ℕ}
∪ {(a_{n}, c_{m}), (a_{n}, e_{m}), (b_{n}, c_{m}), (b_{n}, e_{m}) | f(m) = i}.

By (ii), let X_{0} ∪ X_{1} be an unfriendly partition of V. By Δ_{1}^{0} comprehension, let Z = {n ∈ ℕ | (a_{n} ∈ X_{0} ∧ b_{n} ∈ X_{0}) ∨(a_{n} ∈ X_{1} ∧ b_{n} ∈ X_{1})}. If ¬∃m(f(m) = n), then a_{n} and b_{n} are only adjacent to each other and so must be in opposite sides of the partition. Hence n ∉ Z. Conversely, suppose f(m) = n, and suppose for a contradiction that a_{n} and b_{n} are in opposite sides of the partition. Say a_{n} ∈ X_{0} and b_{n} ∈ X_{1}. The set {a_{n}, b_{n}, c_{m}, d_{m}, e_{m}} is a connected component because f is an injection. Vertices c_{m} and e_{m} cannot both be in X_{0} because otherwise a_{n} has two neighbors in X_{0} and only one neighbor in X_{1}. Similarly, c_{m} and e_{m} cannot both be in X_{1}. Thus one of c_{m} and e_{m} is in X_{0} and the other is in X_{1}. Say c_{m} ∈ X_{0} and e_{m} ∈ X_{1}. Then d_{m} cannot be in X_{0} because this would give c_{m} two neighbors (a_{n} and d_{m}) in X_{0} and only one neighbor (b_{n}) in X_{1}. Similarly, d_{m} cannot be in X_{1}. This contradicts the fact that X_{0} ∪ X_{1} is an unfriendly partition of V. Thus a_{n} and b_{n} are either both in X_{0} or both in X_{1}. Hence n ∈ Z.

We see from the proof of Theorem 6.3.1 (i) ⇒ (ii) that WKL_{0} proves that every bounded graph has an unfriendly partition. In that proof, ACA_{0} is used to produce the bounding function f, and the rest of the proof goes through in WKL_{0}. Unfortunately we do not have a reversal for the bounded version of Theorem 6.3.1.

Shelah and Milner’s method of producing unfriendly 3-partitions in [58] goes through in ACA_{0} in the countable case. It is convenient to make the following definition.

Definition 6.3.2. Let G = (V, E) be a graph. A function f: V → ℕ is unfriendly at v ∈ V if and only if |{u ∈ N(v) | f(u) = f(v)}| ≤ |{u ∈ N(v) | f(u) ≠ f(v)}|. A function f: V →{0, 1, 2} is an unfriendly 3-partition of G if and only if f is unfriendly at every v ∈ V.

Lemma 6.3.3 (in ACA_{0}). Let G = (V, E) be a graph. Let A, B ⊆ V be such that A∩B = ∅ and (∀v ∈ A)(N(v)∩(A∪B) is finite). Then any function g: B →{0, 1, 2} can be extended to a function h: A ∪ B →{0, 1, 2} such that h(A) ⊆ {0, 1} and h is unfriendly at every v ∈ A in the subgraph induced by A ∪ B.

Proof.First assume that A is finite. Choose g′: A →{0, 1} so that h = g ∪ g′ maximizes the number n = |{(v, u) ∈ E | v ∈ A ∧ u ∈ A ∪ B ∧ h(v) ≠ h(u)}|. Then h is unfriendly at each v ∈ A in the subgraph induced by A ∪ B. If not, then there is a v ∈ A such that

|{u ∈ N(v) ∩ (A ∪ B) | h(u) = h(v)}| > |{u ∈ N(v) ∩ (A ∪ B) | h(u) ≠ h(v)}|.

Thus we could increase n by replacing g′(v) with 1 − g′(v), contradicting that h maximizes n.

If A is infinite, let A = {a_{i} | i ∈ ℕ}, and let A_{n} = {a_{i} | i < n}. For σ ∈ 2^{<ℕ} with |σ| = n, define g′_{σ}: A_{n} →{0, 1} by (∀i < n)(g′_{σ}(a_{i}) = σ(i)). Let h_{σ} = g ∪ g′_{σ}. By arithmetical comprehension, define a tree T ⊆ 2^{<ℕ} by

σ ∈ T ⇔(∀i < |σ|)(N(a_{i}) ∩ A ⊆ A_{|σ|} →
h_{σ} is unfriendly at a_{i} in the subgraph induced by A_{|σ|} ∪ B ).

The finite case implies that T is infinite, so by König’s lemma let p: ℕ →{0, 1} be a path through T. Define f: A →{0, 1} by f(a_{n}) = p(n) for each n. We show that h = g∪f is unfriendly at every a_{n} ∈ A in the subgraph induced by A∪B. Let a_{n} ∈ A, and let m be large enough so that N(a_{n}) ∩ A ⊆ A_{m}. Let σ = ⟨ p(0), …, p(m) ⟩. Then h_{σ} is unfriendly at a_{n} in the subgraph induced by A_{m} ∪ B because σ ∈ T. However, N(a_{n}) ∩ (A ∪ B) ⊆ N(a_{n}) ∩ (A_{m} ∪ B), and h agrees with h_{σ} on A_{m} ∪ B. Thus h is unfriendly at a_{n} in the subgraph induced by A ∪ B.

Theorem 6.3.4. The following statement is provable in ACA_{0}. Every graph has an unfriendly 3-partition.

Proof.Let G = (V, E) be a graph. By arithmetical comprehension, let X = {v ∈ V | N(v) is finite}, and let Y = {v ∈ V | N(v) ∩ X is infinite}. Let Z = V \ (X ∪ Y). Define g: Y ∪ Z →{0, 1, 2} by setting g(Y) = 2 and by defining g(Z) ⊆ {0, 1} by the following recursive procedure. Let Z = {z_{i} | i ∈ ℕ}. Initialize an empty priority queue. For n = 0, 1, 2, … do the following. If z_{n} is not a neighbor any z in the queue, then set g(z_{n}) = 0. Otherwise, let z be the highest priority neighbor of z_{n}. Set g(z_{n}) = 1 − g(z). If z is at the front of the queue (i.e., has highest priority), then move z to the back of the queue. If N(z_{n}) ∩ Y is finite, then add z_{n} to the back of the queue. This completes the definition of g.

By Lemma 6.3.3, let h be a function extending g such that h(X) ⊆ {0, 1} and that h is unfriendly at every v ∈ X. We show that h is an unfriendly 3-partition of G. We already know that h is unfriendly at every v ∈ X. The function h is also unfriendly at every v ∈ Y because h(v) = 2 for every v ∈ Y and h(u) is either 0 or 1 for every u ∈ X. Let z ∈ Z. If N(z) ∩ Y is infinite, then h is unfriendly at z because h(z) ∈ {0, 1} and h(Y) = 2. Suppose N(z) ∩ Y is finite. Then N(z) ∩ Z must be infinite. Every time z reaches the front of the queue, we eventually define g(z′) ≠ g(z) for some neighbor of z′ of z. Thus it suffices to show that z reaches the front of the queue infinitely many times. If not, then z never reaches the front of the queue after some step n. Since the priority of z either increases or stays the same at each step after n, there is some step m after which the priority of z never changes. This contradicts the fact that the vertex with priority one at step m has infinitely many neighbors in Z. Thus h is our desired unfriendly partition.

We do not have a reversal for Theorem 6.3.4, but we can prove that RCA_{0} is not strong enough to produce unfriendly 3-partitions.

Theorem 6.3.5. There is a recursive locally finite graph with no recursive unfriendly 3-partition. It follows that the statement “every graph has an unfriendly 3-partition” is not provable in RCA_{0}.

Proof.Our graph G = (V, E) has V = ℕ, and we construct E in stages ⟨ e, s ⟩. For fixed e, in stages ⟨ e, s ⟩ we diagonalize against Φ_{e} in the e^{th} column of ℕ. This diagonalization occurs in three phases. Together phases I and II force Φ_{e} to take on the values 0, 1, and 2 four times each. Phase III consists of several subphases which finish the diagonalization. Call column e good at stage ⟨ e, s ⟩ if the diagonalization succeeds at this stage. Once column e is good at stage ⟨ e, s ⟩, the construction is finished in the e^{th} column and we skip all subsequent stages ⟨ e, t ⟩.

Initially, E has edge (⟨ e, i, 0, 0 ⟩, ⟨ e, i, 1, 0 ⟩) for each e and for each i = 0, 1, 2, 3. Each column begins in phase I. At stage ⟨ e, s ⟩ do the following.

• If column e is in phase I, then do the following.

– If Φ_{e,s}(⟨ e, i, j, 0 ⟩)↑ for some i = 0, 1, 2, 3 and some j = 0, 1, then go to the next stage.

– Otherwise, column e is good at ⟨ e, s ⟩ if either of the following conditions are met:

(i) Φ_{e,s}(⟨ e, i, j, 0 ⟩) > 2 for some i = 0, 1, 2, 3 and some j = 0, 1, or (ii) Φ_{e,s}(⟨ e, i, 0, 0 ⟩) = Φ_{e,s}(⟨ e, i, 1, 0 ⟩) for some i = 0, 1, 2, 3.

If column e is not good, then add edges (⟨ e, i, j, s ⟩, ⟨ e, i, k, 0 ⟩) for i = 0, 1, 2, 3, j = 2, 3, and k = 0, 1, and add edges (⟨ e, i, 4, s ⟩, ⟨ e, i, j, s ⟩) for i = 0, 1, 2, 3 and j = 2, 3. Enter phase II.

• If column e is in phase II, then let ⟨ e, t ⟩ be the stage at which column e entered phase II and do the following.

– If Φ_{e,s}(⟨ e, i, j, t ⟩)↑ for some i = 0, 1, 2, 3 and some j = 2, 3, 4, then go to the next stage.

– Otherwise, column e is good at ⟨ e, s ⟩ if either of the following conditions are met:

(i) Φ_{e,s}(⟨ e, i, j, t ⟩) > 2 for some i = 0, 1, 2, 3 and some j = 2, 3, 4, or (ii) for some i = 0, 1, 2, 3, Φ_{e,s} takes on only two values on the five

vertices ⟨ e, i, 0, 0 ⟩, ⟨ e, i, 1, 0 ⟩, ⟨ e, i, 2, t ⟩, ⟨ e, i, 3, t ⟩, and ⟨ e, i, 4, t ⟩. If column e is not good, then add edge (⟨ e, 4, 0, s ⟩, ⟨ e, 4, 1, s ⟩). Enter phase III.

• Phase III consists of sixteen subphases and begins in subphase 1. Let ⟨ e, r ⟩ be the stage at which column e entered phase III, and do the following.

– If column e is in phase III, subphase 1, do the following.

∗ If Φ_{e,s}(⟨ e, 4, j, r ⟩)↑ for some j = 0, 1, then go to the next stage.

∗ Otherwise, column e is good at ⟨ e, s ⟩ if either of the following conditions are met:

(i) Φ_{e,s}(⟨ e, 4, j, r ⟩) > 2 for some j = 0, 1, or (ii) Φ_{e,s}(⟨ e, 4, 0, r ⟩) = Φ_{e,s}(⟨ e, 4, 1, r ⟩). If column e is not good, then Φ_{e,s}(⟨ e, 4, 0, r ⟩) = x ≤ 2 and Φ_{e,s}(⟨ e, 4, 1, r ⟩) = y ≤ 2, where x ≠ y. Let z ≤ 2 be the number not equal to x or y. For each i ≤ 3, let b_{i} be the least of ⟨ e, i, 0, 0 ⟩, ⟨ e, i, 1, 0 ⟩, ⟨ e, i, 2, t ⟩, ⟨ e, i, 3, t ⟩, and ⟨ e, i, 4, t ⟩ such that Φ_{e,s}(b_{i}) = z, where t is such that column e entered phase II at stage ⟨ e, t ⟩ (the b_{i} exist by the fact that column e was not good in phase II). Add edges (⟨ e, 4, j, s ⟩, ⟨ e, 4, k, r ⟩) for j = 2, 3 and k = 0, 1, and add edges (⟨ e, 4, j, s ⟩, b_{i}) for j = 2, 3 and i = 0, 1, 2. Enter subphase 2.

– If column e is in phase III, subphase 2, do the following. Let ⟨ e, t ⟩ be the stage at which column e entered subphase 2.

∗ If Φ_{e,s}(⟨ e, 4, j, t ⟩)↑ for some j = 2, 3, then go to the next stage.

∗ Otherwise, column e is good at ⟨ e, s ⟩ if any of the following conditions are met:

(i) Φ_{e,s}(⟨ e, 4, j, t ⟩) > 2 for some j = 2, 3, (ii) Φ_{e,s}(⟨ e, 4, j, t ⟩) = z for some j = 2, 3 (where z is as in subphase

1), or (iii) Φ_{e,s}(⟨ e, 4, 2, t ⟩) = Φ_{e,s}(⟨ e, 4, 3, t ⟩). If column e is not good, then Φ_{e,s} is x on one of the two vertices ⟨ e, 4, 2, t ⟩ and ⟨ e, 4, 3, t ⟩, and Φ_{e,s} is y on the other vertex. Let v_{x} be the vertex on which Φ_{e,s} is x, and let v_{y} be the vertex on which Φ_{e,s} is y. Add edges (⟨ e, 4, 4, s ⟩, ⟨ e, 4, k, r ⟩) for k = 0, 1 and edges (⟨ e, 4, 4, s ⟩, b_{i}) for i = 0, 1, 2. Enter subphase 3.

– If column e is in phase III, subphase 2n + 1 for 1 ≤ n ≤ 7, then do the following. Let ⟨ e, t ⟩ be the stage at which column e entered subphase 2n + 1.

∗ If Φ_{e,s}(⟨ e, 4, 2n + 2, t ⟩)↑ then go to the next stage.

∗ Otherwise, column e is good at ⟨ e, s ⟩ if either of the following conditions are met:

(i) Φ_{e,s}(⟨ e, 4, 2n + 2, t ⟩) > 2, or (ii) Φ_{e,s}(⟨ e, 4, 2n + 2, t ⟩) = z. If column e is not good, then add edges (⟨ e, 4, 2n + 3, s ⟩, ⟨ e, 4, k, r ⟩) for k = 0, 1 and edges (⟨ e, 4, 2n+3, s ⟩, b_{i}) for i = 0, 1, 2, 3. Additionally, if Φ_{e,s}(⟨ e, 4, 2n + 2, t ⟩) = x, then add edge (⟨ e, 4, 2n + 3, s ⟩, v_{y}), and if Φ_{e,s}(⟨ e, 4, 2n + 2, t ⟩) = y, then add edge (⟨ e, 4, 2n + 3, s ⟩, v_{x}). Enter subphase 2n + 2.

– If column e is in phase III, subphase 2n + 2 for 1 ≤ n ≤ 7, then do the following. Let ⟨ e, t ⟩ be the stage at which column e entered subphase 2n + 2, and let ⟨ e, p ⟩ be the stage at which column e entered subphase 2n + 1.

∗ If Φ_{e,s}(⟨ e, 4, 2n + 3, t ⟩)↑, then go to the next stage.

∗ Otherwise, column e is good at ⟨ e, s ⟩ if any of the following conditions are met:

(i) Φ_{e,s}(⟨ e, 4, 2n + 3, t ⟩) > 2, (ii) Φ_{e,s}(⟨ e, 4, 2n + 3, t ⟩) = z, (iii) Φ_{e,s}(⟨ e, 4, 2n + 3, t ⟩) = Φ_{e,s}(⟨ e, 4, 2n + 2, p ⟩), or (iv) this is subphase 16. If column e is not good, then add edges (⟨ e, 4, 2n + 4, s ⟩, ⟨ e, 4, k, r ⟩) for k = 0, 1, and add edges (⟨ e, 4, 2n + 4, s ⟩, b_{i}) for i = 0, 1, 2. Enter subphase 2n + 3.

This completes the construction of G. Suppose Φ_{e} is a recursive function ℕ → {0, 1, 2}. We show that Φ_{e} is not an unfriendly 3-partition of G.

Φ_{e} is total, so the construction in column e will progress through the phases until column e becomes good. Suppose column e becomes good at stage ⟨ e, s ⟩. If this stage is in phase I, then Φ_{e}(⟨ e, i, 0, 0 ⟩) = Φ_{e}(⟨ e, i, 1, 0 ⟩) for some i = 0, 1, 2, 3. These two vertices only have each other as neighbors, so Φ_{e} is not an unfriendly partition.

If column e is good at stage ⟨ e, s ⟩ in phase II, then, for some i = 0, 1, 2, 3, Φ_{e} uses only two values on the five vertices ⟨ e, i, 0, 0 ⟩, ⟨ e, i, 1, 0 ⟩, ⟨ e, i, 2, t ⟩, ⟨ e, i, 3, t ⟩, ⟨ e, i, 4, t ⟩ (where t is such that column e entered phase II at stage ⟨ e, t ⟩), and Φ_{e}(⟨ e, i, 0, 0 ⟩) ≠ Φ_{e}(⟨ e, i, 1, 0 ⟩). In this case these five vertices have no other neighbors, and one checks that Φ_{e} cannot be an unfriendly partition of the subgraph induced by these vertices.

If column e is good at stage ⟨ e, s ⟩ in phase III, subphase 1, then Φ_{e}(⟨ e, 4, 0, r ⟩) = Φ_{e}(⟨ e, 4, 1, r ⟩) (where r is such that column e entered phase III, subphase 1 at stage ⟨ e, r ⟩). These two vertices only have each other as neighbors, so Φ_{e} is not an unfriendly partition.

If column e is good at stage ⟨ e, s ⟩ in phase III, subphase 2, then either Φ_{e}(⟨ e, 4, j, t ⟩) = z for some j = 2, 3, or Φ_{e}(⟨ e, 4, 2, t ⟩) = Φ_{e}, (⟨ e, 4, 3, t ⟩) (where t is such that column e entered phase III, subphase 2 at stage ⟨ e, t ⟩). If Φ_{e}(⟨ e, 4, j, t ⟩) = z then ⟨ e, 4, j, t ⟩ has five neighbors, and Φ_{e} is z on three of these neighbors. Hence Φ_{e} is not an unfriendly partition. If Φ_{e}(⟨ e, 4, 2, t ⟩) = Φ_{e}(⟨ e, 4, 3, t ⟩) ≠ z, then the value of Φ_{e} on these two vertices is the same as the value of Φ_{e} on ⟨ e, 4, j, r ⟩ for either j = 0 or j = 1 (where r is such that column e entered phase III, subphase 1 at stage ⟨ e, r ⟩). For the witnessing j, ⟨ e, 4, j, r ⟩ has three neighbors, and Φ_{e} agrees with Φ_{e}(⟨ e, 4, j, r ⟩) on two of these neighbors. Hence Φ_{e} is not an unfriendly partition.

If column e is good at stage ⟨ e, s ⟩ in phase III, subphase 2n + 1, then Φ_{e}(⟨ e, 4, 2n + 2, t ⟩) = z (where t is such that column e entered phase III, subphase 2n + 1 at stage ⟨ e, t ⟩). In this case ⟨ e, 4, 2n + 2, t ⟩ has five neighbors, and Φ_{e} is z on three of these neighbors. Thus Φ_{e} is not an unfriendly partition.

If column e is good at stage ⟨ e, s ⟩ in phase III, subphase 2n + 2, then either Φ_{e}(⟨ e, 4, 2n + 3, t ⟩) = z, Φ_{e}(⟨ e, 4, 2n + 3, t ⟩) = Φ_{e}(⟨ e, 4, 2n + 2, p ⟩), or this is subphase 16 (where t and p are such that column e entered phase III, subphase 2n + 2 at stage ⟨ e, t ⟩ and entered phase III, subphase 2n + 1 at stage ⟨ e, p ⟩). If Φ_{e}(⟨ e, 4, 2n + 3, t ⟩) = z, then ⟨ e, 4, 2n + 3, t ⟩ has seven neighbors, and Φ_{e} is z on four of these neighbors. Hence Φ_{e} is not an unfriendly partition. If Φ_{e}(⟨ e, 4, 2n+3, t ⟩) = Φ_{e}(⟨ e, 4, 2n+2, p ⟩), then for either j = 0 or j = 1 the vertex ⟨ e, 4, j, r ⟩ has 2n + 3 neighbors, and Φ_{e} agrees with Φ_{e}(⟨ e, 4, j, r ⟩) on n + 2 of these neighbors (where r is such that column e entered phase III, subphase 1 at stage ⟨ e, r ⟩). Hence Φ_{e} is not an unfriendly partition. If this is subphase 16, and Φ_{e}(⟨ e, 4, 2n + 3, t ⟩) is neither z nor Φ_{e}(⟨ e, 4, 2n + 2, p ⟩), then either Φ_{e} is x on over half of the neighbors of v_{x}, or Φ_{e} is y on over half of the neighbors of v_{y}. In either case Φ_{e} is not an unfriendly partition.

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