Math. Res. Lett. 18 (2011), no. 00, 10001–10014 \begin{matrix}c \\ ◯ \end{matrix} International Press 2011

THERE MAY BE NO MINIMAL NON σ-SCATTERED LINEAR ORDERS

Hossein Lamei Ramandi and Justin Tatch Moore

Abstract. In this paper we demonstrate that it is consistent, relative to the existence of a supercompact cardinal, that there is no linear order which is minimal with respect to being non σ-scattered. This shows that a theorem of Laver, which asserts that the class of σ-scattered linear orders is well quasi-ordered, is sharp. We also prove that PFA^{+} implies that every non σ-scattered linear order either contains a real type, an Aronszajn type, or a ladder system indexed by a stationary subset of ω_{1}, equipped with either the lexicographic or reverse lexicographic order. Our work immediately implies that CH is consistent with “no Aronszajn tree has a base of cardinality ℵ_{1}.” This gives an affirmative answer to a problem due to Baumgartner.

1. Introduction

In [9], Laver verified a longstanding conjecture of Fraïssé: the countable linear orders are well quasi-ordered by embeddability. That is to say if L_{i} (i < ∞) is an infinite sequence of countable linear orderings, then there is an i < j such that L_{i} is embeddable into L_{j}. In fact, Laver proved the following stronger result.

Theorem 1.1. [9] The class ℳ of σ-scattered linear orders is well quasi-ordered by embeddability.

Recall that a linear order is scattered if it does not contain an isomorphic copy of the linear order (ℚ, ≤) and is σ-scattered if it is a union of countably many scattered suborders.

In the final paragraph of [9], Laver writes, “Finally, the question arises as to how the order types outside of ℳ behave under embeddability.” For instance, is there a class of linear orders which is closed under taking suborders, which properly includes the class of σ-scattered linear orders, and which is well quasi-ordered by embeddability? Cast in another way, is there a non σ-scattered linear order which embeds into all of its non σ-scattered suborders?

Already in [1], Baumgartner proved that it is consistent that any two ℵ_{1}-dense sets of reals are isomorphic; in fact this conclusion is a consequence of the Proper Forcing Axiom (PFA). Here a linear order is κ-dense if all of its intervals have cardinality κ. It is not difficult to show that any suborder of ℝ of cardinality ℵ_{1} is bi-embeddable with an ℵ_{1}-dense set of reals and thus in Baumgartner’s model, any set of reals of cardinality ℵ_{1} is minimal with respect to being non σ-scattered. On the other hand, it follows easily from work of Dushnik and Miller [5] that the Continuum Hypothesis (CH) implies that there are no minimal uncountable linear orders which are separable.

10001

2010 Mathematics Subject Classification. Primary: 03E35; Secondary: 03E05, 03E57.

Key words and phrases. proper forcing, σ-scattered linear order.

10002 H. Lamei Ramandi and J. Tatch Moore

(In fact Dushnik and Miller show in ZFC that there is no minimal separable linear order of cardinality continuum.)

The main result of this paper is that Theorem 1.1 is consistently sharp.

Theorem 1.2. If there is a supercompact cardinal, then there is a forcing extension which satisfies CH in which there are no minimal non σ-scattered linear orders.

This result builds on work of Moore [13] and Ishiu-Moore [7]. In [13] it was proved that it is consistent with CH that ω_{1} and −ω_{1} are the only minimal uncountable linear orderings. In fact, this conclusion is derived from the conjunction of CH and a certain combinatorial consequence (A) of PFA. Notice that if ω_{1} and −ω_{1} are the only minimal uncountable linear orders, then any minimal non σ-scattered linear order must have the property that it does not contain an uncountable separable suborder or an Aronszajn suborder. Here an Aronszajn line is an uncountable linear order which does not contain uncountable separable or scattered suborders.

In [7] it was proved that PFA^{+}, a strengthening of PFA, implies that every minimal non σ-scattered linear order is either isomorphic to a set of reals of cardinality ℵ_{1} or else is an Aronszajn line. Moreover, Martinez-Ranero [10], building on work of Moore [12] [14] proved that that PFA implies that the class of Aronszajn lines is well quasiordered by embeddability. In [7], it was pointed out that if the consequences of PFA^{+} needed to carry out the analysis in that paper were consistent with the conjunction of (A) and CH, then one could establish the consistency of “there are no minimal non σ-scattered linear orders.” In fact these consequence of PFA^{+} followed from a weaker axiom CPFA^{+} which had been expected to be consistent with CH; this was later refuted in [15]. The strategy of the present paper for proving Theorem 1.2 also utilizes the combination of (A) and CH, but involves a re-examination of the hypotheses sufficient to obtain the results of [7].

In addition to proving Theorem 1.2, we will also establish a result concerning the structure of non σ-scattered linear orders under the assumption of PFA^{+}. Baumgartner proved in [3] that there exist non σ-scattered linear orders which do not contain real or Aronszajn types. His construction can be described as the lexicographic ordering on a family {x_{α} : α ∈ S} where S ⊆ ω_{1} is stationary and x_{α} is a cofinal strictly increasing ω-sequence in α for each α in S. We will refer to such a linear ordering as a Baumgartner type and we will refer to S as its index set.

Theorem 1.3. Assume PFA^{+} and let X ⊆ ℝ have cardinality ℵ_{1} and C be a Countryman line. If L is a non σ-scattered linear order, then L contains an isomorphic copy of one of the following linear orders: X, C, −C, a Baumgartner type or its reverse.

The proof of Theorem 1.2 immediately yields the following result.

Theorem 1.4. It is consistent with CH that no Aronszajn tree has a base of cardinality ℵ_{1}.

Here a collection B of uncountable downward closed subtrees of an Aronszajn tree T is called a base if whenever U is an uncountable downward closed subtree of T, there is V ∈ B such that V ⊆ U. This answers a problem posed in [2], where it is proved that every Aronszajn tree has a base of cardinality ℵ_{1} after Levy collapsing an inaccessible cardinal to ℵ_{2}.

NON σ-SCATTERED ORDERS 10003

The paper will be organized as follows. Section 2 will review some notation, definitions, and results concerning linear orders. In Section 3 we will prove Theorem 1.3. Section 4 contains the analysis needed to derive the conclusion of Theorem 1.2 from a list of axioms. Section 5 gives a proof that the collection of axioms used in Section 4 is consistent. This section also includes a proof of theorem 1.4 as a remark. The paper closes with some open problems in Section 6.

2. Preliminaries

This section is devoted to some background and conventions on trees, linearly ordered sets and forcing axioms. More discussion can be found in [7], [11], [13], [16] and [17]. We will also introduce two set-theoretic axioms (∗) and (†) which will play an important role in the proofs of Theorem 1.2 and 1.3.

We first recall the notion of a forcing axiom associated to a class of partial orders.

Notation 2.1. If 𝔓 is a class of partial orders, then by FA(𝔓) we mean the forcing axiom for the class 𝔓: whenever P is in 𝔓 and D is a collection of ℵ_{1}-many dense subsets of P, there is a filter G ⊆ P which intersects all of the dense sets in D. FA^{+}(𝔓) is the assertion that whenever P is in 𝔓, D is a collection of ℵ_{1}-many dense subsets of P, and Ṡ is a name for a stationary subset of ω_{1}, then there is a filter G ⊆ P which intersects all the dense sets in D and satisfies that the set

{ξ ∈ ω_{1} : ∃p ∈ G(p ⊩ ξ̌ ∈ Ṡ)}

is stationary.

The following axiom is a consequence of FA^{+}(σ-closed) and will play an important role in our analysis of non σ-scattered linear orders of cardinality ℵ_{1}.

Definition 2.2. (†) is the assertion that if S ⊆ ω_{1} is stationary and for each α ∈ S, U_{α} ⊆ α is open, then there a club E ⊆ ω_{1} such that for stationarily many α ∈ S ∩ E there is an ᾱ < α such that either E ∩ (ᾱ, α) ⊆ U_{α} or E ∩ (ᾱ, α) ∩ U_{α} = ∅.

Let P be the poset consisting of all countable closed subsets of ω_{1}, ordered by end extension and let Ė be the P -name for the union of the generic filter. By using the arguments of [11], it is possible to show that if S ⊆ ω_{1} is stationary and ⟨U_{α} : α ∈ S⟩ is as in the formulation of (†), then every condition forces Ė satisfies the conclusion of (†) for ⟨U_{α} : α ∈ S⟩. In particular FA^{+}(σ-closed) implies (†). Moreover, (†) holds in the model obtained by adding ℵ_{2} Cohen subsets of ω_{1} to a model of GCH. It should be noted that while this shows that it is easy to obtain models of (†) and CH, it remains an open problem whether the strengthening of (†) in which a relative club of α ∈ S ∩ E are required to satisfy the conclusion is consistent with CH (see [6]).

It will often be convenient to let, for each set X, θ_{X} denote the least regular cardinal such that all finite iterates of the power set applied to X are in H(θ_{X}), the collection of sets of hereditary cardinality less than θ_{X}. Let E(X) denote the collection of all countable elementary submodels of H(θ_{X}) which have X as an element.

We will now recall a number of definitions from [7]. For a linearly ordered set L we will use L̂ to denote the completion of L. Formally this is the set of all Dedekind cuts of L with z identified with the cut {x ∈ L : x < z}. The purpose of the following definitions is to abstractly recover the set of indices from a Baumgartner type, purely from its order-theoretic properties.

Definition 2.3. Whenever L is a linearly ordered set and Z is some arbitrary set we say that Z captures x ∈ L if there is a z ∈ Z ∩ L̂ such that there is no element of Z ∩ L which is strictly in between z and x.

Fact 2.4. [7] Suppose L is a linear order and let λ be a regular cardinal such that L̂ is in H(λ). If M is a countable elementary submodel of H(λ) with L ∈ M and x ∈ L̂ ∖ M, then M captures x if and only if there is a unique z ∈ L̂ ∩ M such that there is no element of M ∩ L which is strictly in between x and z. In this case we say M captures x via z.

Definition 2.5. [7] If L is a linear order, define Γ(L) to be the set of all countable subsets Z of L̂ such that for some x ∈ L, Z does not capture x. (This is the relative complement of the set Ω(L) in [7].)

If B = ⟨x_{α} : α ∈ S⟩ is a Baumgartner type and M is a countable elementary submodel of H(θ) for some regular cardinal θ ≥ ω_{2}, then M ∈ Γ(B) if and only if M ∩ ω_{1} ∈ S. This is because M captures all elements of B except x_{δ}, where δ = M ∩ ω_{1}. So Γ(B) is equivalent to S modulo the equivalence induced by the following quasi-order.

Definition 2.6. Let A, B be two collections of countable sets and X = ⋃ A, Y = ⋃ B. we say B ≤ A if there is an injection ι : X → Y such that for club many M in [Y]^{ω}, if M ∈ B then ι^{−1}M is in A. We let B < A if B ≤ A but not A ≤ B; A and B are equivalent if A ≤ B and B ≤ A.

The following results summarize the properties of the map L ↦ Γ(L) and the quasi-order ≤.

Theorem 2.7. [7] A linear order L is not σ-scattered if and only if Γ(L) is stationary.

Proposition 2.8. [7] If L_{0} and L are linearly ordered sets and L_{0} embeds into L, then Γ(L_{0}) ≤ Γ(L).

A key feature of Baumgartner types L is that it is always possible to find a non σ-scattered suborder L_{0} such that Γ(L_{0}) < Γ(L). This is not always possible in the more general class of non σ-scattered orders as the next proposition shows.

Proposition 2.9. [7] If a linear order L contains a real or Aronszajn type, then Γ(L) contains a club.

Definition 2.10. If L is a linear order and M is in E(L), then we say that an element x of L is internal (respectively external) to M, if there is a club E ⊆ [L̂]^{ω} in M such that every (respectively no) element of E ∩ M captures x.

The next definition will play a central role in the proofs of our results. It abstracts the property of Baumgartner types needed to allow us to decrease Γ by thinning out the linear order.

Definition 2.11. A linear order L is said to be amenable if whenever M is in E(L) and x ∈ L, then x is internal to M.

Observe that by Theorem 2.7, σ-scattered linear orders are amenable. It is also true that Baumgartner types are amenable.

Proposition 2.12. [7] If L is a non σ-scattered amenable linear order of cardinality ℵ_{1} and S ⊆ Γ(L) is stationary, then there is a non σ-scattered L_{0} ⊆ L such that Γ(L_{0}) ≤ S.

In particular, non σ-scattered amenable linear orders of cardinality ℵ_{1} are not minimal. The next theorem shows that the existence of external elements of a linear order characterizes the presence of either a real or Aronszajn suborder. In particular amenable linear orders do not contain real or Aronszajn types.

Theorem 2.13. [7] The following are equivalent for a linear order L:

• L contains a real or Aronszajn type.

• There are M in E(L) and x ∈ L such that x is external to M.

We are now ready to formulate the other set-theoretic hypothesis which will be needed in our analysis.

Definition 2.14. (∗) is the assertion that for every non σ-scattered linear order L there is a continuous ∈-chain ⟨M_{ξ} : ξ ∈ ω_{1}⟩ in E(L) such that:

• the set of all ξ ∈ ω_{1} such that M_{ξ} ∩ L̂ ∈ Γ(L) is a stationary set,

• L̂_{0} ⊆ ⋃_{ξ∈ω_{1}} M_{ξ}, where L_{0} = L ∩ (⋃_{ξ∈ω_{1}} M_{ξ}),

• for every ξ if M_{ξ} ∩ L̂ ∈ Γ(L) then there is an x ∈ L_{0} such that M_{ξ} does not capture x.

Observe that if L_{0} ⊆ L are as in the statement of (∗), then L_{0} is also non σscattered. Thus (∗) implies every non σ-scattered linear order contains a non σscattered suborder of cardinality ℵ_{1}. Also, if we apply (∗) to a linear order of cardinality at most ℵ_{1}, then L ⊆ ⋃_{ξ∈ω_{1}} M_{ξ} and consequently L̂ ⊆ ⋃_{ξ∈ω_{1}} M_{ξ}. This gives the following fact.

Fact 2.15. Assume (∗). If L is a linear order of cardinality at most ℵ_{1} which does not contain a real type, then |L̂| ≤ ℵ_{1}.

In particular (∗) implies that CH is true. A consequence of the work in [5] and [13] is that by iterating certain forcings over a model of CH, it is possible to obtain a generic extension in which there is no minimal real or Aronszajn type. We briefly review this result and recall some of the relevant definitions and terminology. If T is an Aronszajn tree, then a subtree of T is an uncountable downward closed subset of T.

Notation 2.16. If T is a tree, t ∈ T and α is an ordinal, then t ↾ α is defined to be t if α is greater than the height of t otherwise it is the unique s ≤ t with height α.

Definition 2.17. A sequence ⟨f_{α} : α ∈ lim(ω_{1})⟩ is called ladder system coloring if the ⟨dom(f_{α}) : α ∈ ω_{1}⟩ forms a ladder system and the range of each f_{α} is contained in ω.

Definition 2.18. If T is an ω_{1}-tree, then a ladder system coloring ⟨f_{α} : α ∈ lim(ω_{1})⟩ can be T -uniformized if there is a subtree U of T and function from ϕ : U → ω such that whenever height of u ∈ U is a limit ordinal α, f_{α} agrees with ξ ↦ ϕ(u ↾ ξ) at all except for finitely many ξ ∈ dom(f_{α}).

Definition 2.19. (A) is the assertion that every ladder system coloring can be T - uniformized for every Aronszajn tree T.

The significance of (A) lies in the following theorem, along with the fact that it is consistent with CH.

Theorem 2.20. [13] Assume (A) and 2^{ℵ_{0}} < 2^{ℵ_{1}}. There is no minimal Aronszajn line.

In [13], a forcing Q_{T,f̄} was introduced which T -uniformizes a given ladder system coloring f̄. We will recall the definition of this poset in Section 5 when we need to analyze it, but for now we will simply summarize its important properties.

While (< ω_{1})-properness and complete properness play a role in the proof of the main result of this paper, they can be treated as black boxes via the following results, along with the straightforward fact that σ-closed posets are both (< ω_{1})-proper and completely proper.

Lemma 2.21. [13] For every ladder system coloring f̄ and Aronszajn tree T, the forcing Q_{T,f̄} is completely proper and (<ω_{1})-proper.

Theorem 2.22. [16] A countable support iteration of (<ω_{1})-proper, completely proper forcing is proper and does not introduce new real numbers.

We will also need the following iteration theorem of Shelah.

Theorem 2.23. [16, III.8.5] If the iterands of a countable support iteration are proper and don’t add new uncountable branches to ω_{1}-trees, then the iteration is proper and does not add uncountable branches to ω_{1}-trees.

3. A rough classification of non σ-scattered orders

In [7] it was shown that under PFA^{+}, every non σ-scattered linear order contains an amenable non σ-scattered suborder of cardinality ℵ_{1}. In this section we prove that under a fragment of PFA^{+} every non σ-scattered amenable linear order contains a copy of a Baumgartner type or its reverse. Taken together, these results determines a basis for the class of non σ-scattered linear orders under PFA^{+}: if X is any set of reals of cardinality ℵ_{1} and C is any Countryman type, then any non σ-scattered linear order must contain an isomorphic copy of either X, C, −C, or a Baumgartner type of cardinality ℵ_{1} or its reverse.

Theorem 3.1. Assume the conjunction of MA_{ω_{1}} and (†). If L is an amenable non σ-scattered linear order of size ℵ_{1}, then it contains a copy of a Baumgartner type or its reverse.

First we will prove the following lemma.

Lemma 3.2. Suppose that L is a an amenable linear order of cardinality ℵ_{1}. If ⟨M_{ξ} : ξ ∈ ω_{1}⟩ is a continuous ∈-chain of elements of E(L) which is in N ∈ E(L) and N ∩ ω_{1} = δ, then M_{δ} and N capture the same elements of L.

Proof. First observe that by continuity of the ∈-chain and the fact that {ν ∈ ω_{1} : M_{ν} ∩ ω_{1} = ν} is a club in N, M_{δ} ⊆ N and M_{δ} ∩ ω_{1} = δ. Next observe that since L has cardinality ℵ_{1}, N ∩ L = M_{δ} ∩ L and thus any element of L captured by M_{δ} is captured by N. Now suppose that N captures x ∈ L and let z ∈ L̂ ∩ N be such that there is no element of N ∩ L which is strictly in between x and z. Since L is amenable, there is a club E ⊆ [L̂]^{ω} in M_{δ} such that for all Z ∈ M_{δ} ∩ E, Z captures x. Let λ ∈ θ_{L} ∩ M_{0} be a regular cardinal such that the powerset of [L̂]^{ω} is in H(λ). Let \overline{M} ∈ N be a countable elementary submodel of H(λ) such that ⟨M_{ξ} ∩ H(λ) : ξ ∈ ω_{1}⟩, E, and z are in \overline{M}. Observe that for sufficiently large ξ < δ, M_{ξ} ∩ L̂ is in E and if ν = \overline{M} ∩ ω_{1} then L ∩ \overline{M} = M_{ν} ∩ L. Notice that \overline{M} captures x via z. Since M_{ν} ∩ L̂ is in E ∩ M_{δ}, M_{ν} also captures x. By Fact 2.4, it must be that z is in M_{ν} and hence M_{δ}. □

Proof of Theorem 3.1. Now let ⟨M_{ξ} : ξ ∈ ω_{1}⟩ be a continuous ∈-chain of elements of E(L). Since L is amenable it does not contain any real types, there is a countable set X_{ξ} ⊆ L such that if M_{ξ} ∩L ⊆ X_{ξ} and if y ∈ L\M_{ξ}, there is a unique x ∈ X_{ξ} \M_{ξ} such that x ≠ y. Let x : ω × ω_{1} → L be such that for all ξ ∈ ω_{1}, X_{ξ} = {x(n, M_{ξ} ∩ ω_{1}) : n ∈ ω}. Now let ⟨N_{ξ} : ξ ∈ ω_{1}⟩ be a continuous ∈-chain of elements of E(L) such that ⟨M_{ξ} : ξ ∈ ω_{1}⟩ and x are in N_{0}. Note that there is a club of ξ in ω_{1} such that M_{ξ} ∩ ω_{1} = ξ = N_{ξ} ∩ ω_{1} and hence M_{ξ} and N_{ξ} capture the same elements of L. Since Γ(L) is stationary, then by applying the pressing down lemma there is a stationary set S_{0} ⊆ ω_{1}, an n ∈ ω, and a club E ⊆ [L̂]^{ω} such that if ξ ∈ S_{0}:

• M_{ξ} ∩ ω_{1} = ξ = N_{ξ} ∩ ω_{1};

• x(n, ξ) is not captured by N_{ξ};
• E is in N_{ξ} and if Z is in N_{ξ} ∩ E, then Z captures x(n, ξ).

Set y_{ξ} = x(n, ξ) for all ξ ∈ S_{0}. Now it is easy to see that for all ξ and η in S_{0}, N_{ξ} captures y_{η} if and only if ξ ≠ η.

Let z_{ξ} (ξ ∈ ω_{1}) be an enumeration of all z ∈ L̂ for which there are η ∈ ω_{1} and α ∈ S_{0} such that N_{η} captures y_{α} via z. We can assume without loss of generality that this enumeration is in N_{0}. For every α ∈ S_{0} define g_{α} : α → {z_{ξ} : ξ ∈ ω_{1}} by letting g_{α}(ξ) be the unique z ∈ N_{ξ} such that N_{ξ} captures y_{α} via z. Note that if g_{α}(ξ) = z_{η} then η ∈ α.

Claim 3.3. The following are true for α, β ∈ S_{0}:

(1) {ξ ∈ α : g_{α}(ξ) ≠ g_{α}(ξ + 1)} has order type ω and supremum α

(2) If y_{α} < y_{β}, then g_{α}(ξ) ≤ g_{β}(ξ) for all ξ < min(α, β).
(3) If α < β, then there is a ξ < α such that g_{α}(ξ) ≠ g_{β}(ξ).
(4) If ξ < η < min(α, β) and g_{α}(ξ) ≠ g_{β}(ξ), then g_{α}(η) ≠ g_{β}(η).

Proof. First observe that for each α ∈ S_{0} and limit η ∈ α there is an η̄ ∈ η such that g_{α} ↾ (η̄, η] is constant. On the other hand N_{α} does not capture y_{α} and therefore the set {ξ ∈ α : g_{α}(ξ) ≠ g_{α}(ξ + 1)} must have ordertype ω and supremum α. This proves (1); the remainder of the items follow easily from (1) and the definitions. □

Define C_{α} to be the set of all ξ ∈ α such that z_{ξ} is in the range of g_{α} and equip the set {C_{α} : α ∈ S_{0}} with the lexicographic order. For each α ∈ S_{0} let U_{α} = {ξ ∈ α : g_{α}(ξ) < y_{α}} and observe that U_{α} is an open subset of α. So by (†) there is a stationary set S ⊆ S_{0} such that either

• for every α ∈ S and ξ ∈ S ∩ α, g_{α}(ξ) > y_{α} or,

• for every α ∈ S and ξ ∈ S ∩ α, g_{α}(ξ) < y_{α}.

Without loss of generality assume that for every α ∈ S and ξ ∈ α ∩ S, g_{α}(ξ) > y_{α}. Define S′ to be the set of all elements of S which are limit points of elements of S.

Let Q be the set of all finite p ⊆ S′ such that whenever α ≠ β are in p, C_{α} <_{lex} C_{β} if and only if y_{α} < y_{β}. We will prove that Q is c.c.c..

Suppose for a contradiction that X is an uncountable antichain in Q. By applying the Δ-System Lemma and removing the root if necessary, we may assume that X is pairwise disjoint and consists of elements of some fixed cardinality n. Let M be an element of E(Q) such that X, L, x, ⟨N_{ξ} : ξ ∈ ω_{1}⟩, ⟨y_{ξ} : ξ ∈ S⟩, and ⟨z_{ξ} : ξ ∈ ω_{1}⟩ are all in M. Set δ = M ∩ ω_{1} and let p = {α_{1}, ..., α_{n}} be in X such that δ < α_{i} for all i ≤ n. Let ζ ∈ δ ∩ S be such that:

• if i, j ≤ n, then g_{α_{i}} ↾ δ ≠ g_{α_{j}} ↾ δ implies g_{α_{i}}(ζ) ≠ g_{α_{j}}(ζ);

• the range of g_{α_{i}} ↾ ζ + 1 coincides with the range of g_{α_{i}} ↾ δ for each i ≤ n (i.e.

C_{α_{i}} ∩ δ ⊆ ζ + 1 for each i ≤ n).

Notice that the existence of ζ follows from the observation that if g_{α}(ξ) ≠ g_{β}(ξ), then g_{α}(η) ≠ g_{β}(η) for all η > ξ. By elementarity of M there exists a p′ = {α′_{1}, ..., α′_{n}} in M ∩ X such that:

• for all i, j ≤ n, y_{α_{i}} < y_{α_{j}} if and only if y_{α′_{i}} < y_{α′_{j}};

• if i ≤ n, then g_{α_{i}}(ζ) = g_{α′_{i}}(ζ).

We will now show that p ∪ p′ ∈ Q. Let i, j ≤ n. There are two cases, depending on whether g_{α_{i}}(ζ) and g_{α_{j}}(ζ) are the same. If g_{α_{i}}(ζ) ≠ g_{α_{j}}(ζ), then observe that g_{α_{j}}(ζ) = g_{α′_{j}}(ζ) and

C_{α_{i}} ∩ (ζ + 1) ≠ C_{α_{j}} ∩ (ζ + 1) = C_{α′_{j}} ∩ (ζ + 1).

Since p and p′ are both in Q, it follows that y_{α_{i}} < y_{α′_{j}} is equivalent to C_{α_{i}} <_{lex} C_{α′_{j}}

If g_{α_{i}}(ζ) = g_{α_{j}}(ζ), then observe that g_{α_{i}} ↾ δ = g_{α_{j}} ↾ δ and thus that C_{α_{i}} ∩ δ = C_{α_{j}} ∩ δ. Observe that

g_{α′_{j}} ↾ ζ = g_{α_{j}} ↾ ζ = g_{α_{i}} ↾ ζ

and that g_{α_{i}} is constant on the interval [ζ, δ). Also, g_{α′_{j}} is not constant on [ζ, δ) by Claim 3.3. Observe that there is a ξ ∈ S such that ζ < ξ < α′_{j} and

g_{α′_{j}}(ξ) < g_{α′_{j}}(ζ) = g_{α_{j}}(ζ) = g_{α_{j}}(ξ) = g_{α_{i}}(ξ)

It follows that y_{α_{i}} > y_{α′_{j}}. On the other hand,

C_{α_{i}} ∩ δ = C_{α_{i}} ∩ (ζ + 1) = C_{α′_{j}} ∩ (ζ + 1) ≠ C_{α′_{j}} ∩ δ

and consequently C_{α′_{j}} <_{lex} C_{α_{i}}. Since i, j ≤ n were arbitrary, p and p′ are compatible and thus Q is c.c.c..

By applying MA_{ℵ_{1}} to the finite support product Q^{<ω} of countably many copies of Q, it is possible to find a partition of S into countably many pieces such that whenever α and β are in the same piece of the partition, C_{α} <_{lex} C_{β} if and only if y_{α} < y_{β}. Since there is a piece of this partition which is stationary, it shows that L contains a Baumgartner type. □

We finish this section by noting if we add a Cohen real r to a model of ZFC, then Theorem 3.1 will not hold in the resulting generic extension. To see this, suppose that r ∈ 2^{ω} and ⟨x_{ξ} : ξ ∈ lim(ω_{1})⟩ is such that x_{ξ} : ω → ξ is increasing and has cofinal range for each ξ. Define a linear ordering on lim(ω_{1}) by ξ <_{r} η if and only if

x_{ξ}(n) < x_{η}(n) is equivalent to r(n) = 0

where n is minimal such that x_{ξ}(n) ≠ x_{η}(n). It is left to the reader to check that if S ⊆ lim(ω_{1}) is stationary, then there is a comeager set of r such that (S, <_{r}) contains both a copy of ω_{1} and of −ω_{1}. Furthermore, if S is non-stationary, then (S, <_{r}) is σ-scattered and thus not a Baumgartner type. On the other hand, it is not hard to show that every uncountable subset of a Baugmartner type contains a copy of ω_{1}; in particular, Baumgartner types do not contain −ω_{1}. Since every stationary subset of ω_{1} in the generic extension by a Cohen real contains a ground model stationary set, this proves the claim.

4. An axiomatic analysis of non σ-scattered orders

In this section we will prove the following proposition.

Proposition 4.1. Assume (†) and (∗). If L is a non σ-scattered linear order which does not contain a real or Aronszajn type, then there is a non σ-scattered suborder L′ ⊆ L with Γ(L′) < Γ(L).

Proof. As noted in Section 2, (∗) implies that L contains a non σ-scattered suborder L_{0} such that L̂_{0} has cardinality ℵ_{1}. We may therefore assume without loss of generality that |L| = |L̂| = ℵ_{1}. This implies, in particular that if M and N are in E(L) and M ∩ ω_{1} = N ∩ ω_{1}, then M ∩ L̂ = N ∩ L̂. If Z ⊆ L̂ is countable, let {x(n, Z) : n ∈ ω} ⊆ L be such that Z ∩ L ⊆ {x(n, Z) : n ∈ ω} and if y ∈ L \ Z, then there is an n such that no element of L ∩ Z is between x(n, Z) and y. This is possible since L does not contain a real type. Let ⟨N_{ξ} : ξ ∈ ω_{1}⟩ be a continuous ∈-chain in E(L) with the map Z ↦ {x(n, Z) : n ∈ ω} in N_{0}. Since L is not σ-scattered, there is an n ∈ ω such that

S_{0} = {ξ ∈ ω_{1} : N_{ξ} ∩ ω_{1} = ξ and N_{ξ} does not capture x(n, N_{ξ} ∩ L̂)}

is stationary. Fix such an n and set x_{ξ} = x(n, N_{ξ} ∩ L̂). For each α ∈ S_{0} let U_{α} be the set of all ξ ∈ α such that N_{ξ} captures x_{α}. Clearly U_{α} is an open subset of α so by (†) there is a stationary subset S ⊆ S_{0} and a club E ⊆ ω_{1} such that for every α ∈ S there is an ᾱ ∈ α such that either E ∩ (ᾱ, α) ⊆ U_{α} or E ∩ (ᾱ, α) ∩ U_{α} = ∅. The second alternative can only happen for at most nonstationary many α ∈ S, because L has no external element by Theorem 2.13. By applying the Pressing Down Lemma and thinning S down if necessary, we can assume that for every α, β ∈ S, N_{α} captures x_{β} if and only if α ≠ β.

Now let S′ ⊆ S be stationary such that S ∖ S′ is also stationary and define L′ = {x_{ξ} : ξ ∈ S′}. We will show that L′ is not σ-scattered and that Γ(L′) < Γ(L).

Fix an M ∈ E(L) which has ⟨N_{ξ} : ξ ∈ ω_{1}⟩, S, and S′ as elements and has the property that M ∩ ω_{1} = δ ∈ S′. To show that L′ is not σ-scattered we prove that M does not capture x_{δ} in L′. Suppose for contradiction that M captures x_{δ} in L′ via z ∈ L̂ ∩ M. By replacing L with −L if necessary, we may assume that z < x_{δ}. Let

A = {x_{ξ} : ξ ∈ S′ and z < x_{ξ}}.

Observe that A is in M and hence inf(A) is also in M. Since inf(A) ≤ x_{δ} and x_{δ} is not in M, it follows that inf(A) < x_{δ}. Since M does not capture x_{δ} in L, there is y ∈ L∩M such that z ≤ inf(A) < y < x_{δ}. By elementarity of M, there is a ξ ∈ S′∩M such that z < x_{ξ} < y < x_{δ}. But this contradicts our assumption that M captures x_{δ} in L′ via z.

To see that Γ(L) ≰ Γ(L′) it suffices to show that the set of all M ∈ E(L) which capture all elements of L′ but does not capture some elements of L forms a stationary set. To this end let M ∈ E(L) with L′ ∈ M and M ∩ ω_{1} ∈ S ∖ S′ and observe that M does not capture x_{δ} in L but it captures all elements of L′. □

5. The consistency of the axioms

In this section we will prove that if there is a supercompact cardinal, then there is a forcing extension with the same reals which satisfies (∗), (†), and (A). By results of the previous section this will finish the proof of Theorem 1.2. Our forcing construction will resemble the consistency proof of PFA^{+} and will involve a countable support iteration of forcings which are completely proper, (< ω_{1})-proper, and which do not add new uncountable branches through ω_{1}-trees. By results of Shelah discussed in the introduction, the resulting iteration will not introduce new reals or uncountable branches through ω_{1}-trees.

All of the iterands used in building the iteration will either be σ-closed or else be of the following form.

Definition 5.1. [13] For an Aronszajn tree T and ladder system coloring f̄ let Q_{T,f̄} be the set of all conditions q = (ϕ_{q}, U_{q}) such that:

• ϕ_{q} is a function from X_{q} ⊆ T into ω such that X_{q} is a countable downward closed subset of T which has a last level of height α_{q},

• if t ∈ X_{q} has limit height δ, f_{δ} agrees with ξ ↦ ϕ_{q}(t ↾ ξ) at all ξ ∈ C_{α} except for finitely many ξ ∈ C_{α}.

• U_{q} is a non-empty countable collection of pruned subtrees of T^{[n]} for some n.

• for every U ∈ U_{q} there is some σ ∈ U which is a subset of the last level of X_{q}.

(T^{[n]} is the collection of all weakly increasing n-tuples from some level of T, regarded as a tree with the coordinatewise order.) We let p ≤ q, in Q if X_{p} ↾ α_{q} = X_{q}, U_{q} ⊆ U_{p}, and ϕ_{p} ↾ X_{q} = ϕ_{q}.

Remark 5.2. A simplification of this type of forcings can be used to prove Theorem 1.4. For an Aronszajn tree T let Q_{T} be the set of all conditions as above forgetting the information about the ladder system coloring. More precisely Q_{T} consists of all conditions q = (X_{q}, U_{q}) such that,

• X_{q} is a countable downward closed subset of T which has a last level of height α_{q},

• U_{q} is a non-empty countable collection of pruned subtrees of T^{[n]} for some n.

• for every U ∈ U_{q} there is some σ ∈ U which is a subset of the last level of X_{q}. We let p ≤ q, in Q if X_{p} ↾ α_{q} = X_{q}, U_{q} ⊆ U_{p}. It is easy to see that the forcing Q_{T,f̄} projects onto Q_{T} for every Aronszajn tree T, so by the work in [13], Q_{T} is completely proper, < ω_{1}-proper and satisfies proper isomorphism condition. Now let ℙ be the countable support iteration of all posets of Q_{T} of length ω_{2} such that whenever T is an Aronszajn tree in some intermediate model, Q_{T} is repeated in the iteration cofinally often. Let 𝕍 be a model satisfying 2^{ω} = ω_{1} + 2^{ω_{1}} = ω_{2}, and let G be ℙ-generic over 𝕍. Then it is easy to see that ω_{2} is preserved and in 𝕍[G]

• 2^{ω} = ω_{1} + 2^{ω_{1}} = ω_{2},

• if T is an Aronszajn tree, there is a sequence ⟨V_{i} : i ∈ ω_{2}⟩ of uncountable downward closed subtrees of T such that whenever i ∈ j, V_{i} contains no subtree of V_{j}.

This proves Theorem 1.4.

The following lemma asserts that these forcings Q_{T,f̄} do not add new uncountable branches to ω_{1}-trees.

Lemma 5.3. Suppose T is Aronszajn and S is an ω_{1}-tree, and f̄ is a ladder system coloring. Then Q_{T,f̄} does not add new uncountable branches to S. Consequently, if L is a linear order of size ℵ_{1}, then forcing with Q_{T,f̄} does not introduce new elements to L̂.

Proof. Let Q denote Q_{T,f̄} and let ḃ be a Q-name which is forced by some p ∈ Q to be an uncountable branch in S which is not in the ground model. If q is in Q and σ is in T^{[n]} for some n, then we say that σ is consistent with q if the range of σ ↾ α_{q} is contained in X_{q}.

Let M ∈ E(Q) with p, ḃ ∈ M and set δ = M ∩ ω_{1}.

Claim 5.4. If σ ∈ T_{δ}^{[n]} is consistent with p and s ∈ S_{δ}, then there is a condition q ≤ p in M ∩ Q such that q ⊩ š ∉ ḃ and such that σ is consistent with q.

Proof. By Lemma 5.5 in [13] we can find a decreasing sequence ⟨p_{k} : k ∈ ω⟩ in M such that:

• p_{0} = p,

• p_{k+1} decides ḃ ↾ α_{p_{k}},

• σ is consistent with p_{k} for all k,

• ⟨p_{k} : k ∈ ω⟩ has a lower bound in M. Thus without loss of generality we can assume that p forces š ↾ α̌_{p} ∈ ḃ.

Suppose for contradiction that for every q ≤ p in M ∩ Q, if q ⊩ š ∉ ḃ, then σ is not consistent with q. Define W to be the set of all τ ∈ T^{[n]} which are compatible with σ ↾ α_{p} and such that there exists an s̄ ∈ S_{ht(τ)} compatible with s ↾ α_{p} and for all q ≤ p, if q ⊩ s̄ ∉ ḃ and α_{q} ≤ ht(τ), then range(τ ↾ α_{q}) ⊈ X_{q}. Since W is definable from parameters in M, it is in M. Observe that W is downwards closed and that s witnesses that σ is in W. Hence by elementarily of M, W is uncountable. Let U be the set of all τ ∈ W which have uncountably many extensions in W. Notice that σ ↾ α_{p} is in U and thus p′ = (φ_{p}, X_{p}, U_{p} ∪ {U}) is a condition in Q.

For each τ ∈ U_{δ} and t ∈ S_{δ}, let φ(τ, t) be the assertion: “whenever r ≤ p′ is (M, Q)-generic with range(τ) ⊆ X_{r}, r ⊩ ť ∈ ḃ.” Notice that if r is (M, Q)-generic, then so is r ↾ δ. It is easy to see that for every τ ∈ W_{δ} there exists a unique t ∈ S_{δ} which extends s ↾ α_{p} such that φ(τ, t). Moreover, observe that if τ_{1} and τ_{2} are in U_{δ} and s_{1}, s_{2} are such that ϕ(τ_{1}, s_{1}) and ϕ(τ_{2}, s_{2}), then we can find an (M, Q)-generic condition r ≤ p′ which is consistent to both τ_{1} and τ_{2}. This implies r ⊩ š_{1} = š_{2}. Thus s ∈ S_{δ} satisfies that ϕ(τ, s) holds for every τ ∈ U_{δ}.

We now claim that p′ ⊩ š′ ∈ ḃ for all s′ < s. Since such an s′ is necessarily in M, by elementarity it suffices to show that if p′′ ≤ p′ is in M, then p′′ has an extension r which forces that š′ ∈ ḃ. Because Q is proper, p′′ has an (M, Q)-generic extension r. Let τ ∈ U_{δ} be such that range(τ) ⊆ X_{r}. Since r ≤ p′ and φ(τ, s′) holds, r ⊩ š′ ∈ ḃ. Thus we have established that p′ ⊩ š′ ∈ ḃ for all s′ < s. By elementarily, {t ∈ S : p′ ⊩ ť ∈ ḃ} is uncountable, which implies that p′ decides ḃ, a contradiction. □

Returning to the main proof, by the claim we can find a condition p̄ ≤ p such that p̄ ⊩ S_{δ} ∩ ḃ = ∅. □

Theorem 5.5. Assume there is a supercompact cardinal. Then there is forcing extension in which (A), (†), and (∗) hold.

Proof. Let V be a ground model with a supercompact cardinal κ. By performing some preparatory forcing if necessary, we may assume that CH is true. Mimicking the consistency proof of PFA (see [4] or [8]), use a Laver function ψ to build a countable support iteration ⟨P_{α}, Q̇_{α} : α ∈ κ⟩ such that:

• Q̇_{α} is a P_{α}-name in V_{κ} for a partial order which is either σ-closed or of the form Q_{Ṫ ,f̄};

• if ψ(α) is a P_{α}-name and p ∈ P_{α} forces that ψ(α) either σ-closed or of the form Q_{Ṫ ,f̄}, then p forces Q̇_{α} = ψ(α).

Let G ⊆ P_{κ} be a V -generic filter. It is immediate that V [G] satisfies (A). By Lemma 2.21 and Theorem 2.22 the iteration does not add new reals and thus V [G] satisfies CH. By Lemmas 2.23 and 5.3, every final segment of the iteration does not add new uncountable branches to ω_{1}-trees. Arguing as in [4], V [G] satisfies FA^{+}(σ-closed) and in particular (†).

We will now show that (∗) holds in V [G]. Fix for a moment a non σ-scattered linear order L in V [G] and let Q be the set of all countable continuous ∈-chains in E(L) ordered by end extension. It is obvious that Q is σ-closed and easily verified that

Ṡ = {(ξ̌, q) : ξ ∈ dom(q) and q(ξ) ∩ L̂ ∈ Γ(L)}

is a Q-name for a stationary subset of ω_{1}. Since Q is countably closed, it does not add new elements to L̂. Thus if H ⊆ Q is a V [G]-generic filter, then V [G][H] contains the desired witness to (∗) for L. Moreover, this witness is preserved in any further generic extension by a proper forcing in which L̂ does not gain new elements.

The proof that FA^{+}(σ-closed) holds in V [G] can now be applied in this situation to show that (∗) holds in V. The only difference is that while in the verification of FA^{+}(σ-closed) it is sufficient to know that the factor forcings are proper, in our setting it is necessary to know that, additionally, the factor forcings do not add new elements to the completions of linear orders. As noted already, this follows from Lemmas 2.23 and 5.3. □

6. Open questions

We will conclude this paper by mentioning some open questions which are natural in light of the results obtained here and in [7]. The first is a problem of Galvin.

Question 6.1. (see [3, Problem 4]) Must every minimal non σ-scattered linear order be a real type nor an Aronszajn type?

Of course it is consistent that this question has positive answer (this was first shown in [7]), but at present it seems possible that this question could have a positive answer in ZFC (we conjecture, however, that a negative answer is consistent). We also don’t know the answer to the following question.

Question 6.2. Must every minimal non σ-scattered linear order have cardinality ℵ_{1}?

Notice that if κ > ℵ_{1} is a regular cardinal and L = {x_{α} : α ∈ S} is a ladder system indexed by a nonreflecting stationary set S ⊆ κ. consisting of ordinals of countable cofinality, then the lexicographical ordering on L is non σ-scattered but has no σscattered suborder of cardinality ℵ_{1} (of course this example fails to be minimal).

Finally, it is unclear whether Theorem 1.3 can be sharpened so that the Baumgartner types are all realized as suborders of a single Baumgartner type.

Question 6.3. Assume PFA^{+}. If two Baumgartner types are indexed by a common stationary subset of ω_{1}, must there be a non σ-scattered order which embeds into both of them?

Acknowledgements

The research presented in this paper was supported in part by NSF grants DMS-1262019 and DMS-1600635.

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Department of Mathematics, Cornell University, Ithaca, NY 14853–4201, USA

E-mail address: hossein@math.cornell.edu
E-mail address: justin@math.cornell.edu