FORCING AXIOMS AND THE CONTINUUM HYPOTHESIS, PART II: TRANSCENDING ω_{1}-SEQUENCES OF REAL NUMBERS

JUSTIN TATCH MOORE

Dedicated to Fennel, Laurel, and Stephanie.

Abstract. The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the Continuum Hypothesis. This answers a longstanding problem of Shelah.

1. Introduction

In his seminal analysis of the cardinality of infinite sets, Cantor demonstrated a procedure which, given an ω-sequence of real numbers, produces a new real number which is not in the range of the sequence. He then posed the Continuum Problem, asking whether the cardinality of the real line was ℵ_{1} or some larger cardinality. This problem was eventually resolved by the work of Gödel and Cohen who proved that the equality |ℝ| = ℵ_{1} was neither refutable nor provable, respectively, within the framework of ZFC. In particular, there is no procedure in ZFC which takes an ω_{1}-sequence of real numbers and produces a new real number not in the range of the original sequence.

The purpose of this article is to revisit this type of diagonalization and to demonstrate that models of the Continuum Hypothesis exhibit a strong form of incompactness. The primary motivation for the results in this paper is to answer a question of Shelah [9, 2.18] concerning which forcing axioms are consistent with the Continuum Hypothesis (CH). The forcing axiom for a class ℭ of partial orders is the assertion that if D is a collection of ℵ_{1} cofinal subsets of a partial order, then there is an upward directed set G which intersects each element of D. The well known Martin’s Axiom for ℵ_{1} dense sets (MA_{ℵ_{1}}) is the forcing axiom for the class of c.c.c. partial orders. Foreman, Magidor, and Shelah have isolated the largest class of partial orders ℭ for which a forcing axiom is consistent with ZFC. Shelah has established certain sufficient conditions on a class ℭ of partial orders in order for the forcing axiom for ℭ to be consistent with CH. His question concerns to what extent his result was sharp. The main result of the present article, especially when combined with those of [1] show that this is essentially the case.

2000 Mathematics Subject Classification. 03E50, 03E57.

Key words and phrases. completely proper forcing, Continuum Hypothesis, forcing axiom, iterated forcing.

The research of the author presented in this paper was supported by NSF grant DMS–0757507. Any opinions, findings, and conclusions or recommendations expressed in this article are those of the author and do not necessarily reflect the views of the National Science Foundation.

The solution to Shelah’s problem has some features which are of independent interest. Suppose that T is a collection of countable closed subsets of ω_{1} which is closed under taking closed initial segments and that T has the following property:

(1) Whenever s and t are two elements of T with the same supre-

mum δ and lim(s) ∩ lim(t) is unbounded in δ, then s = t. (Here lim(s) is the set of limit points of s.) As usual, we regard T as a set-theoretic tree by declaring s ≤ t if s is an initial part of t. Observe that this condition implies that T contains at most one uncountable path: any uncountable path would have a union which is a closed unbounded subset of ω_{1} and such subsets of ω_{1} must have an uncountable intersection. In fact it can be shown that this condition implies that

{(s, t) ∈ T^{2} : (ht_{T} (s) = ht_{T} (t)) ∧ (s ≠ t)}

can be decomposed into countably many antichains (T^{2} is special off the diagonal). Jensen and Kunen, in unpublished work, have constructed examples of ω_{1}-Baire trees which are special off the diagonal under the assumption of ♢, but this article represents the first such construction from CH.

The main result of this article is to prove that if the Continuum Hypothesis is true, then there is tree T which satisfies (1) together with the following additional properties:

(2) T has no uncountable path;

(3) for each t in T, there is a closed an unbounded set of δ such that t ∪ {δ} is in T;

(4) T is proper as a forcing notion and remains so in any outer model with the same set of real numbers in which T has no uncountable path. Moreover T is complete with respect to a simple ℵ_{1}-completeness system 𝔻 and in fact is ℵ_{1}-completely proper in the sense of [7].

(5) If X is a countable subset of T, then the collection of bounded chains which are contained in X are Borel as a subset of 𝒫(X).

This provides the first example of a consequence of the Continuum Hypothesis which justifies condition (b) in the next theorem.

Theorem 1.1. [8] [3] Suppose that ⟨P_{α}; Q_{α} : α ∈ θ⟩ is a countable support iteration of proper forcings which:

(a) are complete with respect to a simple 2-completeness system 𝔻;

(b) satisfy either of the following conditions:

(i) are weakly α-proper for every α ∈ ω_{1}; (ii) are proper in every proper forcing extension with the same set

of real numbers.

Then forcing with P_{θ} does not introduce new real numbers.

Results of Devlin and Shelah [2] have long been known to imply that condition (a) is necessary in this theorem. The degree of necessity of condition (b) — and (bi) in particular — has long been a source of mystery, however. Shelah has already shown [8, XVIII.1.1] that there is an iteration of forcings in L which satisfies (a) and which introduces a new real at limit stage ω^{2}. This construction, however, has two serious limitations: it is not possible in the presence of modest large cardinal hypotheses — such as the existence of a measurable cardinal — and it does not refute the consistency of the corresponding forcing axiom with the Continuum Hypothesis. Still, this construction is starting point for the results in the present article.

Properties (3) and (4) of T imply that forcing with T adds an uncountable path through T and does not introduce new real numbers. Thus T can have an uncountable branch in an outer model with the same real numbers. By condition (1), this branch is in a sense unique, in that once such an uncountable branch exists, there can never be another. (Of course different forcing extensions will have different cofinal branches through T and in any outer model in which ω_{1}^{V} is countable, there are a continuum of paths through T whose union is cofinal in ω_{1}^{V}.)

Property (5) of the tree T is notable because it causes a number of related but formally different formulations of completeness to be identified [7]. This is significant because it was demonstrated in [1] that in general different notions of completeness can give rise to incompatible iteration theorems and incompatible forcing axioms.

The existence of the tree T can be seen as the obstruction to a diagonalization procedure for ω_{1}-sequences of reals. For each ω_{1}-sequence of reals r, there will be an associated sequence T⃗^{r} of trees of length at most ω^{2} which satisfy (1). These sequences are such that they have positive length exactly when (ω_{1})^{L[r]} = (ω_{1})^{V} and if ξ is less than the length of the sequence, then:

(6) T_{ξ}^{r} has an uncountable path if and only if it is not the last entry in the sequence;

(7) if ξ′ ∈ ξ, then every element of T_{ξ} is contained in the set of limit points of E_{ξ′}, where E_{ξ′} is the union of the uncountable path through T_{ξ′}^{r};

(8) if T_{ξ}^{r} is the last entry in the sequence, then either L[r] contains an real number not in the range of r or else T_{ξ}^{r} is completely proper in every outer model with the same real numbers.

(9) if the length of the sequence is ω^{2} and δ is in the intersection of ⋂_{ξ∈ω^{2}} E_{ξ}, then ⟨E_{ξ} ∩ δ : ξ ∈ ω^{2}⟩ is not in L[r]. (In particular, r is not an enumeration of ℝ.)

Moreover the (partial) function r ↦ T⃗_{ξ}^{r} is Σ_{1}-definable for each ξ ∈ ω^{2}. Thus in any outer model, the sequence T⃗^{r} may increase in length, but it maintains the entries from the inner model. Observe that if r is an enumeration of ℝ in ordertype ω_{1}, then T⃗^{r} has successor length η_{r} + 1.

Shelah has proved (unpublished) that an iteration of completely proper forcings of length less than ω^{2} does not add new real numbers. Thus if r is a well ordering of ℝ in type ω_{1} and ξ ∈ ω^{2}, then it is possible to go into a forcing extension with the same reals such that ξ ≤ η_{r}. The above remarks show that this result is optimal. Also, for a fixed ξ ∈ ω^{2}, the assertion that there is an enumeration r of ℝ in type ω_{1} with η_{r} at least ξ is expressible by a Σ_{1}^{2}-formula. By Woodin’s Σ_{1}^{2}-Absoluteness Theorem (see [6]), this means that, in the presence of a measurable Woodin cardinal, CH implies that for each ξ ∈ ω^{2}, there is an enumeration r of ℝ in type ω_{1} such that η_{r} is at least ξ.

While the construction of the sequence of trees is elementary, the reader is assumed to have a solid background in set theory at the level of [5]. Knowledge of proper forcing will be required at some points, although the necessary background will be reviewed for the readers convenience; further reading can be found in [3] [4] [8]. Notation is standard and will generally follow that of [5].

2. Background on complete properness

In this section we will review some definitions associated to proper forcing, culminating in the definition of complete properness. This is not necessary to understand the main construction; it is only necessary in order to understand the verification of (4). A forcing notion is a partial order with a least element. Elements of a forcing notion are often referred to as conditions. If p and q are conditions in a forcing, then p ≤ q is often read “q extends p” or “q is stronger than p.” If to conditions have a common extension, they are compatible; otherwise they are incompatible.

Let H(θ) denote the collection of all sets of hereditary cardinality at most θ. If Q is a forcing notion in H(θ), then a countable elementary submodel M of H(θ) is suitable for Q if it contains the powerset of Q. If M is a suitable model for Q and q̄ is in Q, then we say that q̄ is (M, Q)-generic if whenever q̄ ≤ r and D ⊆ Q is a dense subset of Q in M, r is compatible with some element of D ∩ M. A forcing notion Q is proper whenever M is suitable for M and q is in Q ∩ M, there is an extension of q which is (M, Q)-generic.

If M and N are sets, then \overrightarrow{MN} will denote a tuple (M, N, ϵ) where ϵ is an elementary embedding ϵ of (M, ∈) into (N, ∈) such that the range of ϵ is a countable element of N. Such a tuple is will be referred to as an arrow. I will write M → N to mean \overrightarrow{MN} and also to indicate “\overrightarrow{MN} is an arrow.” If M → N and X denotes an element of M, then X^{N} will be used to denote ϵ(X) where ϵ is the embedding corresponding to M → N.

If Q is a forcing, M is suitable for Q, and M → N, then a filter G ⊆ Q∩M is \overrightarrow{MN}-prebounded if whenever N → P and the image G′ of G under the composite embeddings is in P, P satisfies “G′ has a lower bounded.” If Q is a forcing notion, then a collection of embeddings M → N_{i} (i ∈ I) will be referred to as a Q-diagram. A forcing Q is λ-completely proper if whenever M → N_{i} (i ∈ γ) is a Q-diagram for γ ∈ 1 + λ, there is a (M, Q)-generic filter G ⊆ Q ∩ M which is \overrightarrow{MN_{i}}-prebounded for all i ∈ γ.

Observe that as λ increases, λ-complete properness becomes a weaker condition. In [7] it was shown that λ-complete properness implies 𝔻completeness with respect to a simple λ-completeness system 𝔻 in the sense of Shelah (see [7] for undefined notions). Moreover the converse is true for forcing notions Q with the property that whenever Q_{0} ⊆ Q is countable {G ⊆ Q_{0} : G has a lower bound in Q} is Borel.

3. The construction

Assume CH and fix a bijection ind : H(ω_{1}) → ω_{1} such that if x is a countable subset of ω_{1}, then sup(x) ∈ ind(x). If x, y are in H(ω_{1}), I will abuse notation and write ind(x, y) for ind((x, y)). Fix a Σ_{1}-definable map ξ ↦ ξ^{∗} which maps the countable limit ordinals into ω_{1} such that if ϱ ∈ ω_{1}, then {ξ ∈ lim(ω_{1}) : ξ^{∗} = ϱ} is uncountable (for instance define ξ^{∗} to be the unique ordinal ϱ such that for some ς, ξ = ω^{ς} + ω · ϱ and ω · ϱ ∈ ω^{ς}).

If δ is a limit ordinal, let C_{δ} denote the cofinal subset of δ of ordertype ω which minimizes ind; set C_{α+1} = {α}. Let e_{β} : β → ω be defined by e_{β}(α) = ϱ̄_{1}(α, β) where ϱ̄_{1} is defined from ⟨C_{α} : α ∈ ω_{1}⟩ as in [10]. In what follows, we will only need that the sequence ⟨e_{β} : β ∈ ω_{1}⟩ is determined by ind, e_{β} : β → ω is an injection, and if β ∈ β′ ∈ ω_{1}, then

{α ∈ β : e_{β}(α) ≠ e_{β′}(α)}

is finite.

Let E ⊆ ω_{1} be a club. Define T = T_{E} = T_{E}^{ind} to be all t which are countable closed sets of limit points of E such that:

• if ν is a limit point of t, then t ∩ ν has finite intersection with every ladder in ν which either has index less than ind(E ∩ ν) or else is C_{ν};

• if ν is a limit point of t, then min(E \ ν) ∈ ind(t ∩ ν);

• for all α ∈ β, ind(t ∩ β + 1 \ α) ∈ min(t \ β + 1). Define T̃ = T̃_{E}^{ind} ⊆ T by recursion. Begin by declaring ∅ ∈ T̃. Now

suppose that we have defined T̃ ∩ 𝒫(ξ + 1) for all ξ ∈ δ. Before defining T̃ ∩ 𝒫(δ + 1), we need to specify a family of logical formulas which will be needed in the definition. I will use T̃ ↾ ν to denote {t ∈ T̃ : ind(t) ∈ ν}. Let β be a fixed ordinal less than δ and let t ∈ T̃ be such that t ⊆ β. Consider the following recursively defined formulas about a closed subset x of β:

θ_{0}^{δ}(x, t, β): max(t) ∈ min(x), t ∪ x is in T̃ ↾ β, and

otp(E ∩ min(x))^{∗} = ind(t, n)

for some n ∈ ω; θ_{1}^{δ}(x, t, β): if D is a dense subset of T̃ ↾ ν for some limit ordinal

ν ∈ β, ind(D) ∈ β, and

otp(E ∩ min(x))^{∗} = ind(t, e_{δ}(ind(D))),

then t ∪ x is in D. θ_{2}^{δ}(x, t, β): if y ⊆ β, e_{δ}(min(y)) ∈ e_{δ}(min(x)) and

θ_{0}^{δ} ∧ θ_{1}^{δ} ∧ θ_{2}^{δ} ∧ θ_{3}^{δ}(y, t, β),

then x ∩ y ⊆ {min(x)}. θ_{3}^{δ}(x, t, β): if s, z ⊆ β, min(z) = min(x), and

θ_{0}^{δ} ∧ θ_{1}^{δ} ∧ θ_{2}^{δ}(z, s, β),

then ind(x) ≤ ind(z).

The truth of θ_{i}^{δ}(x, t, β) is defined by recursion on the tuple

(e_{δ}(min(x)), ind(x), i)

equipped with the lexicographical ordering (δ is fixed). Observe that since e_{δ} is injective, there is at most one set D which satisfies the hypotheses of θ_{1}^{δ}(x, t, β). In particular, if θ_{1}^{δ}(x, t, β) holds, then θ_{1}^{δ}(x, t, β′) holds for all β ∈ β′ ∈ δ.

Now if t is an element of T with sup(t) = δ and δ is not a limit point of t, define t to be in T̃ if and only if t ∩ δ is in T̃. If t is an element of T with sup(t) = δ and δ a limit point of t, then we define t to be in T̃ if t ∩ α + 1 is in T̃ for all α ∈ δ and if, for all but finitely many consecutive pairs α ∈ β in C_{δ} for which (α, β] ∩ t is non empty,

⋀_{i∈4} θ_{i}^{δ}(t ∩ (α, β], t ∩ α + 1, β).

Lemma 3.1. If E and E′ are clubs such that for some δ ∈ ω_{1}, E′ ∩δ = E ∩ δ, then T̃_{E} ∩ 𝒫(δ + 1) = T̃_{E′} ∩ 𝒫(δ + 1).

Proof. This follows from the observation that, under the hypotheses of the lemma, T_{E} ∩ 𝒫(δ + 1) = T_{E′} ∩ 𝒫(δ + 1) and the fact that T̃_{E} ∩𝒫(δ+1) is defined by recursion from T_{E} ∩𝒫(δ+1) and T̃_{E} ↾ δ. □

Lemma 3.2. If t and t′ are in T̃ and δ ∈ ω_{1} is a limit point of lim(t)∩ lim(t′), then t ∩ δ = t′ ∩ δ.

Proof. Let t and t′ be in T̃ and δ be a limit point of lim(t) ∩ lim(t′). It is sufficient to show that t ∩ α = t′ ∩ α for cofinally many α ∈ δ. Let δ_{0} ∈ δ be arbitrary and let α ∈ β be consecutive elements of C_{δ} such that δ_{0} ∈ α, (α, β] ∩ lim(t) ∩ lim(t′) is non empty, and such that

θ_{i}^{δ}(t ∩ (α, β], t ∩ α + 1, β)
θ_{i}^{δ}(t′ ∩ (α, β], t′ ∩ α + 1, β)

holds for all i ∈ 4. Without loss of generality, we may assume that

e_{δ}(ind(t ∩ (α, β])) ∈ e_{δ}(ind(t′ ∩ (α, β])).

Define x = t′ ∩ (α, β] and y = t ∩ (α, β]. Observe that since x and y have common limit points, x ∩ y is in particular not contained in {min(x), min(y)}. Since θ_{2}^{δ}(x, t ∩ α, β) is true, it follows that min(x) = min(y) and hence, by θ_{0}^{δ}(y, t ∩ α + 1, β) and θ_{0}^{δ}(x, t′ ∩ α + 1, β), that t ∩ α = t′ ∩ α. □

Lemma 3.3. The tree {(s, t) ∈ T^{2} : (ht_{T} (s) = ht_{T} (t)) ∧ (s ≠ t)} is a union of countably many antichains.

Proof. Let U denote those (s, t) ∈ T^{2} such that ht_{T} (s) = ht_{T} (t), s ≠ t, and max(s) ≤ max(t). By symmetry it is sufficient to prove that U is a countable union of antichains. For each (s, t) in U, define Osc(s, t) be the set of all ξ ≥ min(s△t) such that either

• ξ is in s and min(t \ ξ + 1) ∈ min(s \ ξ + 1) or

• ξ is in t and min(s \ ξ + 1) ∈ min(t \ ξ + 1).

Let osc(s, t) denote the ordertype of Osc(s, t), observing that Lemma 3.2 implies that osc(s, t) ∈ ω^{2} for all (s, t) in U. If max(s) ∈ max(t), define

β(s, t) = min{β ∈ t : max(s) ∈ β}
n(s, t) = e_{β(s,t)}(max(s))

and set n(s, t) = ω if max(s) = max(t). Observe that if (s, t) and (s′, t′) are in U and s < s′ and t < t′, then Osc(s, t) is an initial part of Osc(s′, t′) and that no element of Osc(s, t) is greater than max(s). Consequently either min(s′ \ s) is in Osc(s′, t′) \ Osc(s, t) and hence osc(s, t) ≠ osc(s′, t′) or else β(s, t) = β(s′, t′) and n(s, t) ≠ n(s′, t′). It follows that, for each ξ ∈ ω^{2} and k ∈ ω

{(s, t) ∈ U : (osc(s, t) = ξ) ∧ (n(s, t) = k)}

is an antichain. Since ω^{2} × ω is countable, this finishes the proof. □

Lemma 3.4. Suppose that M is a countable elementary submodel of H(2^{ω_{1}}^{+}) with T̃ in M and suppose that t_{i} (i ∈ n) is a sequence of elements of T̃ such that there is a club C ⊆ ω_{1} in M such that C ∩M ⊆ ⋃_{i∈n} t_{i}. Then T̃ has an uncountable chain.

Proof. Let M, t_{i} (i ∈ n), and C be as in the statement of the lemma. By replacing C with its limit points if necessary, we may assume that C ∩ M ⊆ ⋃_{i∈n} lim(t_{i}). Without loss of generality t_{i} (i ∈ n) are such that if i ≠ j, then t_{i}∩M ≠ t_{j} ∩M. By Lemma 3.2, there is a ζ ∈ M ∩ω_{1} such that if ν is a limit point of lim(t_{i}) ∩ lim(t_{j}) ∩ M, then ν ∈ ζ. Let i ∈ n be such that lim(t_{i}) ∩ C′ ∩ M is non-empty for every club C′ in M. Such an i exists since otherwise there would exist C′_{i} (i ∈ n), clubs in M such that C ∩ ⋃_{i∈n} C′_{i} is empty.

Let N be a countable elementary submodel of H(ω_{2}) in M such that C and ζ are in N and δ = N ∩ ω_{1} is in lim(t_{i}). Since δ is not in lim(t_{j}) for all j ≠ i, there is a ζ′ ∈ δ such that t_{j} ∩ δ ⊆ ζ′ whenever j ≠ i. Let C′ be the set of elements of C which are greater than ζ′. If ν is in C′ ∩ N, then ν is in lim(t_{i}) \ lim(t_{j}) for each j ∈ n which is different from i. Define b to be the set of p ∈ T̃ such that C′ ∩ sup(p) is an infinite cofinal subset of lim(p). Since it is definable from parameters in N, b is in N. Furthermore, b is uncountable since it is not contained in N — it has t_{i} ∩ δ + 1 as an element. By Lemma 3.2, b is a chain in T̃ and hence satisfies the conclusion of the lemma. □

Lemma 3.5. Suppose that E ⊆ ω_{1} is club and T̃_{E} does not contain an uncountable chain. Then T̃_{E} is completely proper.

Proof. Let T̃ denote T̃_{E}. Suppose that M → N_{i} (i ∈ ω) is a T̃-diagram and t_{0} is in T̃ ∩ M. Define δ = M ∩ ω_{1} and let η ∈ ω_{1} be an upper bound for ω_{1}^{N_{i}} for each i ∈ ω. In particular, if X ⊆ M ∩ ω_{1} is in N_{i}, then ind^{N_{i}}(X) ∈ η. Let D_{n} (n ∈ ω) enumerate the dense subsets of T̃ in M and let X_{n} (n ∈ ω) enumerate the collection of all cofinal subsets X of δ of ordertype ω which are in N_{i} for some i ∈ ω. We will construct t_{n} (n ∈ ω), α_{n} (n ∈ ω), and β_{n} (n ∈ ω) by induction so that for all n ∈ ω:

(10) t_{n+1} extends t_{n} and is in D_{n} ∩ M;

(11) α_{n} ∈ β_{n} ∈ α_{n+1} ∈ δ;

(12) if i ∈ n then t_{n+1} \ t_{n} contained in β_{n} \ α_{n} and is disjoint from X_{i} if i ∈ n;

(13) if i ∈ n then e_{δ}^{N_{i}}(ξ) = e_{δ}(ξ) whenever ξ ∈ β_{n} \ α_{n};

(14) if i ∈ n, then there are consecutive elements ᾱ ∈ β̄ of C_{δ}^{N_{i}} such that ᾱ ∈ α_{n} ∈ β_{n} ∈ β̄;

(15) there is a limit ordinal ν such that max(t_{n+1}) ∈ ν ∈ β_{n+1} and otp(E ∩ min(t_{n+1} \ t_{n}))^{∗} = ind(t_{n}, e_{δ}(ind(D ↾ ν))).

(16) if i ∈ n, then

N_{i} ⊧ ⋀_{i∈4} θ_{i}^{δ}(t_{n+1} \ t_{n}, t_{n}, β_{n+1}).

Assuming that this can be accomplished, then define t̄ = ⋃_{n} t_{n} ∪ {δ}. It follows that, if i ∈ ω and N_{i} → Ñ with t̄ ∈ Ñ, then t̄ is in T Ñ. If t̄ is in T Ñ then moreover we have arranged that

N_{i} ⊧ ⋀_{i∈4} θ_{i}^{δ}(t̄∩ (α, β], t̄∩ α + 1, β)

whenever α ∈ β are consecutive elements of C_{δ}^{N_{i}} such that max(t_{i}) ∈ α. In particular t̄ is in T̃ Ñ. Thus t_{n} (n ∈ ω) is \overrightarrow{MN_{i}}-prebounded in T̃ for each i ∈ ω.

Now suppose that t_{n} is given. Let M′ be a countable elementary submodel of H(2^{ω_{1}}^{+}) such that:

• M′ is in M;

• D_{n} is in M′;

• M′ is an increasing union of an ∈-chain of elementary submodels of H(2^{ω_{1}}^{+}).

Set β_{n+1} = M′ ∩ ω_{1} and let F ⊆ M′ be a finite set such that if i ∈ n, then

C_{δ}^{N_{i}} ∩ M′ ⊆ F
X_{i} ∩ M′ ⊆ F
{ξ ∈ M′ ∩ ω_{1} : e_{δ}^{N_{i}}(ξ) ≠ e_{δ}(ξ)} ⊆ F

Fix a M′′ which is an elementary submodel of H(2^{ω_{1}}^{+}) such that F ⊆ M′′ ∈ M′. Set ζ = ind(D_{n} ∩ M′′), ν = M′′ ∩ ω_{1}, and let α_{n+1} ∈ ν be such that max F ∈ α_{n+1}. Observe that by elementarity, ζ is in M′ since it is definable from ν, D_{n}, and ind. Furthermore, ν ≤ ζ since otherwise D_{n} ∩ M′′ would be in M′′. Observe that e_{δ}^{N_{i}}(ζ) = e_{δ}(ζ) for all i ∈ n. Let ξ be a limit point of E such that α_{n+1} ∈ ξ ∈ ν and

otp(E ∩ ξ)^{∗} = ind(t_{n}, e_{δ}(ζ))

Let y_{i} (i ∈ l) list the closed subsets y of δ such that:

• for some j ∈ n, y ⊆ (ᾱ, β̄] where ᾱ ∈ β̄ are the consecutive elements of C_{δ}^{N_{j}} with ᾱ ≤ α_{n+1} ∈ β_{n+1} ≤ β̄;

• e_{δ}^{N_{j}}(min y) ∈ e_{δ}^{N_{j}}(ξ);

• ⋀_{i∈4} θ_{i}^{δ}(y, t_{n}, β̄).

Observe that θ_{3}^{δ} ensures that there are only finitely many such y’s. Furthermore, Lemma 3.4 and our hypothesis implies that ⋃_{i∈l} y_{i} does not contain a set of the form C ∩ M′′ where C is a club in M′′. Thus there is a countable elementary submodel M′′′ of H(ω_{2}) in M′′ such that E, T̃, D_{n}, ξ, and ind are in M′′′ and M′′′ ∩ω_{1} is not in ⋃_{i∈l} y_{i}. Let η be a limit point of E which is in M′′′ such that ξ and max(⋃_{i∈l} y_{i}) are less than η. Since D_{n} is dense, elementarity of M′′′ ensures that there is an extension of t_{n} ∪ {ξ, η} which is in D_{n} ∩ M′′′. Such an extension t̃ necessarily satisfies that t̃\ t_{n} is disjoint from y_{i} \ {ξ} for all i ∈ l. We have therefore arranged that θ_{0}^{δ} ∧ θ_{1}^{δ} ∧ θ_{2}^{δ}(t̃\ t_{n}, t_{n}, β_{n+1}). Let t_{n+1} the element of T̃ be such that

min(t_{n+1} \ t_{n}) = ξ,
θ_{0}^{δ} ∧ θ_{1}^{δ} ∧ θ_{2}^{δ}(t_{n+1} \ t_{n}, t_{n}, β_{n+1}),

and ind(t_{n+1} \ t_{n}) is minimized. It follows that θ_{3}^{δ}(t_{n+1} \ t_{n}, t_{n}, β_{n+1}). This finishes the inductive construction and, therefore, the proof. □

Theorem 3.6. Assume CH. Then there is a club E such that T̃_{E} has no uncountable branch. In particular, CH implies the negation of CPFA.

Proof. Suppose for contradiction that CH holds and T̃_{E} contains an uncountable branch for every club E ⊆ ω_{1}. Let A ⊆ ω_{1} be such that ind is in L[A]. In particular, ℝ ⊆ L[A] and ω_{1} = ω_{1}^{L[A]}. Inductively construct clubs E_{ξ} (ξ ∈ ω^{2}) as follows. Since L[A] satisfies ♢, there is a function h : ω_{1} → ω_{1} such that if E is a club in L[A], then for some limit point δ of E, there is a ladder X ⊆ δ with X ∩ E infinite and ind(X) ∈ h(δ). Let E_{0} be the <_{L[A]}-least club in L[A] such that for all δ, h(δ) ∈ min(E_{0} \ δ + 1). Given E_{ξ}, let E_{ξ+1} be the union of the branch through T̃_{E_{ξ}}. If E_{ξ} (ξ ∈ η) has been defined for η a limit less than ω^{2}, define E_{η} = ⋂_{ξ∈η} E_{ξ}. Observe that E_{2} is not in L[A] since whenever δ is a limit point of E_{1},

h(δ) ∈ min(E_{0} \ δ + 1) ∈ ind(E_{1} ∩ δ).

and hence E_{2} has the property that whenever δ is a limit point of E_{2}, E_{2} ∩ δ is disjoint from X whenever X is a ladder in δ with index less than h(δ). Observe that if ξ ∈ η, then E_{η} is contained in the limit points of E_{ξ}. Furthermore if δ is a limit point of E_{η}, then

min(E_{ξ} \ δ + 1) ∈ ind(E_{η} ∩ δ) ∈ min(E_{η} \ δ + 1)

In particular, if δ ∈ ⋂_{ξ∈ω^{2}} E_{ξ}, then for all k ∈ ω

sup_{i∈ω} ind(E_{ω·k+i} ∩ δ) ∈ E_{ξ}

whenever ξ ∈ ω · (k + 1).

Let δ be the least element of ⋂_{ξ∈ω^{2}} E_{ξ}. Observe that ⟨E_{ξ} ∩ δ : ξ ∈ ω^{2}⟩ is in L[A]. We will obtain a contradiction once we show that ⟨E_{ξ} : ξ ∈ ω^{2}⟩ is in L[A]. Working in L[A], define ν_{α} (α ∈ ω_{1}) and t_{α}(ξ) (ξ ∈ ω^{2} ∧ α ∈ ω_{1}) by simultaneous recursion as follows. The sets t_{α}(ξ) will satisfy that they are E_{ξ} ∩ ν_{α} and the ordinals ν_{α} will each be elements of ⋂_{ξ∈ω^{2}} E_{ξ}. Set ν_{0} = δ and t_{0}(ξ) = E_{ξ} ∩ δ. Given ν_{α} and t_{α}(ξ) (ξ ∈ ω^{2}), define

ν_{α+1,k} = sup_{ξ∈ω·k} ind(t_{α}(ξ))

ν_{α+1} = sup_{k∈ω} ν_{α+1,k}.

Next we define t_{α+1}(ξ) (ξ ∈ ω^{2}) by recursion on ξ. t_{α+1}(0) = E_{0} ∩ν_{α+1}. Given t_{α+1}(ξ), define t_{α+1}(ξ + 1) to be the unique element t of T̃_{E_{ξ}} ∩ 𝒫(ν_{α+1} + 1) such that sup t = ν_{α+1} and ν_{α+1,k} ∈ t for all k. (here we are employing Lemmas 3.1 and 3.2). If η ∈ ω^{2} is a limit ordinal, set

t_{α+1}(η) = ⋂_{ξ∈η} t_{α+1}(ξ).

If ν_{α} has been defined for all α ∈ β, set ν_{β} = sup_{α∈β} ν_{α} and t_{β}(ξ) = ⋃_{α∈β} t_{α}(ξ).

It is now easily seen that t_{α}(ξ) = E_{ξ} ∩ ν_{α} and therefore that E_{ξ} is in L[A] for all ξ ∈ ω^{2}. This is a contradiction, however, since E_{2} is not in L[A]. □

I will finish by remarking that if r is an ω_{1} sequence of reals, then the sequence T⃗^{r} described in the introduction is defined as follows. If (ω_{1})^{L[r]} < ω_{1}, then define T⃗^{r} to be the empty sequence. Otherwise, let ind be the <_{L[r]}-least bijection between H(ω_{1}) ∩ L[r] and ω_{1}. Let A ⊆ ω_{1} be such that L[A] = L[r] and define T_{ξ}^{r} = T_{E_{ξ}}^{ind} as detailed in the proof of Theorem 3.6.

References

[1] David Asperó, Paul B. Larson, and Justin Tatch Moore. Forcing axioms and the continuum hypothesis, part I. to appear in Acta Mathematica.

[2] Keith Devlin and Saharon Shelah. A weak version of ♢ which follows from 2^{ℵ_{0}} < 2^{ℵ_{1}}. Israel Journal of Math, 29(2–3):239–247, 1978.

[3] Todd Eisworth and Peter Nyikos. First countable, countably compact spaces and the continuum hypothesis. Transactions of the American Mathematical Society, 357(11):4269–4299, 2005.

[4] Todd Eisworth, David Milovich, and Justin Tatch Moore. Iterated forcing and the Continuum Hypothesis. In James Cummings and Ernest Schimmerling, editors, Appalachian Set Theory 2006–2012, London Math Society Lecture Notes series, Cambridge University Press (2013).

[5] Kenneth Kunen. An Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1983.

[6] Paul B. Larson. The stationary tower, volume 32 of University Lecture Series. American Mathematical Society, Providence, RI, 2004. Notes on a course by W. Hugh Woodin.

[7] Justin Tatch Moore. ω_{1} and −ω_{1} may be the only minimal uncountable order types. Michigan Mathematical Journal, 55(2):437–457, 2007.

[8] Saharon Shelah. Proper and Improper Forcing. Springer-Verlag, Berlin, second edition, 1998.

[9] Saharon Shelah. On what I do not understand (and have something to say): part I. Fundementa Mathematicae, 166(1-2):1–82, 2000. Saharon Shelah’s anniversary issue.

[10] Stevo Todorcevic. Walks on ordinals and their characteristics, volume 263 of Progress in Mathematics. Birkhäuser, 2007.

Justin Moore, Department of Mathematics, Cornell University, Ithaca, NY 14853-4201

E-mail address: justin math.cornell.edu