BAUMGARTNER’S ISOMORPHISM PROBLEM FOR ℵ_{2}-DENSE SUBORDERS OF ℝ

JUSTIN TATCH MOORE AND STEVO TODORCEVIC

Dedicated to the memory of James Baumgartner

Abstract. In this paper we will analyze Baumgartner’s problem asking whether it is consistent that 2^{ℵ_{0}} ≥ ℵ_{2} and every pair of ℵ_{2}- dense subsets of ℝ are isomorphic as linear orders. The main result is the isolation of a combinatorial principle (∗∗) which is immune to c.c.c. forcing and which in the presence of 2^{ℵ_{0}} ≤ ℵ_{2} implies that two ℵ_{2}-dense sets of reals can be forced to be isomorphic via a c.c.c. poset. Also, it will be shown that it is relatively consistent with ZFC that there exists an ℵ_{2} dense suborder X of ℝ which cannot be embedded into −X in any outer model with the same ℵ_{2}.

1. Introduction

In one of the first results concerning the structure of abstract linear orders, Cantor showed that any two countable dense linear orders are isomorphic. Here a linear order is dense if it has no first or last elements and between any two elements there is a third. For uncountable linear orders, the situation is more complicated. First, there is the trivial observation that two uncountable linear orders of the same cardinality might have the property that one has a countable interval while the other does not. For this reason, one usually focuses attention on κ-dense linear orders, for some cardinal κ — linear orders with the property that every nonempty interval contains exactly κ elements. Even in the class of ℵ_{1}-dense linear orders, however, there are many

2000 Mathematics Subject Classification. 03E05, 03E35. Key words and phrases. ℵ_{2}-dense, linear order. The research presented in this paper was initiated during the Fields Institutes

fall 2012 thematic program on Forcing and its Applications; we would like to thank the institute for its generous hospitality. The first author’s research was supported in part by NSF grant DMS–1262019; the second author’s research was supported in part by grants from CNRS and NSERC. We would like to thank Garrett Ervin and Hossein Lamei Ramandi for carefully reading earlier version of this paper. We would also like to thank Itay Neeman for pointing out a serious error in an earlier version of this paper.

have qualitative differences, the most notable of which is separability. For the purpose of this paper, we will focus our attention on separable linear orders which are κ-dense (see, e.g., [14] for a discussion of the more general setting).

Dushnik and Miller showed that the class of 2^{ℵ_{0}}-dense linear orderings is already extremely complicated.

Theorem 1.1. [5] If X ⊆ ℝ has cardinality 2^{ℵ_{0}}, then there is a Y ⊆ X of cardinality 2^{ℵ_{0}} such that if f is a partial monotone function from Y to Y, then f differs from the identity on a set of cardinality less than the continuum. In particular such a Y satisfies that any two distinct suborders of Y of cardinality 2^{ℵ_{0}} are not isomorphic.

Hence if 2^{ℵ_{0}} = ℵ_{1}, then there are 2^{ℵ_{1}} pairwise nonisomorphic ℵ_{1}- dense suborders of ℝ. Baumgartner showed, however, that it is consistent that every two ℵ_{1}-dense suborders of ℝ be isomorphic.

Theorem 1.2. [4] Assume 2^{ℵ_{0}} = ℵ_{1} < 2^{ℵ_{1}} = ℵ_{2}. Then there is a poset which satisfies the countable chain condition and which forces the statement “every two ℵ_{1}-dense suborders of ℝ are isomorphic.”

In particular, the Proper Forcing Axiom implies that every two ℵ_{1}- dense suborders of ℝ are isomorphic. In the same paper, Baumgartner asked whether it was consistent that 2^{ℵ_{0}} ≥ ℵ_{2} and every two ℵ_{2}-dense suborders of ℝ are isomorphic. For ease of writing, we will let BA_{κ} denote the assertion that 2^{ℵ_{0}} ≥ κ and every two κ-dense suborders of ℝ are isomorphic.

While this problem has been popularized by Shelah and others (see e.g. [10, 2.20]), little progress has been made on the problem until now. Notice that, by Dushnik and Miller’s result, any model of BA_{ℵ_{2}} would necessarily satisfy 2^{ℵ_{0}} > ℵ_{2}. Moreover, the second author has proved the following result.

Theorem 1.3. [12] If there is an unbounded chain in (ω^{ω}, <^{∗}) of cofinality κ, then BA_{κ} is false.

Here <^{∗} is the order of eventual domination on integer sequences.

Since the methods for building models of 2^{ℵ_{0}} > ℵ_{2} are relatively limited, the difficulty of this problem has often been attributed to the more general difficulty of obtaining models of set theory with a “large continuum” while exerting more control than is allowed with c.c.c. forcings. The purpose of this article is to relate Baumgartner’s problem to a seemingly different problem of a more combinatorial nature. Consider the following assertions:

ℵ_{2}-DENSE SUBORDERS OF ℝ 3

(∗): If F is a collection of one-to-one functions from ω_{2} to ω_{2} and |F| ≤ ℵ_{2}, then there is a g : ω_{2} → ω_{2} which is one-to-one such that for every f in F, {α ∈ ω_{2} : f(α) = g(α)} is countable. (∗∗): If F is a collection of one-to-one functions from ω_{2} to ω_{2} and |F| ≤ ℵ_{2}, then there is a g : ω_{2} → ω_{2} which is one-to-one such that:

• for every f in F, {α ∈ ω_{2} : f(α) = g(α)} is countable and • for every f in F, there is a countable set D ⊆ ω_{2} such that

if α ≠ β ∈ ω_{2} \ D, then f(g(α)) ≠ g(β).

Notice that both of these principles are preserved in c.c.c. forcing extensions.

The following are the main results of the present paper.

Theorem 1.4. If MA_{ℵ_{2}}(σ-linked) is true and (∗) is false, then there are two ℵ_{2}-dense suborders of ℝ which are not isomorphic.

Theorem 1.5. Assume (∗∗) in conjunction with 2^{ℵ_{0}} ≤ ℵ_{2} and 2^{ℵ_{2}} = ℵ_{3}. There is a poset which satisfies the countable chain condition and which forces BA_{ℵ_{1}}, BA_{ℵ_{2}}, and MA_{ℵ_{2}}.

Notice the analogy between Theorems 1.2 and 1.5: the only difference is that in the case of ℵ_{2}-dense suborders of ℝ, we must adequately prepare the ground model first, beyond what is provided by cardinal arithmetic alone. It should be noted that the hypothesis 2^{ℵ_{2}} = ℵ_{3} is only needed in Theorem 1.5 in order to obtain a c.c.c. forcing; we can always collapse 2^{ℵ_{2}} to ℵ_{3} while not adding subsets of ℵ_{2} (and thus preserving both (∗∗) and 2^{ℵ_{0}} ≤ ℵ_{2}).

It should be noted that it would be completely unexpected if BA_{ℵ_{2}} is consistent but implies the failure of MA_{ℵ_{2}}(σ-linked). Also, the existing techniques which seem most relevant for obtaining the consistency of (∗∗) involve iterating σ-closed posets, necessarily resulting in a model of 2^{ℵ_{0}} = ℵ_{1}. Thus we have shown that obtaining the consistency of the conjunction of BA_{ℵ_{2}} and MA_{ℵ_{2}}(σ-linked) lies somewhere between obtaining the consistency of (∗) and the consistency of (∗∗) with 2^{ℵ_{0}} ≤ ℵ_{2}. While (∗∗) is formally stronger than (∗), the exact relationship between these principles is at present unclear; see the discussion in the concluding remarks section concerning some recent developments.

The significance of Theorem 1.4 is amplified by a result of Abraham and Shelah [2] which we will see implies that there are models of the failure of (∗) in which the witness to the failure of (∗) persists in any outer model with the same ℵ_{2}. In fact the proof of Theorem 1.4 will show that there are models of set theory in which there are ℵ_{1}-dense and ℵ_{2}-dense sets of reals which are not isomorphic to their reverse and which retain this property in any outer model with the same ℵ_{2}.

This paper is organized as follows. Section 2 will recall some of the work of Abraham and Shelah on closed unbounded subsets of ω_{1} and relate it to (∗). Section 3 will give a proof of Theorem 1.4. Finally, Section 4 will give a proof of Theorem 1.5.

2. Fast clubs, (∗), and collapsing ℵ_{2}

Suppose that E is a collection of closed unbounded subsets of ω_{1}. A club C ⊆ ω_{1} is fast with respect to E if C \ E is countable whenever E is in E. If there is a fast club with respect to E, then we will say that E can be diagonalized. It is a well known result of Jensen that if 2^{ℵ_{0}} = ℵ_{1}, then there is a σ-closed forcing which is ℵ_{2}-c.c. — and in particular preserves ℵ_{2} — and which adds a club which is fast with respect to the ground model club filter.

Notice that if E is a collection of clubs of cardinality ℵ_{1}, then there is a club C which is fast with respect to E. In particular, if E is any family of clubs, we can force to add a club which is fast with respect to E by collapsing the cardinality of E to be ℵ_{1} while preserving ℵ_{1}. Abraham and Shelah have proved the following result which shows that this is necessary in some cases.

Theorem 2.1. [2] Assume CH. There is a proper cardinal preserving forcing extension in which there is a family E of closed unbounded subsets of ω_{1} such that E has cardinality ℵ_{2} and in any outer model with the same ℵ_{1}, the intersection of every uncountable subset of E is finite. In particular in any outer model of the generic extension with the same ℵ_{2}, E is not diagonalized.

Next we will relate (∗) to the assertion that every collection of clubs of cardinality at most ℵ_{2} can be diagonalized. Notice that this latter assertion is equivalent to the assertion that every subset of ω_{1}^{ω_{1}} of cardinality ℵ_{2} is bounded by a single function in the order of eventual dominance.

Fix a sequence ⟨e_{β} : β ∈ ω_{2}⟩ such that for each β ∈ ω_{2} \ ω_{1}, e_{β} : β → ω_{1} is a bijection.

Proposition 2.2. (∗) is equivalent to the following statement: whenever F is a collection of at most ℵ_{2} many one-to-one functions from ω_{1} to ω_{1}, there is a countable-to-one g : ω_{2} → ω_{2} such that whenever β is closed under g and f is in F, there is a countable D ⊆ β such that if xi ∈ β \ D,

f(min(e_{β}(ξ), e_{β}(g(ξ))) < max(e_{β}(ξ), e_{β}(g(ξ))).

In particular if there is a family of ℵ_{2}-many closed unbounded subsets of ω_{1} which cannot be diagonalized, then (∗) is false.

Proof. First assume (∗) and let F ⊆ ω_{1}^{ω_{1}} have cardinality ℵ_{2}. For each f in F, let F_{f} be a countable collection of one-to-one functions which cover

{(ξ, η) ∈ ω_{1}^{2} : max(ξ, η) ≤ f(min(ξ, η))}.

Define H to be the set of all functions h such that for some f in F, f′ ∈ F_{f} and β ∈ ω_{2},

h(ξ) = \begin{cases}e_{β}^{−1}(f′(e_{β}(ξ))) & if ξ ∈ β \\ ξ & otherwise. \end{cases}

Now suppose that g : ω_{2} → ω_{2} is such that, for every h in H, {ξ ∈ ω_{2} : g(ξ) = h(ξ)} is countable. Then for each β ∈ ω_{2} which is closed under g and each f in F, there are at most countably many ξ ∈ β such that

max(e_{β}(ξ), e_{β}(g(ξ))) ≤ f(min(e_{β}(ξ), e_{β}(g(ξ))).

Next suppose that the hypothesis stated in the lemma holds and that H ⊆ ω_{2}^{ω_{2}} is as in the statement of (∗). Define F to be the set of all functions of the form ξ ↦ e_{β}(h(e_{β}^{−1}(ξ))) such that h is in H and β is uncountable and closed under h. It is easily checked that if g : ω_{2} → ω_{2} satisfies the conclusion of the hypothesis in the proposition with respect to F, then g satisfies the conclusion of (∗) with respect to H. □

The following proposition has a similar proof which is left to the reader.

Proposition 2.3. (∗∗) is equivalent to the following statement: whenever F is a collection of at most ℵ_{2} many one-to-one functions from ω_{1} to ω_{1}, there is a one-to-one g : ω_{2} → ω_{2} such that whenever β is closed under g and f is in F, there is a countable D ⊆ β such that:

• if ξ ∈ β \ D, then
f(min(e_{β}(ξ), e_{β}(g(ξ)))) < max(e_{β}(ξ), e_{β}(g(ξ))),
• if ξ ≠ η ∈ β \ D, then
f(min(e_{β}(g(ξ)), e_{β}(g(η)))) < max(e_{β}(g(ξ)), e_{β}(g(η))), .

3. Robust counterexamples to BA_{κ}

In this section, we will show how the results discussed in the previous section can be used to build robust counterexamples to BA_{ℵ_{1}} and BA_{ℵ_{2}}. This will demonstrate limitations as to what methods could be used to establish the consistency of BA_{ℵ_{2}}. It will also give a new proof of an old theorem of Abraham and Shelah that BA_{ℵ_{1}} is not a consequence of MA_{ℵ_{1}}.

In this section we will examine the following combinatorial statement for cardinals μ ≤ κ ≤ λ:

ED(κ, λ, μ): If F ⊆ κ^{κ} consists of countable-to-one functions and |F| ≤ λ, then there exists a countable-to-one g ∈ κ^{κ} such that for all f in F, {α ∈ κ : f(α) = g(α)} has cardinality less than μ.

Thus (∗) is just the assertion ED(ω_{2}, ω_{2}, ω_{1}). Also, it is not difficult to show that ED(ω_{1}, λ, ω_{1}) is equivalent to the assertion that every subset of ω_{1}^{ω_{1}} of cardinality at most λ is bounded in the order of eventual dominance.

Proposition 3.1. Assume MA_{λ}(σ-linked). If κ ≤ λ are cardinals, F ⊆ κ^{κ} consists of partial one-to-one functions, and |F| ≤ λ, then there is an enumeration r : κ → 2^{ω} of a κ-dense subset of 2^{ω} such that, for each f in F, the function r(α) ↦ r(f(α)) can be covered by countably many increasing functions. In particular if ED(κ, λ, ω_{1}) is false and MA_{λ}(σ-linked) is true, then there is a κ-dense X ⊆ ℝ such that X is not isomorphic to a suborder of −X.

Proof. Let F = {f_{ξ} : ξ ∈ λ} be given. Define a poset Q as follows. The underlying set consists of all pairs q = (r_{q}, X_{q}) such that:

(1) for some finite subset D_{q} of κ and l_{q} ∈ ω, r_{q} is a one-to-one function from D_{q} into 2^{l_{q}};

(2) X_{q} is a function from λ × ω into the powerset of D_{q} and

{(ξ, k) ∈ λ × ω : X_{q}(ξ, k) ≠ ∅}

is finite;

(3) if (ξ, k) ∈ λ × ω and α ∈ X_{q}(ξ, k) is in the domain of f_{ξ}, then f_{ξ}(α) is in D_{q}. Similarly, if α ∈ X_{q}(ξ, k) is in the range of f_{ξ}, then f_{ξ}^{−1}(α) is in D_{q};

(4) if (ξ, k) ∈ λ × ω and α ≠ α′ ∈ X_{q}(ξ, k), then r_{q}(α) <_{lex} r_{q}(α′) if

and only if r_{q}(f_{ξ}(α)) <_{lex} r_{q}(f_{ξ}(α′)). Define q ≤ p for p, q ∈ Q to mean:

(5) D_{p} ⊆ D_{q} and l_{p} ≤ l_{q}; (6) if α is in D_{p}, then r_{p}(α) is an initial part of r_{q}(α);

(7) if (ξ, k) ∈ λ × ω, then X_{p}(ξ, k) ⊆ X_{q}(ξ, k). If q is in Q, let D_{q}(i) (i < m_{q}) be an increasing enumeration of D_{q} and let F_{q}(j) (j < n_{q}) be an increasing enumeration of

F_{q} = {ξ ∈ λ : ∃k ∈ ω(X_{q}(ξ, k) ≠ ∅)}

If i < m_{q}, define r̄_{q}(i) = r_{q}(D_{q}(i)). If j < n_{q} and k < ω, define

X̄_{q}(j, k) = {i < m_{q} : D_{q}(i) ∈ X_{q}(F_{q}(j), k)}.

If i, i′ < m_{q} and j < n_{q}, set f̄_{q}(i, j) = i′ if f_{F_{q}(j)}(D_{q}(i)) = D_{q}(i′); f̄_{q}(i, j) is undefined if no such i′ exists. Notice that by our hypothesis, λ ≤ 2^{ℵ_{0}} and hence there exists a sequence s_{ξ} (ξ ∈ λ) of distinct elements of 2^{ω}. Let s̄_{q} denote the pair

(⟨s_{D_{q}(i)} ↾ k : i < m_{q}⟩, ⟨s_{F_{q}(i)} ↾ k : i < n_{q}⟩)

where k is minimal such that no element of 2^{k} occurs more than once in either coordinate. The tuple (m_{q}, n_{q}, r̄_{q}, X̄_{q}, f̄_{q}, s̄_{q}) will be referred to as the type of q.

Claim 3.2. Q is σ-linked.

Remark 3.3. It is unclear whether Q is in fact σ-centered or whether MA_{λ}(σ-centered) is sufficient as a hypothesis is Theorem 1.4.

Proof. It suffices to show that if p, p′ ∈ Q have the same type, then p and p′ are compatible. Given such p and p′, define q as follows:

D_{q} = D_{p} ∪ D_{p′} X_{q}(ξ, j) = X_{p}(ξ, j) ∪ X_{p′}(ξ, j).

If α is in D_{p}, define r_{q}(α) = r_{p}(α)^{⌢}0; if α is in D_{p′} \ D_{p}, define r_{q}(α) = r_{p′}(α)^{⌢}1. First observe that if α is in D_{p} ∩D_{q}, then since the types of p and p′ are in the same, there is an i < m such that D_{p}(i) = α = D_{p′}(i). In particular, r_{p}(α) = r̄(i) = r_{p′}(α). Thus it is sufficient to verify that q is in Q as it will then follow immediately that q ≤ p, p′.

Since both p and p′ are in Q, the only condition which is nontrivial to check is (4). Toward this end, suppose that (ξ, k) ∈ λ × ω, α ≠ α′ ∈ X_{q}(ξ, k). Without loss of generality, we may assume that α ∈ D_{p} \ D_{p′} and α′ ∈ D_{p′} \ D_{p}. Notice that since p and p′ satisfy condition (3), we have that f_{ξ}(α) is in D_{p} \ D_{p′} and f_{ξ}(α′) is in D_{p′} \ D_{p}. Let i, i′ < m be such that D_{p}(i) = α and D_{p′}(i′) = α′. Notice that since the types of p and p′ coincide, there is a j < n such that F_{p}(j) = Fp̄(j) = ξ.

First suppose that i = i′. In this case r_{q}(α) = r̄(i)^{⌢}0 and r_{q}(α′) = r̄(i)^{⌢}1. Furthermore,

r_{q}(f_{ξ}(α)) = r̄(f̄(i, j))^{⌢}0 r_{q}(f_{ξ}(α′)) = r̄(f̄(i′, j))^{⌢}1.

It follows that r_{q}(α) <_{lex} r_{q}(α′) and r_{q}(f_{ξ}(α)) <_{lex} r_{q}(f_{ξ}(α′)).

Next, suppose that i ≠ i′ and let β = D_{p}(i′). Since α is in X_{p}(ξ, k) and α′ is in X_{p′}(ξ, k), both i and i′ are in X̄(j, k). It follows that β is in X_{p}(ξ, k). Then

r_{p}(β) = r_{p′}(α′) = r̄(i′) ≠ r̄(i) = r_{p}(α)
r_{p}(f_{ξ}(β)) = r_{p′}(f_{ξ}(α′)) = r̄(f̄(i′, j)) ≠ r̄(f̄(i, j)) = r_{p}(f_{ξ}(α))

and hence r_{q}(α) <_{lex} r_{q}(α′) is equivalent to r_{p}(α) <_{lex} r_{p}(β). Similarly

r_{q}(f_{ξ}(α)) <_{lex} r_{q}(f_{ξ}(α′))

is equivalent to

r_{p}(f_{ξ}(α)) <_{lex} r_{p}(f_{ξ}(β)).

Since p is in Q, this establishes that r_{q}(α) <_{lex} r_{q}(α′) is equivalent to r_{q}(f_{ξ}(α)) <_{lex} r_{q}(f_{ξ}(α′)). □

Notice that, for each α ∈ κ and l ∈ ω, the set of all q in Q such that α is in D_{q} and l ≤ l_{q} is dense. Furthermore, for each α ∈ κ and ξ ∈ λ, the set of q in Q for which there is a j such that α is in X_{q}(ξ, j) is dense in Q. Finally, if s is a finite binary sequence and α ∈ κ, then the set of q such that there is a β ∈ D_{q} with r_{q}(β) extending s and α ∈ β is dense in Q. Let G ⊆ Q be a filter which meets each of these dense sets and define, for each α ∈ κ, ξ ∈ λ, and j ∈ ω,

r(α) = ⋃_{q∈G} r_{q}(α) X(ξ, j) = ⋃_{q∈G} X_{q}(ξ, j)

It follows that r enumerates a κ-dense subset of 2^{ω} and that for all ξ ∈ λ, ⋃_{j=0}^{∞} X(ξ, j) = κ. Moreover, if α ≠ β are in X(ξ, j) for some ξ ∈ λ and j ∈ ω, then r(α) <_{lex} r(β) if and only if r(f(α)) <_{lex} r(f(β)). □

Notice that from this we can readily prove that MA_{ℵ_{1}} does not imply BA_{ℵ_{1}}: observe that in any c.c.c. forcing extension of a model of 2^{ℵ_{1}} = ℵ_{2}, ED(ω_{1}, ω_{2}, ω_{1}) fails. Thus if we force MA_{ℵ_{2}} by a c.c.c. forcing over any model of 2^{ℵ_{1}} = ℵ_{2} (which is possibly by [11]), we obtain a model of MA_{ℵ_{1}} in which BA_{ℵ_{1}} fails. This should be compared with the more involved iterated forcing construction of [3].

4. A strategy for obtaining the consistency BA_{ℵ_{2}}

In this section we will prove Theorem 1.5. The bulk of the work is in showing that if X and Y are subsets of ℝ of cardinality ℵ_{2}, (∗) is true, and 2^{ℵ_{0}} ≤ ℵ_{2}, then there is a c.c.c. forcing of cardinality ℵ_{2} which forces that there is a function f : X → Y which is a countable union of increasing subfunctions. The main ingredient in the proof is the following lemma which is essentially a consequence of the proof of [13, 4.2].

Lemma 4.1. Suppose that x_{α} (α ∈ ω_{1}) and y_{α} (α ∈ ω_{1}) are two given ω_{1}-sequences of elements of ℝ. There exists a family F ⊆ ω_{1}^{ω_{1}} consisting of countable-to-one functions such that |F| = 2^{ℵ_{0}} and if g : ω_{1} → ω_{1} is a countable-to-one function such that for every f in F there is a countable D ⊆ ω_{1} such that

• for all α ∈ ω_{1} \ D, f(min(α, g(α))) < max(α, g(α)) and

• for all α ≠ β ∈ ω_{1} \ D, f(min(g(α), g(β))) < max(g(α), g(β)), then the poset Q consisting of all finite q ⊆ ω_{1} such that {(x_{α}, y_{g(α)}) : α ∈ q} is increasing is powerfully c.c.c.

Proof. For each continuous function F from a Borel subset ℝ^{2n−1} into ℝ for some n, define f_{F} : ω_{1} → ω_{1} by setting f_{F} (α) equal to the least upper bound of the set of all β such that there exist ξ_{i} (i < n) and η_{i} (i < n) less than α such that one of the following holds:

F (x_{ξ_{0}}, y_{η_{0}}, …, y_{η_{n−2}}, x_{ξ_{n−1}}) = y_{β}
F (x_{ξ_{0}}, y_{η_{0}}, …, y_{η_{n−2}}, y_{η_{n−1}}) = x_{β}

Let F denote the collection of all such f_{F}. The proof of [13, 4.2] shows that this F satisfies the conclusion of the lemma. □

Lemma 4.2. Assume (∗∗) and 2^{ℵ_{0}} ≤ ℵ_{2}. If X and Y are two subsets of ℝ of cardinality ℵ_{2}, then there is a c.c.c. forcing Q of cardinality ℵ_{2} such that, after forcing with Q, there is a one-to-one function from X to Y which is a countable union of increasing subfunctions.

Proof. Fix, for each uncountable β < ω_{2}, a bijection e_{β} : β → ω_{1}. Let x_{α} (α ∈ ω_{2}) and y_{α} (α ∈ ω_{2}) be enumerations of X and Y, respectively, without repetition. For each uncountable β < ω_{2}, let F_{β} ⊆ ω_{1}^{ω_{1}} satisfy the conclusion of Lemma 4.1 for the sequences ⟨x_{e_{β}^{−1}(ξ)} : ξ ∈ ω_{1}⟩ and ⟨y_{e_{β}^{−1}(ξ)} : ξ ∈ ω_{1}⟩. Set F = ⋃_{β∈ω_{2}} F_{β}, noting that by our assumption, |F| ≤ ℵ_{2}. Applying (∗∗), there is a countable-to-one function g : ω_{2} → ω_{2} such that which satisfies the conclusion of Lemma 2.3 with respect to F.

Define Q to consist of all finite subsets q of ω_{2} such that {(x_{α}, y_{g(α)}) : α ∈ q} is strictly increasing as a partial function from ℝ to ℝ; we view Q as a poset with the order of reverse inclusion. By Lemma 4.1, every suborder of Q of cardinality ℵ_{1} is powerfully c.c.c. and hence Q is powerfully c.c.c.. It is easily verified that the countable (finite support) power of Q forces that x_{α} ↦ y_{g(α)} is a union of countably many increasing subfunctions. □

We also recall the following technique of the second author (see the proof of Corollary 8.3 of [13]) for proving the results in Baumgartner’s [4].

Lemma 4.3. [13] Suppose that MA_{κ}(σ-centered) holds and whenever X, Y ⊆ ℝ with |X| = |Y | = κ, there is a σ-increasing function from X into Y. Then every two κ-dense subsets of ℝ are order isomorphic.

Theorem 4.4. Assume (∗), 2^{ℵ_{0}} ≤ ℵ_{2}, and 2^{ℵ_{1}} = 2^{ℵ_{2}} = ℵ_{3}. There is a c.c.c. poset which forces:

• MA_{ℵ_{2}};

• if κ ≤ ℵ_{2}, then every two κ-dense sets of reals are isomorphic.

Proof. First observe that (∗) is preserved by c.c.c. forcing, since every countable-to-one function from ω_{2} to ω_{2} in a generic extension can be covered by countably many such ground model functions. Using Lemma 4.2 and standard bookkeeping arguments (see [11] or [7]), build a finite support iteration of c.c.c. forcings ⟨P_{α}; Q̇_{α} : α ∈ ω_{3}⟩ such that:

• if Q̇ is a P_{ω_{3}}-name for a c.c.c. forcing of cardinality at most ℵ_{2}, then there is an α ∈ ω_{3} such that Q̇_{α} = Q̇.

• if κ ≤ ℵ_{2} and Ẋ and Ẏ are P_{ω_{2}}-names for subsets of ℝ of cardinality κ, then there is an α ∈ ω_{3} such that every condition of Q̇_{α} forces that there is an at most countable-to-one σ-increasing function from Ẋ into Ẏ.

It follows from Lemma 4.3 that the resulting model satisfies MA_{ℵ_{2}} and both BA_{ℵ_{1}} and BA_{ℵ_{2}}. □

In order to see how to obtain the relative consistency of BA_{ℵ_{1}} with 2^{ℵ_{0}} large, start with a model of GCH and let κ ≥ ℵ_{2} be a cardinal. First iterate Jensen’s fast club forcing with countable support to obtain a forcing extension with the same cardinals such that every subset of ω_{1}^{ω_{1}} of cardinality at most κ is bounded in the order of eventual dominance — i.e. that ED(ω_{1}, κ, ω_{1}) holds. Notice that ED(ω_{1}, κ, ω_{1}) is preserved by c.c.c. forcing and if 2^{ℵ_{0}} ≤ κ, then by Lemma 4.1, it implies that if X and Y are two ℵ_{1}-dense suborders of ℝ, then there is a c.c.c. forcing of cardinality ℵ_{1} which makes X and Y isomorphic as linear orders. Now construct a finite support iteration of c.c.c. forcings of length κ^{+} such that every intermediate stage of the construction forces 2^{ℵ_{0}} ≤ κ and such that, by the end of the iteration, BA_{ℵ_{1}} holds (of course we can arrange MA_{κ} as well). (This is essentially the same proof as given in [1], although it is cast in a different language.)

5. Concluding remarks and recent developments

The original submission of this paper was made in March 2014 and contained only the combinatorial statement (∗) and a flawed proof of the version of Theorem 1.5 in which (∗∗) was replaced with (∗). In late September 2014, Itay Neeman pointed out the error (which was in an analog of Lemma 4.1). At a workshop at CIRM Luminy the following week he announced that under a suitable large cardinal assumption, there is a forcing extension by a σ-closed forcing in which (∗) and CH both hold. The combinatorial statement (∗∗) was formulated later by the authors following discussions with Neeman at the time of the workshop; it is natural strengthening of (∗) which is needed to make the original arguments valid. Neeman has since adapted his methods to yield the consistency of (∗∗) relative to a large cardinal hypothesis.

The following problems are, at least at present, left open by Neeman’s work.

Problem 5.1. Does there exist an n such that either 𝔱 ≤ ℵ_{n} or else there is a κ ≤ ℵ_{n} such that BA_{κ} fails?

Problem 5.2. Does the conjunction of BA_{ℵ_{2}} and MA_{ℵ_{2}} imply that there is a cardinal which is inaccessible in the constructable universe?

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

Stevo Todorcevic, Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

Stevo Todorcevic, Institute de Mathématique de Jussieu, UMR 7586, Case 247, 4 Place Jussieu, 75252 Paris Cedex, France

  E-mail address: justin@math.cornell.edu
  E-mail address: stevo@math.utoronto.ca
  E-mail address: stevo@math.jussieu.fr