Chang's Conjecture, The Weak Reflection Principle and the Tree Property at $\omega_2$

We prove that a strong version of Chang's Conjecture, equivalent to the Weak Reflection Principle at $\omega_2$, together with $2^\omega=\omega_2$, imply there are no $\omega_2$-Aronszajn trees.


Introduction
We say that ω 2 has the tree property if every tree of height ω 2 and levels of size ℵ 1 has a cofinal branch, and it is usually abbreviated by TP(ω 2 ).A tree of height ω 2 with levels of size at most ℵ 1 with no cofinal branches are usually called an ℵ 2 -Aronszajn tree.
In these notes we work with a strong version of Chang's Conjecture (see Definition 3.1).On one hand, Todorcevic and Torres-Perez proved that under a stronger version of CC * , the negation of CH implies there are no special ℵ 2 -Aronszajn trees (see [9]).
On the other hand, Sakai and Velickovic proved that under SSR, the negation of CH together with MAω 1 (Cohen) imply the strong tree property at ω 2 and so in particular they imply TP(ω 2 ) (see [6]).

Some preliminaries
We denote by pred T (t) the set of all the < T -predecessors of t in T, and by ht T (t) = o.t.(pred T (t)).We will denote by T ξ = {t ∈ T : ht T (t) = ξ}.Often we will just drop off the subindex T if the context is clear.
For A, B ⊆ T we denote by A ⊥ B if for every s ∈ A and every t ∈ B, s and t are not comparable.Similarly, for s, t ∈ T and A ⊆ T , let s ⊥ t and s ⊥ A iff {s} ⊥ {t} and {t} ⊥ A respectively.
The following fact is used: Fact 2.1 (Shelah [7], Chapter XIII, 1.6 and 1.10).Assume that κ is a strongly compact cardinal.Let (Pα, Qβ : α ≤ κ, β < κ) be a revised countable support iteration of semi-proper posets of size Assume that κ is strongly compact in V .Let (Pα, Qβ : α ≤ κ, β < κ) be the countable support iteration of random forcing.Here recall that a revised countable support iteration coincides with a countable support iteration for proper posets.Note also that κ = ω 2 in V Pκ .Hence SSR holds in V Pκ by the above fact.Moreover, MAω 1 (Cohen) fails in V Pκ as adding random reals makes non(B) into ω 1 .

Main Theorem
Theorem 3.1.Under CC * , ¬CH is equivalent to the tree property at ω 2 .
We follow very closely the proof of Theorem 2.2 in [9].Observe that it is already known that TP(ω 2 ) implies ¬CH.
We will need the following Proposition for the proof of Lemma 3.1, namely in Claim 3.1.Proposition 3.1.Let T be a κ-Aronszajn tree (κ a regular cardinal).Given a regular cardinal µ < κ, consider a family of collection of nodes A ξ : ξ ∈ X such that X contains a stationary set consisting of ordinals of cofinality at least µ, A ξ ⊆ T ξ and |A ξ | < µ for every ξ ∈ X.Then for every λ large enough such that {κ, T, X, A ξ : ξ ∈ X , . ..} ⊂ H λ and for every elementary submodel N ≺ H λ ; ∈, <, κ, T, X, A ξ : ξ ∈ X , . . .such that A ξ ⊆ N for every ξ ∈ X ∩ N , then for every t ∈ T of height at least sup(N ∩ κ) there are unboundedly many (in sup(N ∩ κ)) ξ ∈ X ∩ N such that every s ∈ A ξ is incomparable with t.
Proof.Suppose otherwise, and take t ∈ T of height at least sup(N ∩ κ) and α ∈ N such that for all ξ ∈ X ∩ N \ α, there is a node t ξ ∈ A ξ such that t ξ ≤ T t.Without loss of generality, we can suppose that X is a stationary set consisting of ordinals of cofinality at least µ.
Since |A ξ | < µ for any ξ ∈ X ∩ N \α, there is an ordinal β ξ < ξ such that for any s, s By elementarity and using Fodor's Lemma, we can find β ∈ N ∩ X and a stationary set S ∈ N such that for any ξ ∈ S, s Then for every s ∈ A β , we can define a function fs : S → T such that fs(ξ) is the unique and therefore fs is defined in N .However, by our initial assumption, fs(ξ) = t ξ for every ξ ∈ S ∩ N , and so fs defines in N a cofinal branch of T , contradiction.
Let T be an ℵ 2 -Aronszajn tree.In order to simplify the proof, without loss of generality, we suppose that T ⊆ ω 2 and let e : ω 2 × ω 1 → T be a bijective function such that e(δ, ξ) ∈ T δ for every (δ, ξ) ∈ ω 2 × ω 1 .Let θ be sufficiently large such that T , e and all relevant parameters are members of H θ .Lemma 3.1.Assume CC * and that T is a ℵ 2 -Aronszajn tree.For every M ≺ H θ countable, and for every η 0 , η 1 ∈ ω 2 , we can find M 0 , M 1 ≺ H θ countable such that: Proof.Fix λ >> θ sufficiently large such that CC * holds in H λ and M, η 0 , η 1 and all relevant parameters are in Fix M 1 witnessing CC* for M and γ.
We need the following Claim: Claim 3.1.For every t ∈ T of height at least γ, there is Proof.Assume otherwise, and take t ∈ T of height at least γ such that for every Working in N and using that CC * holds in N , build a sequence of models Mη : η ∈ ω 2 such that Mη ⊒ M and Mη ∩ ω 2 \ η = ∅ for every η ∈ ω 2 .Let β ξ be the minimum β ∈ ω 2 \ ξ such that there is η ∈ ω 2 such that β ξ = min(Mη ∩ ω 2 \ η).Let η ξ be the minimum η ∈ ω 2 such that Define A ξ : ξ ∈ ω 2 by setting A ξ to be the set of nodes r in T ξ with r ≤ s for some s ∈ Mη ξ ∩ T β ξ .Remark that since Mη ξ is countable, so is A ξ .By Proposition 3.1, there are unboundedly many ξ ∈ N ∩ ω 2 such that t ⊥ A ξ , so choose one of such ξ's.Then there is s ∈ Mη ξ ∩ T β ξ such that s < T t.Thus there is r ∈ A ξ such that r ≤ T s < T t, contradicting that r and t are incomparable.
Take s ∈ T δ 0 ∩ M 0 and t ∈ T δ 1 ∩ M 1 .In particular, there is n ∈ ω and β ∈ M 0 n ∩ ω 2 such that t = tn and t ⊥ T β ∩ M 0 n .Since β ∈ M 0 n ⊆ M 0 , we have s↾ β ∈ M 0 .More ever, since the enumeration function e ∈ M 0 n ⊆ M 0 and M 0 n ∩ ω 1 = M 0 ∩ ω 1 , we have T β ∩ M 0 = T β ∩ M 0 n and so s↾ β ∈ M 0 n .Therefore s↾ β is not comparable with t, and so neither are s and t.
This finishes the proof of Lemma 3.1.
Proof.Let f : [ω 2 ] <ω → ω 2 such that the set C f of closure points of f (i.e.X ∈ C f iff for every e ∈ [X] <ω , f (e) ∈ X) is disjoint with S T .We can suppose that T ⊆ ω 2 and e : ω 1 × ω 2 → T is a bijection such that e(δ, β) ∈ T δ .Let λ be sufficiently large such that T, S T , f, e and all relevant parameters are members of H λ .
Using previous Lemma, build a binary tree Mσ σ∈2 <ω of countable elementary submodels of H λ with the property that for every σ ∈ 2 <ω (1) ).Let δ be the common supremum of every Mr ∩ ω 2 , r ∈ 2 ω .Then for every r ∈ 2 ω , there is tr ∈ T δ ∩ Mr such that for every pred(tr ) ∩ Mr is unbounded in δ.
Claim 3.2.The application r → tr is an injection from 2 ω to T δ (and so CH does not hold).
Proof.Let r 0 , r 1 ∈ 2 ω with r 0 = r 1 and denote by t i the node tr i for i ∈ {0, 1}.We will find two predecessors of t 0 and t 1 that are incomparable.
Let n ∈ ω such that r 0 ↾n= r 1 ↾n= σ, and By the construction of our binary tree, we can take . Therefore, t 0 ↾ δ 0 and t 1 ↾ δ 1 are incomparable, and so t 0 = t 1 .
This finishes the proof of Lemma 3.2.
We are now ready to finish the proof of our Theorem.From the previous lemma we know that the set S T is stationary in [ω 2 ] ℵ 0 .Let S ′ T = S T ∩ Ce, where Ce is the club of all countable subsets of ω 2 closed under the level enumeration function e of T. Definition 3.2.WRP(λ) is the following statement: For any stationary subset S of [λ] ω , there is X ⊂ λ such that We now use that CC * is equivalent to WRP(ω 2 ) (See [5] and [8]).Take X ⊆ ω 2 of size ℵ 1 such that ω 1 ⊆ X and where S ′ T ∩ [X] ω is stationary.Take t ∈ T of height at least sup(X).

Some applications
We mention two applications of our main theorem.The Strong reflection principle (SRP) is a very strong form of reflection principle introduced by Todorcevic.We quote an equivalent definition due to Feng-Jech [3]: For λ ≥ ω 2 , the Strong Reflection Principle at λ, (SRP(λ)) is the statement: for every projective stationary P ⊂ [H λ ] ω , there is a continuous elementary chain N ξ |ξ < ω of countable models such that every N ξ is an element of P .
It is well-known (c.f [3]) that SRP(ω 2 ) imply WRP(ω 2 ) and 2 ω = ω 2 .Therefore a direct application of Theorem 3.1 is the following corollary: We now present the second application about forcing tree property on ω 2 .Such forcing was originally constructed by Michell [4].Then Baumgartner and Laver [1] simplified Mitchell's proof iby using instead a weakly compact length countable support iteration of Sacks forcing to obtain the tree property at ω 2 .In fact, Baumgartner showed that the countable support iteration of many other forcings (including Cohen forcing) of weakly compact length produces models of the tree property at TP(ω 2 ).Applying Theorem 3.1, we get a very general improvement of Baumgartner's result.