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«SPENCER UNGER Abstract. From large cardinals we obtain the consistency of the existence of a singular cardinal κ of cofinality ω at which the ...»

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ARONSZAJN TREES AND THE SUCCESSORS OF A SINGULAR

CARDINAL

SPENCER UNGER

Abstract. From large cardinals we obtain the consistency of the existence of

a singular cardinal κ of cofinality ω at which the Singular Cardinals Hypothesis

fails, there is a bad scale at κ and κ++ has the tree property. In particular

this model has no special κ+ -trees.

1. Introduction We prove the following result.

Theorem 1.1.

If κ is supercompact and λ κ is weakly compact, then there is a forcing extension in which κ is a singular strong limit cardinal of cofinality ω, SCH fails at κ, there is a bad scale at κ and the tree property holds at κ++.

To begin we recall some basic definitions.

Definition 1.2. Let κ and λ be cardinals with κ regular.

(1) A κ-tree is a tree of height κ with levels of size less than κ.

(2) A cofinal branch through a tree T is a linearly ordered subset of T whose order type is the height of T.

(3) A κ-Aronszajn tree is a κ-tree with no cofinal branch.

(4) κ has the tree property if and only if there are no κ-Aronszajn trees.

(5) A λ+ -tree T is special if and only if there is a function f : T → λ such that if x and y are comparable in the tree, then f (x) = f (y).

Definition 1.3. For a singular cardinal η, the Singular Cardinal Hypothesis (SCH) at η is the assertion that if η is strong limit, then 2η = η +.

The tree property is well studied. There are many classical results and there has also been some recent research. We review the classical results. The tree property at ℵ0 is precisely K¨nig’s Lemma [7]. Aronszajn [9] constructed an ℵ1 - o Aronszajn tree. Generalizing Aronszajn’s construction Specker [17] proved that if κκ = κ, then there is a special κ+ -tree. In particular CH implies that there is a special ω2 -tree. Mitchell [13] proved that relative to ZFC the tree property at ω2 is equiconsistent with the existence of a weakly compact cardinal. Variations of the forcing from Mitchell’s result play a central role in further forcing results about the tree property.

Date: January 19, 2013.

2010 Mathematics Subject Classification. 03E35,03E55.

Key words and phrases. Large Cardinals, Forcing, Tree Property, Special Tree, Bad Scale.

The results in this paper are to appear in the Author’s PHD thesis under the direction of James Cummings, to whom the author would like express his gratitude.

1

2 SPENCER UNGER

An old question asks whether it is consistent that every regular cardinal greater that ℵ1 can have the tree property. One of the main obstacles is arranging the tree property at the successors of a singular cardinal. Shelah [11] proved that if ν is a singular limit of supercompact cardinals, then ν + has the tree property. The main result of [11] shows that assuming the existence of a little more than a huge cardinal, it is relatively consistent that ℵω+1 has the tree property. Recently Sinapova [16] showed that one can obtain the tree property at ℵω+1 from just ω supercompact cardinals using a very different construction from Magidor and Shelah.

For this paper we are motivated by trying to arrange the tree property at both the successor and double successor of a singular cardinal. For a general singular κ there are two relevant partial results. First, Cummings and Foreman [3] have shown from a supercompact cardinal with a weakly compact cardinal above, it is relatively consistent that there is a singular cardinal of cofinality ω whose double successor has the tree property. Second, using a forcing of Gitik and Sharon [5], Neeman [14] proved that starting from ω supercompact cardinals it is consistent that there is a singular cardinal of cofinality ω at which SCH fails and whose successor has the tree property. Making the singular cardinal into a small cardinal like ℵω or ℵω2 is difficult. Recently, Sinapova [15] was able to define a version of the Gitik-Sharon forcing to obtain the analog of Neeman’s result [14] where κ = ℵω2. The result for κ = ℵω is open.

Our forcing is a combination of the Gitik-Sharon [5] forcing and the forcing from the result of Cummings and Foreman [3]. The forcing from [3] is a variant of a forcing due to Mitchell [13]. In the model for Theorem 1.1, we prove that there is a bad scale at κ. A bad scale at κ is a PCF theoretic object whose existence implies κ+ ∈ I[κ+ ] which in turn implies the failure of weak square, ∗. By a theorem of / κ Jensen [6] weak square is equivalent to the existence of a special Aronszajn tree.

So in particular the model for Theorem 1.1 has no special κ+ -trees. For an account of scales and their use in singular cardinal combinatorics we refer the interested reader to [2].

There is a natural model related to the model for Theorem 1.1 which is a candidate for the full tree property at κ+. We are kept from this further result by difficulties involved in reproducing the argument of Neeman [14]. To illustrate this our presentation of the forcing will take an increasing sequence of regular cardinals κn | n ω as a parameter. If we take κn = κ+n for all n ω as in [5], then we obtain the model for Theorem 1.1. If we instead let each κn be a supercompact cardinal as in [14], then we obtain a model that is a candidate for the full tree property at κ+. We will also prove that there are no special κ+ -trees in the model obtained from letting the κn ’s be supercompact. We include this argument, because it is different from the argument given in the proof of Theorem 1.1.





The paper is organized as follows. In Section 2 we formulate a branch lemma, which will be used in the proof of Theorem 1.1 and has independent interest. We also recall another classical branch lemma needed in the proof below. In Section 3 we prove some preliminary lemmas, which allows us to define the main forcing. In Section 4 we define the main forcing and prove some of its properties. In Section 5 we prove that regardless of the choice of the sequence κn | n ω, the tree property holds at κ++ in the extension. In Section 6 we give the two different models which both have no special κ+ -trees.

ARONSZAJN TREES AND THE SUCCESSORS OF A SINGULAR CARDINAL 3

–  –  –

We will need this stronger lemma to prove that the tree property holds in our model. We will also need the following lemma which is used in [3] and [1]. We refer the interested reader to either paper for a proof.

Lemma 2.5 (Silver).

Suppose that τ, η are cardinals with 2τ ≥ η. If Q is τ + -closed, then forcing with Q cannot add a branch through an η-tree.

3. Preliminaries to the main forcing We give some definitions and results which allow us to define the main forcing.

For the remainder of the paper we work in a ground model V of GCH where κ is a supercompact cardinal which is indestructible under κ-directed closed forcing [10]. For ease of argument we are going to assume that λ κ is measurable with U ∗ a normal measure on λ. Weakening the result to use only weak compactness is straightforward. Let κn | n ω be an increasing sequence of regular cardinals less than λ with κ = κ0. Let ν be the supremum of the κn ’s. Since ν will be collapsed and ν + will be preserved, we let µ = ν +.

Let A = Add(κ, λ). In V A, κ is still supercompact. We let U be a supercompactness measure on Pκ (µ) and for each n ω let Un be the projection of U on to Pκ (κn ). The measures Un concentrate on the sets Xn of x ∈ Pκ (κn ) such that x ∩ κ is an inaccessible cardinal. We define P the diagonal Prikry forcing in the model V A using the measures Un.

Definition 3.1. A condition in P is a sequence p = x0, x1,... xn−1, An, An+1,...

where (1) for all i n, xi ∈ Xi, (2) for all i n − 1, xi ⊆ xi+1 ∩ κi and o. t.(xi ) κ ∩ xi+1 and (3) for all i ≥ n, Ai ∈ Ui and Ai ⊆ Xi.

We call n the length of p and denote it (p). Given another condition q = y0,... ym−1, Bm, Bm+1,...

we define p ≤ q if and only if n ≥ m, for all i m, yi = xi, for all i with m ≤ i n, xi ∈ Bi, and for all i ≥ n Ai ⊆ Bi.

Using the measurability of λ we are going to show that there are many places where the measure U (and hence each Un ) reflects. For α λ let Aα be Add(κ, α).

˙ Let U be an A-name for U. Following the set up of [3] we have the following lemma.

Lemma 3.2.

There is a set B ⊆ λ of Mahlo cardinals with B ∈ V such that ˙ (1) if g is A-generic over V, then for all α ∈ B, ig (U ) ∩ V [g α] ∈ V [g α] and (2) B ∈ U ∗.

ARONSZAJN TREES AND THE SUCCESSORS OF A SINGULAR CARDINAL 5

˙ Proof. Let β λ. For each canonical Aβ -name X for a subset of Pκ (µ) choose ˙ ˙ a maximal antichain of conditions in A deciding the statement “X ∈ U ”. By the κ+ -cc of A and the inaccessibility of λ, the supremum of the domains of conditions in A appearing in any of the above antichains is less than λ. Let F (β) λ be greater than this supremum. The set of limit points of F is club. Let B be the set of Mahlo limit points of F. B is as required for the lemma.

˙ Let g be A-generic over V. For each α ∈ B, let U α =def ig (U ) ∩ V [g α]. It is α clear that U is a supercompactness measure on Pκ (µ) in V [g α]. For α ∈ B we define Pα in V [g α] to be the Diagonal Prikry forcing obtained from U α in the α same way we defined P from U. We call the associated measures Un. Next we note that a Prikry sequence for P gives a Prikry sequence for Pα. This follows from a characterization of genericity for Prikry forcing due to Mathias [12]. The version needed here is that x is P-generic if and only if for all sequences of measure one sets A(n) | n ω, x(n) ∈ A(n) for all sufficiently large n. From this it is easy to see that an A ∗ P-generic object induces a Aα ∗ Pα -generic object for each α ∈ B.

In particular, we just restrict the A-generic object and use the same Prikry generic sequence. It follows that RO(Aα ∗ Pα ) is isomorphic to a complete subalgebra of RO(A ∗ P) where RO(−) denotes the regular open algebra. Our work with the above posets will rely on the notion of a projection.

Definition 3.3. Let P and Q be posets. A map π : P → Q is a projection if (1) π(1P ) = 1Q, (2) for all p, p ∈ P, p ≤ p implies that π(p ) ≤ π(p) and (3) for all p ∈ P and q ≤ π(p), there is p ≤ p such that π(p ) ≤ q.

Definition 3.4. Suppose that π : P → Q is a projection. Then in V Q define ˙ P/Q = {p ∈ P | π(p) ∈ GQ } ordered as a suborder of P. If G is Q-generic, then we may write P/G for P/Q as computed in V [G].

Fact 3.5.

In the context of Definition 3.4, P is isomorphic to a dense subset of Q ∗ P/Q.

We now continue with facts about A ∗ P and related posets.

Lemma 3.6.

For all α ∈ B there is a projection πα : A ∗ P → RO(Aα ∗ Pα ).

Proof. This follows from general considerations about the regular open algebras of posets. First we use the map that takes A ∗ P densely into its regular open algebra.

Then viewing RO(Aα ∗ Pα ) as its isomorphic copy inside RO(A ∗ P), we take the meet over all conditions in RO(Aα ∗ Pα ) that are above a given condition in the range of the first map. It is easy to see that the composition of the above two maps gives a projection.

Note that the projection we get here did not rely on special properties of λ so by a similar argument we have the following lemma.

Lemma 3.7.

For every α, β ∈ B with α β, there is a projection πα,β : Aβ ∗ Pβ → RO(Aα ∗ Pα ) Remark 3.8. In the previous two lemmas we used the fact that there are projections from RO(A ∗ P) to RO(Aβ ∗ Pβ ) and from RO(Aβ ∗ Pβ ) to RO(Aα ∗ Pα ) which we denote σβ and σα,β respectively. We also note that σα,β ◦ πβ = πα,β.

6 SPENCER UNGER

–  –  –

It is easy to see that this proof applies to Aα ∗ Pα for α ∈ B. Before moving on to the definition of the main forcing we record some facts about the extension by A ∗ P. The proofs of properties of P which lead to the following lemma are easy adaptations of the proofs in [5].

Lemma 3.10.

V A∗P satisfies (1) κ is singular strong limit of cofinality ω, (2) κ+ = (ν + )V = µ and (3) 2κ = λ.

–  –  –

respectively.

Using the first projection we see that 2κ ≥ λ and using the second projection we see that each α ∈ B is collapsed to have size µ in the extension by R. As in the Cummings-Foreman paper, we have that the extension by R is contained in an extension by (A ∗ P) × Q where Q is µ-closed.

Definition 4.4. Let Q be the set of third coordinates from R together with the ordering f1 ≤ f2 if and only if dom(f1 ) ⊇ dom(f2 ) and for all α ∈ dom(f2 ), Aα ∗Pα f1 (α) ≤ f2 (α).

Lemma 4.5.

Q is µ-closed and the identity map is a projection from (A ∗ P) × Q to R.

The proof is a straightforward adaptation of Lemma 2.8 of [1].

For suitable choice of generics we have V A∗P ⊆ V R ⊆ V (A∗P)×Q. Using these facts we can prove the following lemma.

Lemma 4.6.

V R satisfies (1) κ is singular strong limit of cofinality ω, (2) ν is collapsed to have size κ and µ is preserved and (3) 2κ = κ++ = λ.

Proof. By Lemma 4.5, every µ sequence from V R is in V A∗P. It follows that κ is singular strong limit of cofinality ω in V R. It also follows that µ is preserved since if it were collapsed then it would have been collapsed by A ∗ P. Since R projects on to A ∗ P, we have that ν is collapsed to have size κ and 2κ ≥ λ. We have that 2κ = λ, since 2κ = λ in V A∗P and every κ sequence from V R is in this model. Finally, each β ∈ B is collapsed to have cofinality µ by Lemma 4.3 and λ is preserved by Lemma 4.2.



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