aa r X i v : . [ m a t h . L O ] J a n SPECIALIZING TREES WITH SMALL APPROXIMATIONS I
RAHMAN MOHAMMADPOUR
Abstract.
We show that under certain appropriate assumptions implied by PFA,every tree of height ω without cofinal branches is specializable via a proper forcing withfinite conditions which has the ω -approximation property. The forcing constructionemploys internally club ω -guessing models as side conditions. Introduction
By the well-known work of Baumgartner, Malitz and Reinhardt [1], under Martin’sAxiom all trees of height and size ω without cofinal branches are special. Unfortunately,the naive generalizations of MA were not thus far capable of specializing trees of heightand size ω without cofinal branches, see [2], [17], and [18]. This has to do with bothtechnical issues in forcing iterations and the nature of trees of height ω . Regrading theconsistency results, there are many interesting results. Extending the well-known workof Laver and Shelah [11] regarding the consistency of ℵ -Suslin Hypothesis with CH froma weakly compact cardinal, Golshani and Hayut in [6] proved, modulo the consistencyof large cardinals, that it is consistent that for every regular cardinal κ , there are κ + -Aronszajn trees and all of them are special. In a collaboration, Golshani and Shelah [7]proved that it is consistent that every tree of height and size κ + , for a prescribed regularcardinal κ , is special. Let us return to forcing axioms, where the focus of attemptswere mostly on countably closed forcings; Neeman’s discovery [15] of generalized sideconditions did shed light on this problem. If the consistency of a higher analogue of PFAis achievable, then the class of posets under consideration is a priori much larger thancountably-closed posets. It is thus natural to rethink about the question, and speculatewhether such forcing axiom can imply that all trees in a reasonable subclass of trees withheight and size ℵ are special. In this direction, as an early application of his method,Neeman attempted to (weakly) specialize trees of height ω with finite conditions [16],where he attaches Baumgartner’s forcing B ω ( T ) (see Definition 4.11.) to his generalizedside conditions to specialize a given tree T of height and size ω , while keeping an eye Mathematics Subject Classification.
Key words and phrases.
Guessing Models, PFA, Side Condition, Trees, Special Trees, Specialization.The author was supported by Austrian Science Fund (FWF): M 3024. on his iterable class of forcings. Though we are going to step in the same direction,our attitude towards the problem is rather different from his; he attempts to construct aforcing notion to specialize a tree of height ω while staying in his iterable class of forcing,and he demonstrates that it is impossible to obtain a full specialization due to the factthat at the same time he is able to add certain weak version of square principle that isstrong enough to produce non-specializable ω -Aronszajn trees yet remaining loyal to hisiterable class. Though we are going to go through the same line, we will not be thinkingabout iteration and forcing axioms, at least in this paper.Let us elaborate further on the reason that the situation is subtly different in the casesof trees of height ω and of height ω . In the case of trees of height ω , not only withoutcertain cardinal arithmetic assumptions the analogue of Baumgartner’s forcing B ω ( ∗ ),say B ω ( T ), collapses cardinals, but also the lack of cofinal branches is not enough toensure that B ω ( T ) preserves ω . As a matter of fact, a cofinal branch is a special case ofthe notion of an ascending path (see e.g [12] for the definition.) whose existence through atree makes it absolutely non-specializable. The basic idea of an ascending path goes backapparently to Laver (see [19]) who isolated the concept in order to construct a non-specialtree of height ω so that it remains non-special in any transitive outer model with correctcomputation of certain cardinals. The earliest example of a non-special Aronszajn ω -tree was constructed by Baumgartner using (cid:3) ω that also was independently discoveredand generalized by Shelah and Stanley [19]. They showed that (cid:3) λ implies the existenceof non-specializable λ + -Aronszajn trees. The connection between the square principlesand their variations, and ascending paths through trees or tree-like systems have beenstudied by several people, just to mention a few: Baumgartner, Brodsky and Rinot [3],Devlin [5], Cummings [4], Lambie-Hanson[9], Lamibie-Hanson and L¨ucke [10], Laver andShelah [11], L¨ucke [12], Neeman [16], Shelah and Stanley [19], Todorˇcevi´c [20].Thus we are in a spot, where both the behavior of the continuum function and theexistence of ascending paths of width ω could prevent us from specializing trees of height ω . In his paper [12], L¨ucke studied these properties of a tree of and established a bridgebetween the existence of ascending paths and the chain condition of B λ ( T ). He thenasked the following questions:(1) Assume PFA. Is it true that every tree of height ω without cofinal branchesspecializable?(2) If T is a tree of height κ + , for an uncountable regular cardinal κ without ascendingpaths of width less than κ , is then T specializable?Our paper stands on five sections. We give the preliminaries in the next section. Thethird section is devoted to the introduction and basic properties of forcing with pure PECIALIZING TREES WITH SMALL APPROXIMATIONS I 3 side conditions, which will participate as a component in our final forcing. In the fourthsection, we shall introduce our main forcing and analyze its basic properties. Finally,in the fifth section we establish our main result, namely Theorem 5.2, which gives thefollowing as a corollary.
Theorem.
Assume
PFA . Every tree of height ω without cofinal branches is specializablevia a proper and ℵ -preserving forcing with finite conditions. Moreover, the forcing enjoysthe ω -approximation property. This theorem answers L¨ucke’s first question in the affirmative. Given a tree T of height ω with no cofinal branches, we shall use ω -guessing models to construct a properforcing notion P T similar to Neeman’s in [16], so that forcing with P T specializes T .Notice that the existence of sufficiently many ω -guessing models of size ℵ implies thefailure of certain versions of the square principle. It is also worth mentioning that anice observation of L¨ucke in the aforementioned paper states that under the existence ofsufficiently many ω -guessing models of size ℵ , and hence under PFA, no tree of height ω without cofinal branches contains an ascending path of width ω . Interestingly, we willnot use this fact, the presence of guessing models in our side conditions suffices to makeour forcing work.We will also settle his second question above consistently in the affirmative for trees ofheight κ ++ without cofinal branches in our forthcoming paper [14], which in particularincludes a proof of the following theorem. Theorem ([14]) . Suppose there are a supercompact cardinal and an inaccessible cardinalabove it. Assume κ is regular cardinal below the supercompact cardinal one. Then ingeneric extensions by some κ -closed forcing notion, κ ≥ κ +++ and every tree of height κ ++ without cofinal branches is specializable. Preliminaries
We will follow standard conventions and notation, but let us recall some of the impor-tant ones. In this paper, by p ≤ q in forcing relations, we mean p is stronger than q ; bya model, we mean a set or class M such that ( M, ∈ ) satisfies a sufficiently rich fragmentof ZFC; for a cardinal θ , H θ denotes the collection of sets whose hereditary size is lessthan θ ; by M ≺ H θ , we always assume that | M | ⊆ M ; for a set X , we let P ( X ) denotethe power-set of X , and if κ is a cardinal, we let P κ ( X ) denote { A ∈ P ( X ) : | A | < κ } ;a set S ⊆ P κ ( H θ ) is called stationary, if for every function F : P ω ( H θ ) → P κ ( H θ ), thereis M ≺ H θ in S such that M is closed under F .Let us recall the definition of a tree and some related concepts. R. MOHAMMADPOUR
Definition 2.1.
Suppose T = ( T, < T ) is a partially ordered set.(1) For every t ∈ T , b t denotes { s ∈ T : s < T t } .(2) T is called a tree if for every node t ∈ T , the ordered set ( b t , < T ) is well-founded.(3) For every t ∈ T , ht T ( t ) denotes the order type of b t .(4) The height of T , denoted by ht( T ) , is sup { ht T ( t ) + 1 : t ∈ T } .(5) T does not split at limits if for every t, s ∈ T of limit height, b t = b s implies t = s .(6) A set b ⊆ T is called a branch through T if it is a downward closed linearlyordered set with respect to < T .(7) A branch b is called a cofinal branch if its order type is the height of T .(8) For every α ≤ ht( T ) , T α denotes the set of nodes of height α . T ≤ α and T <α havethe obvious meanings. Thus T = T < ht( T ) .(9) T is called rooted if T is a singleton. Definition 2.2.
We call a tree T normal if it is rooted, does not split at limits and forevery t ∈ T , there is s ∈ T with t < T s . Notice that normality may require more properties elsewhere. The following is well-known and easy.
Fact 2.3.
Every tree of limit height is isomorphic to a normal tree.
Definition 2.4.
A tree ( T, < T ) of height κ + is called special if there is a mapping f : T → κ which is injective on branches i.e if s < T t , then f ( s ) = f ( t ) . We call f a specializing function of T . Definition 2.5 (Specializability) . A tree T is said to be specializable if there is a notionof forcing P such that P specializes T without collapsing cardinals up to the height of T . Let us now recall also the following closely related definitions from [13] and [8], respec-tively.
Definition 2.6 (Strong properness) . Suppose P is a forcing notion and S is a collectionof sets.(1) Let X ∈ S . A condition p ∈ P is said to be ( X, P )-strongly generic , if for every q ≤ p , there is some q ↾ X ∈ X ∩ P such that every condition r ∈ P ∩ X extending q ↾ X is compatible with q .(2) We say P is S -strongly proper , if for every X ∈ S and every p ∈ P ∩ X , there isan ( X, P ) -strongly generic condition extending p . PECIALIZING TREES WITH SMALL APPROXIMATIONS I 5
It is easily seen that if p is ( X, P )-strongly generic, where X = M ∩ P , for some M ≺ H θ , then p is ( M, P )-strongly generic, and hence ( M, P )-generic. It turns out thatif a forcing notion is S -strongly proper for some stationary set S ⊆ P κ ( H θ ), then P is S -proper, and hence it preserves κ . Definition 2.7 (Approximation property) . Suppose κ is an uncountable regular cardinal.A forcing notion P has the κ -approximation property , if for every V -generic filter G , andevery A ∈ V [ G ] , A is in V if and only if for every a ∈ V of size less than κ , a ∩ A ∈ V . Notice that it is well-known if a forcing notion is strongly proper for sufficiently manymodels in P κ ( H θ ), then it has κ -approximation property.Recall that an elementary submodel M of H θ is called an internally club model (or IC -model for short) if it is the union of a continuous ∈ -sequence ( M α : α < ω ) of countableelementary submodels of H θ .For a set or class M we say that a set x ⊆ M is bounded in M if there is y ∈ M suchthat x ⊆ y . We now recall the definition of a guessing model from [22]. Definition 2.8.
Suppose M is a set. A set x is guessed in M if there is some g ∈ M such that g ∩ M = x ∩ M . Definition 2.9 (Guessing model) . Let γ be a regular cardinal. A set M is said to be γ - guessing if for any x ⊆ M which is bounded in M , if x is γ -approximated in M i.e x ∩ a ∈ M , for all a ∈ M of size less than γ , then x is guessed in M . The following lemma is easy and we leave it without proof.
Lemma 2.10.
Suppose that M ≺ H θ . Then M is a γ -guessing model if and only if forevery ordinal η ∈ M and every function f : M ∩ η → , if f is γ -approximated in M i.efor every a ∈ M of size less than γ , f ↾ a ∈ M , then it is guessed in M . Definition 2.11 (GM ∗ ( ω )) . The principle GM ∗ ( ω ) states that for every sufficientlylarge regular cardinal θ . The set of ω -guessing IC -submodels of H θ of size ω is stationaryin P ω ( H θ ) . The above principle is a stronger version of Weiss’s ISP( ω ), see [23, 24] and [22]. Theonly difference is that we require the models to be IC. Proposition 2.12 (Viale–Weiss, [22]) . PFA implies GM ∗ ( ω ) . In fact, this proposition has not been proved in [22], the authors mentioned the resultwithout providing a proof. We shall sketch a proof, but before that we need the followinglemma which is also crucial in proving our later claims.
R. MOHAMMADPOUR
Lemma 2.13.
Suppose θ is an uncountable regular cardinal. Assume that M ≺ H θ is countable. Let Z ∈ M a set. Suppose that z f z is a function on P ω ( Z ) in M ,where for each z ∈ P ω ( Z ) , f z is a { , } -value function with z ⊆ dom( f z ) . Assume that f : Z ∩ M → is a function which is not guessed in M . Suppose that B ∈ M is a cofinalsubset of P ω ( Z ) . Then there is B ∗ ∈ M cofinal in B such that for every z ∈ B ∗ , f z * f .Proof. For each ζ ∈ Z , and ǫ = 0 ,
1, let A ǫζ = { z ∈ B : f z ( ζ ) = ǫ } . Notice that the sequence h A ǫζ : ζ ∈ Z, ǫ ∈ { , }i belongs to M . We are done if there is some ζ ∈ Z such that both A ζ and A ζ are cofinal in B , as then by elementarity one can find such ζ ∈ M ∩ Z , andthen pick A − f ( ζ ) ζ . Therefore,let us assume that for every ζ ∈ Z , there is an ǫ ∈ { , } , which is necessarily unique,such that A ǫζ is cofinal in B . Now, define h on Z by letting h ( ζ ) be ǫ if and only if A ǫζ iscofinal is B . Clearly h is in M , but then h ↾ M = f since f is not guessed in M . Thus,there exists ζ ∈ M ∩ Z such that h ( ζ ) = f ( ζ ), but it then implies that A − f ( ζ ) ζ is cofinalin B and belongs to M . Let B ∗ be A − f ( ζ ) ζ . Now if z ∈ B ∗ , f z * f . Sketch of a proof of Proposition 2.12.
Assume PFA. Suppose θ ≥ ω is regular. Forevery countable M ≺ H θ , let F ( M ) = { f : M ∩ γ → γ ∈ M is an uncountable ordinal and f is not guessed in M } . As we are going to show that the set of internally club ω -guessing models is stationaryin H θ , we may fix an algebra F : P ω ( H θ ) → H θ .Let P consist of triples p = ( M p , d p , s p ), where(1) M p is a finite ∈ -chain of countable elementary submodels of H θ which are closedunder F .(2) d p : M p → P ω ( H θ ) is a function such that if M ∈ N are in M p , then d p ( M ) ∈ N .(3) s p is a partial function whose domain is a finite subset of S {F ( M ) : M ∈ M p } and whose range is ω . Suppose that M ∈ N are in M p , f ∈ F ( M ) ∩ dom( s p )and g ∈ F ( N ) ∩ dom( s p ). If f ⊆ g , then s p ( f ) = s p ( g ).The ordering is as follows. p ≤ q if and only if M q ⊆ M p , d q ( M ) ⊆ d p ( M ) for every M ∈ M q , and that s q ⊆ s p .Assume for the moment that P is proper, let us show there is an internally club ω -guessing model closed under F . For every filter G ⊆ P , let M G = S p ∈ G M p . We let M ˙ G be the canonical name for M G . It is forced by the maximal condition to be an ∈ -chain PECIALIZING TREES WITH SMALL APPROXIMATIONS I 7 of countable elementary submodels of H θ whose length is ω . Thus, we may consider itas a sequence and let M G ( α ) denote its α -th element.For each α < ω and m, n ∈ ω , let D α,m,n consist of conditions p such that: • p decides M ˙ G ( α ) and M ˙ G ( α + 1). • The m -th element of M ˙ G is included in S { d p ( M ) : M ∈ M p } . • The n -th element of F ( M ˙ G ( α )) ∩ M ˙ G ( α + 1) is in dom( s p )It is easy to see that D α,m,n is dense in P . Set D = { D α,m,n : α < ω , m, n < ω } .By PFA, let G ⊆ P be a D -generic filter. Using the definition of D α,m,n , one can runstandard arguments to show that N = S M G is an elementary submodel of size ℵ which is closed under F , and that M G is a continuous sequence, which in turn impliesthat N is an IC-model. Let M G = ( M α : α < ω ). It is easy to see, using the thirdproperty of the dense sets above, that dom( s G ) = S α<ω F ( M α ) ∩ N . We show that N is an ω -guessing model. Let f be a function with dom( f ) = γ ∈ N , which is countablyapproximated in N . For each α < ω , set f α = f ↾ M α . If for stationary many α , f α isguessed in M α , then the pressing down lemma implies that there is a function g ∈ N such that M ∩ g = M ∩ f . Otherwise, for a club set of ordinals α , f α is not guessed in M α , and thus f α ∈ dom( s G ), but then there should α < β such that s G ( f α ) = s G ( f β ),which is a contradiction as f α ⊆ f β . Claim 2.14. P is proper.Proof. Let θ ∗ > θ be a sufficiently large regular cardinal. Let M ∗ ≺ H θ be countableand contain F, P . Set M = M ∗ ∩ H θ . Suppose that p ∈ M ∗ ∩ P . Let p M = ( M p ∪{ M } , d p ∪ { ( M, ∅ ) } , s p ). We claim that p M is ( M ∗ , P )-generic. Fix q ≤ p M , and set q ↾ M = ( M q ∩ M, d q ↾ M , s q ↾ M ). Let D ∈ M ∗ be a dense subset of P , without loss ofgenerality, we may assume that p ∈ D . Let ( f i : i ≤ k ) be an enumeration of functionsin { f ↾ M : f ∈ F ( N ) for some N, f is bounded in M and M ∈ N } . Let γ i ∈ M be such that dom( f i ) = M ∩ γ i . We may assume that γ i ≤ γ i +1 . Let x q x be a mapping on P ω ( γ k ) in M ∗ , such that • q x ∈ D . • M q x ⊇ M q ↾ M . • The least model of M q x \ M q ↾ M , say M x , contains x and γ , . . . γ k . Moreover, if f ∈ dom( s q x ) \ dom( s q ↾ M ), then M x ⊆ dom( f ). • d q x ( N ) ⊇ d q ( N ), for each N ∈ M q ↾ M • s q x ⊇ s q ↾ M . • | dom( s q x ) | = dom( s q ) | R. MOHAMMADPOUR
Such a mapping exists in M ∗ , as for each x ∈ M ∗ ∩ P ω ( γ k ), x q is a witness. Weclaim that there is some x ∈ M ∗ such that q x is compatible with q . As in q , we mayfix an enumeration ( f xi : i ≤ k ) of functions in dom( s q x ) \ s q ↾ M restricted to M x , so thatdom( f xi ) = γ i Fix i ≤ k , and consider the mapping x f xi . Then applying Lemma 2.13to the triple ( f k , M ∗ , x f xi ), we obtain a set B ki cofinal in P ω ( γ k ) so that for every x ∈ B ki , f xi * f k . Repeating this argument for every i ≤ k , there is a set B k ∈ M ∗ ,cofinal in P ω ( γ k ) such that for every i ≤ k and every x ∈ B k , f xi * f k . Then let C = { x ∩ γ k − : x ∈ B k } , and repeat the argument. To run the argument we may use achoice function in M ∗ , to give us a mapping over P ω ( γ k − ) meaning that if x ∈ C , thenfix some y ∈ B k so that x = y ∩ γ k − . One can then form B k − , C and so on. Finallythere is a set B ∗ ∈ M ∗ cofinal in P ω ( γ k ) such that for every x ∈ B ∗ , every i, j ≤ kf xi * f j . Now pick x ∈ B ∗ ∩ M ∗ . It is easily seen that q x and q are compatible. Pure Side Conditions
This section is devoted to the pure side conditions; a forcing notion which consists ofchains of models of two types. Such a forcing with pure side conditions as well as a finite-support iteration of proper forcings with side conditions were introduced by Neeman in[15]. Though the origin goes back to Neeman, we cannot use his results directly heredue to the fact we work with different models. Instead, our construction is based onVeliˇckovi´c’s presentation [21] of pure side conditions with finite ∈ -chains of models oftwo types in [21], where both types are non-transitive. We will sketch some proofs of theforthcoming facts in this section, but we encourage the reader to consult [21] for morecomprehensive proofs.Let θ be an uncountable regular cardinal. We let E denote a collection of countableelementary submodels of H θ , and let E denote a collection of elementary IC-submodelsof H θ . We further require that for every N ∈ E and every M ∈ E , if N ∈ M , then N ∩ M ∈ E . Definition 3.1.
Assume that
M ⊆ E ∪ E .(1) Suppose that M, N ∈ M . We say M is below N in M , or equivalently N is above M in M , and denote it by M ∈ ∗ N if there is a finite set { M i : i ≤ n } ⊆ M such that M = M ∈ · · · ∈ M n = N .(2) We say M is an ∈ -chain, if for every distinct M and N in M , either M ∈ ∗ N in M or N ∈ ∗ M in M . PECIALIZING TREES WITH SMALL APPROXIMATIONS I 9 (3) We say M is closed under intersections if for every M ∈ M ∩ E , and every N ∈ M ∩ M , N ∩ M belongs to M .(4) If M, N ∈ M ∪ { ∅ , H θ } , then by ( M, N ) M , and intervals of other types, we meanthat the interval is considered in the linearly ordered structure ( M , ∈ ∗ ) . It is easily seen that if M ∈ ∗ N holds in an ∈ -chain M , and that N ∈ E , then M ∈ N .Notice that also if N ∈ E , then M ⊆ N if and only if there is no P ∈ E ∩ M with P ∩ N ∈ ∗ M ∈ ∗ P ∈ N Definition 3.2 (Pure side condition) . We let M ( E , E ) denote the collection of ∈ -chains p = M p ⊆ E ∪ E which are closed under intersections. We consider M ( E , E ) as anotion of forcing equipped with set extension. We simply denote M ( E , E ) by M when there are no confusions. For a condition p ∈ M , we let also E p and E p denote M p ∩ E and M p ∩ E , respectively. We also denotethe interval ( M, N ) M p simply by ( M, N ) p ; such an agreement applies to other types ofintervals as well. Definition 3.3.
Suppose that p ∈ M and M ∈ E ∪ E . We let p M denote the closure of M ∪ { M } under intersections. The following is easy and we leave the proof to the reader.
Fact 3.4.
Suppose that p ∈ M . Assume that M ∈ E ∪ E . Then(1) If M ∈ E , then p M = M ∪ { M } .(2) If M ∈ E , then p M = M ∪ { M } ∪ { N ∩ M : N ∈ E p } .(3) p M is a condition in M extending p . Definition 3.5.
Suppose that p ∈ M , and assume M ∈ M p . We let p ↾ M = M p ∩ M . Notice that p ↾ M is in M as it is a finite subset of M . If M is in E , then p ↾ M is infact the interval ( ∅ , M ) p which is an ∈ -chain, but if M is countable, then it is a unionof intervals. Fact 3.6.
Suppose that p ∈ M . Assume that M ∈ M p is countable. Then M p ↾ M = M p \ [ { [ N ∩ M, N ) p : N ∈ ( E p ∩ M ) ∪ { H θ }} . Proof.
Let P ∈ M p ↾ M . Thus P ∈ M , which implies that P does not belong to theinterval [ M, H θ ) p . Now, let N ∈ E p ∩ M . If N ∈ ∗ P , then P does not belong to theinterval [ N ∩ M, N ) p . Suppose P ∈ ∗ N , then P ∈ N , and hence P ∈ N ∩ M , which in turn implies that P / ∈ [ N ∩ M, N ) p . Therefore, the LHS is a subset of RHS. To see theother direction. Suppose P does not belong to any interval as described in the aboveequation. In particular P ∈ ∗ M . Now, if P / ∈ M , it then means there are some modelsin E p ∩ ( P, M ) p . Let N be the least such model. Then, N ∩ M ∈ ∗ P , since otherwiseby the minimality of N , we should have that P ∈ N ∩ M ⊆ M . Thus P belongs to[ N ∩ M, N ) p , which is a contradiction. It is not hard to see that p ↾ M is an ∈ -chain. Now, the following is immediate. Fact 3.7.
For every condition p ∈ M and M ∈ M p , p ↾ M is a condition and p ≤ p ↾ M . Fact 3.8.
Suppose that p ∈ M and M ∈ E p . Then every condition q ∈ M extending p ↾ M is compatible with q .Proof. Let M r = M p ∪ M q . It is easy to see that M r is closed under intersections.Suppose that P ∈ M p \M q and Q ∈ M q \M p . If P = M we then have that Q ∈ M ∈ ∗ P ,and if P = M , then obviously Q ∈ M . Remark . The above condition is the greatest lower bound of p and q . We denote itby p ∧ q . Notice that M p ∧ q = M p ∪ M q Fact 3.10. M is E -strongly proper, and hence if E is stationary, then M preserves ℵ .Proof. Suppose that M ∈ E . If p ∈ M ∩ M , then by Fact 3.4, p M is a conditionextending p . Let q ≤ p M , then M ∈ M q . By Fact 3.7, q ↾ M is a condition in M ∩ M .Now if r ∈ M ∩ M extends q ↾ M , then q is compatible with r by Fact 3.8. Thus q is( M, M )-strongly generic. Lemma 3.11.
Suppose that p ∈ M . Let M ∈ E p . Then every condition q ∈ M extending p ↾ M is compatible with q .Proof. Let M r be the closure of M p ∪ M q under intersections. Assume that P = Q are in M r . We show that either P ∈ ∗ Q or Q ∈ ∗ P . We discuss only the nontrivial cases. Case 1: P ∈ M p \ M q and Q ∈ M q \ M p . If P is equal to M or is above M in M p ,then clearly Q ∈ ∗ P . Thus assume that P ∈ ∗ M . Since P / ∈ M , there is some N ∈ E p such that P ∈ [ N ∩ M, N ) p . Now, if N ∈ ∗ Q in M q , then P ∈ ∗ Q in M r . If Q ∈ ∗ N ,then Q ∈ N , and hence Q ∈ N ∩ M . This implies that Q ∈ N ∩ M ∈ ∗ P in M r . PECIALIZING TREES WITH SMALL APPROXIMATIONS I 11
Case 2: P ∈ M p ∪ M q and Q = R ∩ N , for some N, R ∈ M p ∪ M q , where R ∈ N are in E and E , respectively. We may assume that P ∈ ∗ R ∈ N . If N ∈ M q , then R ∈ M , and hence Q ∈ M q , which is then as in the previous case. Thus assume that N ∈ M p \ M q . We may also assume that R ∈ M q \ M p . We split the proof into twopieces. sub-case 2A: P ∈ M p \ M q . In this subcase P ∈ ∗ N in M p . If P ∈ N , then P ∈ R ∩ N . If P / ∈ N , then there is some model R ′ ∈ M p ∩ N such that R ′ ∩ N ∈ P .Now if R ∈ R ′ , then Q = R ∩ N ∈ R ∈ R ′ ∩ N ∈ ∗ P . On the other hand, if R ′ ∈ R , then P ∈ R ′ ∈ R ∩ N = Q . sub-case 2B: P ∈ M q \ M p . In this subcase, N is in some interval of the form[ R ′ ∩ M, R ′ ) p . We may assume that R ′ ∈ M . Then, P ∈ R ∈ R ′ ∩ M . Now P ∈ R ∩ R ′ ∩ M = R ′ ∩ M . Thus if R ′ ∩ M ⊆ N , then P ∈ R ∩ N . If this is not the case,then there is some R ′′ ∈ E p such that R ′ ∩ M ∈ R ′′ ∈ N , but R ′′ ∩ N ∈ ∗ R ′ ∩ M . Thus P ∈ ∗ R ∈ R ′′ ∩ N , R ′′ ∩ N ∈ M p , and that R ∩ N = R ∩ ( R ′′ ∩ N ). Now we are as inthe initial move, and this process cannot continue forever. Either R ′′ ∩ N ∈ M , we arethen done, or it falls into another interval. So finally, either P ∈ Q or Q ∈ ∗ P . Remark . As before, the above condition is the greatest lower bound of p and q . Weagain denote it by p ∧ q . Notice that M p ∧ q = M p ∪ M q ∪ { N ∩ M : N, M ∈ M p ∪ M q and N ∈ M } The following is similar to Fact 3.10 in light of Lemma 3.11.
Fact 3.13. M is E -strongly proper. Thus if E is stationary, then M preserves ℵ , andif it contains a club, then M is proper. Forcing Construction
The first subsection here aims to present a phenomenon called overlapping which wasintroduced by Neeman in his paper [16] regarding weak specialization of trees of height ω in the context of his iteration. What Neeman does (we are going to do the same) isattaching B ω ( T ) to side conditions with models of two types: countable and transitive,where he also requires several constraints describing the interaction of the working partsfrom B ω ( T ) and the models in side conditions. He then gives a very fine analysis ofsuch interaction. Our forcing is much simpler than Neeman’s, but the main constrainton forcing conditions is also required in his forcing which, roughly speaking, states if a model in the side conditions overlaps a node outside the model, then the value of thatnode under the working part (partial specialization) is not in the model. Our terminologyis different here, what Neeman describes as “a node being overlapped by a model M ”for us means “a node being guessed in M ”. This terminology is more convenient in thecontext of guessing models. Another small difference is that by “being overlapped by amodel” Neeman requires that the node is not in the model, however for us a node insidea model is always guessed. We then require that if a node in the domain of the workingpart is guessed in a countable model in the side conditions, and whose value of partialspecializing function is in the model, then that node belongs to the model.Throughout this section, we fix a normal tree ( T, < T ) of height ω without cofinalbranches. We also fix a sufficiently large regular cardinal θ such that P ( T ) ∈ H θ . Welet E and E consist of, respectively, countable elementary submodels , and ω -guessingelementary IC -submodels of H θ . We reserve symbols p, q, r for forcing conditions, and s, t, u for nodes in T . Overlaps Between Models and Nodes.
As we mentioned earlier, in this subsection, we study the interaction of models in E ∪ E and nodes in T . We then give the definition of our forcing in the next subsection. Convention 4.1.
A branch through T is called a T -branch. Definition 4.2.
Suppose that M ∈ E ∪ E and t ∈ T . We say t is guessed in M if andonly if there is a T -branch b ∈ M with t ∈ b . It is clear from the above definition that every t ∈ M is guessed in M , and that nonode t with ht( t ) ≥ sup( M ∩ ω ) is guessed in M , since M has no cofinal branches.Moreover, if M ∈ E , then t is guessed in M if and only if t = O M ( t ) ∈ M , but thesituation is different for models in E as if M ∈ E and t ∈ M is of uncountable heightin T , then one can find s ∈ b t \ M . Such s is guessed in M , but does not belong to M .If, in addition, the height of s is sup( M ∩ ht( t )), then s = O M ( t ) does not belong to M .We shall use frequently the following without mentioning. Lemma 4.3.
Suppose that t ∈ T and M ∈ E ∪ E . If there is s ∈ M with t ≤ T s , then t is guessed in M .Proof. By the normality of T , pick u ∈ T with s < T u . Then t ∈ b u . Notation 4.4.
Assume that t ∈ T and M ∈ E ∪ E . Then • η M ( t ) denotes sup(ht( s ) : s ∈ M and s ≤ T t ) . • O M ( t ) denotes the unique node s ∈ T η M ( t ) such that s ≤ T t . PECIALIZING TREES WITH SMALL APPROXIMATIONS I 13 • b M ( t ) denotes b O M ( t ) . Observe that O M ( t ) is always well-defined as T is a rooted tree belonging to everymodel in E ∪ E . By definition, η M ( t ) ≤ sup( M ∩ ω ). We shall see that if M ∈ E , thennot only η M ( t ) is less than M ∩ ω , but also if its cofinality is uncountable, then O M ( t )is in M . It would be useful to have the intuition that the node O M ( t ) is the point, where b t detach from M . In our analysis, we will focus mostly on O M ( t ) rather than t itself.The following lemma sheds light on the phenomenon of overlapping. Lemma 4.5.
Suppose that t ∈ T and M ∈ E ∪ E .(1) If t is guessed in M and η M ( t ) ∈ M , then t ∈ M (2) If t is guessed in M , but η M ( t ) / ∈ M , then ht( t ) ≤ min( M ∩ ω \ η M ( t )) .Proof. Of course, the first item follows form the proof of the second one, but we preferto give independent proofs.(1) Assume that b ∈ M is a T -branch containing t , since η M ( t ) ∈ M , there is s ∈ b ofheight η M ( t ) + 1. Then s < T t is impossible since ht( s ) > η M ( t ), and thus t ≤ T s which implies that t ∈ M as η M ( t ) ≤ ht( t ).(2) We may assume that M is in E as otherwise it is trivial. One observes easilythat η M ( t ) is below sup( M ∩ ω ) since T does not have cofinal branches. Now η ∗ := min( M ∩ ω \ η M ( t )) is an ordinal below ω . Let b ∈ M be a branchcontaining t . If ht( t ) > η ∗ , then there is some node s ∈ b of height η ∗ , thus s < T t , which in turn implies that η M ( t ) ≥ η ∗ > η M ( t ), a contradiction. The following is too easy and so we leave the proof to the readers.
Lemma 4.6.
Suppose that t ∈ T and M ∈ E ∪ E . If η M ( t ) is a successor ordinal, then O M ( t ) is in M . Now we turn our attention to the situation where the overlaps could be more compli-cated when η M ( t ) is a limit ordinal. Lemma 4.7.
Suppose that t ∈ T and M ∈ E . If cof( η M ( t )) is not countable, then O M ( t ) ∈ M .Proof. By Lemma 4.6, we may assume that η M ( t ) is a limit ordinal, and thus of cofinality ω . Let η = η M ( t ). We first assume that η < M ∩ ω . Since M is of size ℵ , η ∈ M and b M ( t ) ⊆ M , if O M ( t ) / ∈ M , then for every countable a ∈ M , the height of nodes in a ∩ b M ( t ) is bounded below η due to the fact that η M ( t ) has uncountable cofinality.Thus b M ( t ) is countably approximated in M . Since M is an ω -guessing model, there is b ∈ M is such that b ∩ M = b M ( t ). Therefore, b M ( t ) = b ∈ M , which in turn impliesthat O M ( t ) ∈ M as it can be read off from b M ( t ) due to the fact that T is normal. Theproof of η = M ∩ ω is similar. In fact this is impossible. Once more, b M ( t ) is countablyapproximated as cof( M ∩ ω ) is uncountable, so it is guessed in M , say b ∩ M = b t ∩ M .By elementarity, b is a cofinal branch through T , which is a contradiction. Corollary 4.8.
Suppose that t ∈ T and M ∈ E . Then η M ( t ) is in M .Proof. By definition η M ( t ) ≤ M ∩ ω , and by Lemma 4.7 η M ( t ) is of countable cofinality,and hence η M ( t ) < M ∩ ω . Therefore, η M ( t ) ∈ M . The following is a master key for our later claims.
Lemma 4.9.
Suppose that N ∈ M are in E and E , respectively. Assume that t ∈ T ∩ N .If t is guessed in M , then it is guessed in N ∩ M .Proof. Let b ∈ M be a T -branch containing t . Let γ = sup(ht( s ) : s ∈ N ∩ b ). Then γ ∈ M ∩ ω . We are done if b ∩ T ≤ γ is guessed in N . If this is not the case, thencof( γ ) = ω , thus by elementarity, there is in N ∩ M a sequence ( s n ) n ∈ ω of nodes in b with s n < T s n +1 whose height is cofinal in γ . If for each n , s n < T t , then t can be readoff from b ∩ T ≤ γ , and hence is in N ∩ M . Otherwise, there is n such that t ≤ T s n , whichmeans that t is guessed in N ∩ M . Lemma 4.10.
Suppose that N ∈ M are in E and E , respectively. Let t ∈ T ∩ N . Then η N ∩ M ( t ) = η M ( t ) , and hence O N ∩ M ( t ) = O M ( t ) .Proof. Since N ∩ M ⊆ M , η N ∩ M ( t ) ≤ η M ( t ). Assume towards a contradiction that theequality fails. Thus, there is some s ∈ M whose height is above η N ∩ M ( t ) such that s ≤ T O M ( t ) ≤ T t . Then s ∈ N as N is a model of size ℵ containing t . Therefore, s ∈ N ∩ M , and hence ht( s ) ≤ η N ∩ M ( t ), a contradiction. Since both O N ∩ M ( t ) and O M ( t )are below t of the same height, they should be equal. Forcing Construction and its Basic Properties.
We are now ready to define our forcing notion P T to specialize T in generic extensions. Definition 4.11.
Let B ω ( T ) denote the forcing notion consisting of finite partial special-izing functions over T equipped with set extension, that is f ∈ B ω ( T ) is a finite partialfunction form T to ω such that if s, t ∈ dom( f ) are comparable in T , then f ( t ) = f ( s ) . PECIALIZING TREES WITH SMALL APPROXIMATIONS I 15
Definition 4.12 ( P T ) . A condition in P T is a pair p = ( M p , f p ) satisfying the followingitems.(1) M p ∈ M := M ( E , E ) .(2) f p ∈ B ω ( T ) .(3) For every M ∈ E p , if t ∈ M , then f p ( t ) ∈ M .(4) For every M ∈ E p and every t ∈ dom( f p ) with f p ( t ) ∈ M , if t is guessed in M ,then t ∈ M .We say p is stronger than q if and only if the following are satisfied.(1) M p ⊇ M q .(2) f p ⊇ f q . Given a condition p in P T and a model M ∈ E ∪E containing p , we define an extensionof p which will turn later to be generic for relevant models. Definition 4.13.
Suppose that M ∈ E ∪ E and p ∈ M ∩ P T . We let p M be defined by ( M Mp , f p ) . Recall that M Mp is the closure of M p ∪ { M } under intersections, see Fact 3.4. Proposition 4.14.
Suppose that M ∈ E ∪ E and p ∈ P T ∩ M . Then p M is a conditionextending p such that M ∈ M p M .Proof. We check Definition 4.12 item by item. Item 1 is essentially Fact 3.4. Item 2 isobvious of course. To see Items 3 and 4 hold true, let N ∈ E p M . We may assume that M ∈ E and that N = P ∩ M for some P ∈ E p . For Item 3, let t ∈ dom( f p M ) ∩ N . Then, f p ( t ) ∈ M since p ∈ M , and f p ( t ) ∈ P since ω ⊆ P . Thus f p ( t ) ∈ N . For Item 4, let t ∈ dom( f p ) be such that f p ( t ) ∈ N . If there is a T -branch b ∈ N with t ∈ b , then t ∈ P ,and hence t ∈ P ∩ M = N .By the construction of p M , M ∈ M p M . Finally, by Fact 3.4 p M extends p . We now define the restriction of a condition to a model in the component of sideconditions.
Definition 4.15 (Restriction) . Suppose that p ∈ P T and M ∈ M p . We let the restriction of p to M be p ↾ M = ( M p ∩ M, f p ↾ M ) . Observe that if M is in E , then by Item 3 of Definition 4.12, f p ↾ M = f p ∩ M . This istrivial for M in E . Proposition 4.16.
Suppose that p ∈ P T and M ∈ M p . Then p ↾ M ∈ P T ∩ M and p ≤ p ↾ M . Proof.
We check Definition 4.12 item by item. By Fact 3.7, M p ∩ M is an ∈ -chain andclosed under intersections, and hence is in M . By Item 3 of Definition 4.12, f p ∩ M is in B ω ( T ), and that p ↾ M is also in M as it is a finite subset of M . Items 3 and 4 remainvalid since all models in M p ↾ M and all nodes in dom( f p ↾ M ) are, respectively, in M p anddom( f p ). It is easy to see that p extends p ↾ M . Notation 4.17.
For conditions p and q in P T , and a model M ∈ M p with q ∈ M extending p ↾ M , we let p ∧ q denote the pair ( M p ∧ M q , f p ∪ f q ) . We warn the reader that in general p ∧ q is not a condition, however we will be usingit as a pair of objects. Notice that M p ∧ q is in M , which is the closure of M p ∪ M q underintersections, see Remark 3.9 and Remark 3.12. Notice that also f p ∧ q is a well-definedfunction due to the fact that p satisfies Item 3 of Definition 4.12. Lemma 4.18.
Suppose p is a condition in P T and M is a model in M p . Assume that q ∈ M ∩ P T extends p ↾ M . Then p ∧ q satisfies Item 3 of Definition 4.12.Proof. Let N be a model in E p ∧ q . Assume that t ∈ dom( f p ) ∪ dom( f q ) is in N . We splitthe proof into two cases. Case 1: M is in E .If N ∈ M q , then t ∈ dom( f q ), and hence f p ∧ q ( t ) = f q ( t ) ∈ N . The only interesting caseis when t ∈ dom( f q ) and N ∈ M p \ M q . In this case, though N is not in M , there is M ′ ∈ E p such that M ⊆ M ′ ∈ N and M ′ ∩ N ∈ M . Then, M ′ ∩ N ∈ M q and t ∈ M ′ ∩ N ,which is then easily implied that f p ∧ q = f q ( t ) ∈ M ′ ∩ N ⊆ N . Case 2: M is in E .It is easily seen that it is enough to assume N ∈ M p ∪ M q . As in the previous case, wemay assume that t ∈ dom( f q ) and N ∈ M p \ M q . Now assume that N ∈ ∗ M . Thusthere is P ∈ E p ∩ M such that N ∈ [ P ∩ M, P ) p . We are done if P ∩ M ⊆ N . If this isnot the case, then there is some Q ∈ N such that Q ∩ N ∈ ∗ P ∩ M ∈ Q . Notice that t ∈ P , and hence t ∈ P ∩ M ⊆ Q . Thus t ∈ Q ∩ N . This shows that if N does notsatisfy our claim, we could assume that it is the minimal counter example. Thus Q ∩ N is in M , and hence it is in M q . Thus f p ∧ q = f q ( t ) ∈ Q ∩ N ⊆ N , a contradiction!Assume now that M ∈ ∗ N . If M ⊆ N , then f q ( t ) ∈ N . Suppose that it is not thecase. Thus there is some P ∈ E p such that P ∩ N ∈ ∗ M ∈ P ∈ N . Notice that t ∈ P ∩ N .Thus this case follows form the previous paragraph. Preserving ℵ . In this subsection we prove that P T preserves ω , we then adapt the idea of our strategyto establish the properness of P T in the subsequent subsection. PECIALIZING TREES WITH SMALL APPROXIMATIONS I 17
Lemma 4.19.
Suppose p is a condition in P T and that M ∈ E p . Assume that q ∈ M isa condition extending p ↾ M . Then p ∧ q satisfies Item 4 of Definition 4.12.Proof. Set r = p ∧ q . Notice that f r is well-defined as a function. Now fix t ∈ dom( f r ) and N ∈ M r so that f r ( t ) ∈ N , we shall show that if t is guessed in N , then t ∈ N . Noticethat since M ∈ E , M r = M p ∪ M q by Remark 3.9. We shall study the situation caseby case, but to avoid the trivial cases, we may assume that either t ∈ dom( f q ) \ dom( f p )and N / ∈ M q , or t ∈ dom( f p ) \ dom( f q ) and N / ∈ M p . Case 1: t ∈ dom( f q ) \ dom( f p ) and N ∈ M p \ M q .In this situation, N is not in M since M q ⊇ M p ∩ M , and hence there is some M ′ ∈ E p with M ⊆ M ′ ∈ N such that M ′ ∩ N ∈ M . By Lemma 4.9, t is guessed in M ′ ∩ N . Onthe one hand, f q ( t ) belongs to M ∩ N ⊆ M ′ ∩ N , and that M ′ ∩ N ∈ M ∩ M p ⊆ M q .On the other hand q is a condition, and hence t ∈ M ′ ∩ N ⊆ N . Case 2: t ∈ dom( f p ) \ dom( f q ) and N ∈ M q \ M p .Since t is guessed in N , there is a T -branch b ∈ N ⊆ M with t ∈ b . This implies that t ∈ M , and thus t ∈ dom( f q ), a contradiction! Definition 4.20.
Assume that p ∈ P T . Suppose that M ∈ E p . We let D ( p, M ) = { t ∈ dom( f p ) : t / ∈ M } . Definition 4.21 ( M -support) . Suppose p is a condition in P T and that M ∈ E p . Wesay a finite set Σ = { ˆ t : t ∈ D ( p, M ) } ⊆ M ∩ T is an M -support for p if the followinghold.(1) If O M ( t ) ∈ M , then ˆ t = O M ( t ) .(2) If O M ( t ) / ∈ M , then ˆ t < T O M ( t ) is such that no node in dom( f p ) has height inthe interval [ht(ˆ t ) , η M ( t )) . Lemma 4.22.
Suppose p is a condition in P T . Assume that M ∈ E p . Then, there is an M -support for p in M .Proof. Let t ∈ D ( p, M ) with O M ( t ) / ∈ M . Notice that for every p ∈ P T , dom( f p ) isfinite, and that if O M ( t ) / ∈ M , then η M ( t ) is a limit ordinal by Lemma 4.6. Thus onemay easily find ˆ t with the above properties. Definition 4.23 ( M -reflection) . Suppose p is a condition in P T , and that M ∈ E p . Acondition q is called an ( M, Σ)-reflection , where Σ is an M -support for p if the followingproperties are satisfied.(1) q ≤ p ↾ M . (2) For every ˆ t ∈ Σ , the following hold:(a) No node in dom( f q ) has height in the interval [ht(ˆ t ) , η M ( t )) .(b) For every s ∈ dom( f q ) , if s < T ˆ t , then f q ( s ) = f p ( t ) .Let R p ( M, Σ) be the set of M -reflections of p with support Σ .Remark . Notice that if M ∗ ≺ H θ ∗ , for some sufficiently large regular cardinal θ ∗ ,which contains T and H θ , and that p is a condition in P T with M := M ∗ ∩ H θ ∗ ∈ E p ,then R p ( M, Σ) ∈ M ∗ , whenever Σ is an M -support for p . Lemma 4.25.
Suppose p is a condition in P T , and that M ∈ E p . Let Σ be an M -supportfor p . Then p ∈ R p ( M, Σ) .Proof. We check the items in Definition 4.23. Item 1 is essentially Proposition 4.16.Item 2a follows from the definition of Σ. Item 2b follows form the fact that p is acondition, and that ˆ t < T t . Lemma 4.26.
Suppose p is a condition in P T , and that M ∈ E p . Let q ∈ M be an ( M, Σ) -reflection of p , for some M -support Σ for p . Let r = p ∧ q . Then f r ∈ B ω ( T ) .Proof. Since q ≤ p ↾ M , f r is well-defined as a function. We shall show that it satisfies thespecializing property. To do this, we only discuss the nontrivial cases by considering twoarbitrary nodes t ∈ dom( f p ) \ dom( f q ) and s ∈ dom( f q ) \ dom( f p ). Assume that they arecomparable in T . We claim that f r ( t ) = f r ( s ). Observe that t ≤ T s is impossible, andthus s < T t . Since q ∈ R p ( M, Σ) ∩ M , the height of s is not in the interval [ht(ˆ t ) , η M ( t )),where ˆ t is the corresponding node of t in Σ. Thus s < T ˆ t . Then Item 2b of Definition 4.23implies that f q ( s ) = f p ( t ). We have now all the necessary tools to prove the preservation of ℵ by P T . Lemma 4.27.
Suppose p is a condition in P T . Assume that θ ∗ is a sufficiently largeregular cardinal, and that M ∗ ≺ H θ ∗ contains the relevant objects. Suppose that M := M ∗ ∩ H θ is in E p . Then, p is ( M ∗ , P T ) -generic.Proof. Let D ∈ M ∗ be a dense subset of P T . We may assume that p ∈ D . By Lemma 4.25,there exists an M -support of p , say Σ, such that p ∈ R p ( M, Σ). Notice that R p ( M, Σ) isin M ∗ , thus by elementarity, there is some q ∈ D ∩ R p ( M, Σ) ∩ M . Set r = p ∧ q . Now,Fact 3.8 and Lemmas 4.26, 4.18 and 4.19 imply that r satisfies Items 1, 2, 3 and 4 ofDefinition 4.12, respectively. It is clear that p ∧ q is a common extension of p and q . Corollary 4.28. P T preserves ℵ . PECIALIZING TREES WITH SMALL APPROXIMATIONS I 19
Proof.
Let θ ∗ be a sufficiently large regular cardinal. It is enough to show that forstationary many models M in H θ ∗ , of size ℵ , every condition in M can be extended toan ( M, P T )-generic condition. Let S = { M ≺ H θ ∗ : E , E , T, θ ∈ M and M ∩ H θ ∈ E } .Clearly S is stationary in P ω ( H θ ∗ ). Now let M ∗ ∈ S and p ∈ P T ∩ M ∗ . Set M = M ∗ ∩ H θ .By Proposition 4.14, p M is a condition and by Lemma 4.27 it is ( M ∗ , P T )-generic. Properness.
This subsection is devoted to the proof of the properness of P T . We try to follow closelythe strategy in the previous subsection. We follow closely the same strategy as in theprevious subsection. We warn the reader that we will be using notation and definitionabout models in E similar to the case of E in the previous subsection, however sincetheses two parts are completely independent, there will be no confusions. Lemma 4.29.
Suppose p is a condition in P T and that M ∈ E p . Assume that q ∈ M isa condition extending p ↾ M . Then p ∧ q satisfies Item 4 of Definition 4.12.Proof. Set r = p ∧ q . Notice that f r is well-defined as a function. Now fix t ∈ dom( f r )and N ∈ M r so that t is guessed in N and f r ( t ) ∈ N , we shall show that t ∈ N .As in Lemma 4.19, we shall study the situation case by case, where again to avoidthe trivial cases, we may assume that either t ∈ dom( f q ) \ dom( f p ) and N / ∈ M q , or t ∈ dom( f p ) \ dom( f q ) and N / ∈ M p . Since M is in E , the proof will consist of threecases as there could be models in M r coming from some intersections not belonging to M p ∪ M q . Recall that by Remark 3.12, M r is the union of M p ∪ M q and the set ofmodels of the form P ∩ Q , where P, Q ∈ M p ∪ M q are in E and E , respectively. Case 1: t ∈ dom( f q ) \ dom( f p ) and N ∈ M p \ M q .In this situation either N is above M on the chain M p , or is in a gap ( P ∩ M, P ] p forsome P ∈ E p ∩ M . To avoid repetition, we can also include P = H θ . Since t is guessedin N , we should have that t ∈ P . We are done if P ∩ M ⊆ N . If not, there is a model Q ∈ E p such that P ∩ M ∈ Q ∈ N ∈ P , and Q ∩ N ∈ ∗ P ∩ M . On the one hand f q ( t ) ∈ P , and thus f q ( t ) ∈ P ∩ M ⊆ Q . Therefore, f q ( t ) ∈ Q ∩ N . On the other hand,by Lemma 4.9 t is guessed in Q ∩ N . This shows that if our claim does not hold true for N , then we can assume that N is the least counter example. Therefore, Q ∩ N ∈ M , andhence Q ∩ N ∈ M q . Since f q ( t ) ∈ Q ∩ N , t should be in Q ∩ N ⊆ N , a contradiction! Case 2: t ∈ dom( f p ) \ dom( f q ) and N ∈ M q \ M p .This case is impossible, as then t is guessed in M , and f p ( t ) ∈ M since N ⊆ M . Thus t ∈ M ∩ dom( f p ) ⊆ dom( f q ). Case 3: t ∈ dom( f r ) and N ∈ M r \ ( M p ∪ M q ).There are P ∈ E and Q ∈ E in M p ∪ M q with P ∈ Q such that N = P ∩ Q . We have seen several times that in this situation t ∈ P . Now t is guessed in Q and f p ( t ) ∈ Q . Bythe two previous cases, t ∈ Q . Thus t ∈ P ∩ Q = N . Notation 4.30.
Assume that p is a condition, and that M ∈ M p .(1) We let D ( p, M ) denote the set of t ∈ dom( f p ) such that t / ∈ M , but f p ( t ) ∈ M .(2) O ( p, M ) := { t ∈ D ( p, M ) : O M ( t ) is not guessed in M and η M ( t ) / ∈ M } . Definition 4.31 ( M -support) . Suppose p is a condition in P T , and that M ∈ E p . Wesay a finite set Σ = { g t : t ∈ D ( p, M ) } ⊆ M is an M -support for p if the following hold.(1) If O M ( t ) is guessed in M , then g t ∈ M is such that M ∩ g t = b M ( t ) .(2) If O M ( t ) is not guessed in M , then g t = b ˆ t for some ˆ t < T t in M such that nodein dom( f p ) has height in the interval [ht(ˆ t ) , η M ( t )) . By elementarity, g t in the above definition is a T -branch, in fact it is a cofinal branchthrough T <η ∗ M ( t ) , where η ∗ M ( t ) = min( M ∩ ω \ η M ( t )). Lemma 4.32.
Suppose p is a condition in P T . Assume that M ∈ E p . Then, there is an M -support for p .Proof. Suppose that t ∈ D ( p, M ). If O M ( t ) is guessed in M , then there is a T -branch b ∈ M such that O M ( t ) ∈ b . Let η ∗ M ( t ) = min( M ∩ ω \ η M ( t )). Now set g t = b ∩ T <η ∗ M ( t ) .It is easily seen that M ∩ g t = b M ( t ) . If O M ( t ) is not guessed in M , then since the conditions are finite, and η M ( t ) is a limitordinal by Lemma 4.6, and that there is a sequence of nodes in M cofinal in O M ( t ),there is some ordinal γ ∈ M , such that there is no node in dom( f p ) whose height is in[ γ, η M ( t )). Choose a node ˆ t of height γ below O M ( t ). Definition 4.33 ( M -reflection) . Suppose p is a condition in P T . Assume that M ∈ E p .Let Σ be an M -support for p . A condition q is called an ( M, Σ)-reflection for p if thefollowing properties are satisfied.(1) q ≤ p ↾ M .(2) The following hold for every g t ∈ Σ .(a) If η M ( t ) ∈ M , no node in dom( f q ) has height in the interval [ht(ˆ t ) , η M ( t )) ,where ˆ t is such that g t = b ˆ t .(b) If s ∈ dom( f q ) with s ∈ g t , then f q ( s ) = f p ( t ) .Let R p ( M, Σ) the set of ( M, Σ) -reflections of p . Notice that as before, if M ∗ ≺ H θ ∗ , for some sufficiently large regular cardinal θ ∗ which contains T and H θ , and p is a condition in P T with M := M ∗ ∩ H θ ∗ ∈ E p , then R p ( M, Σ) ∈ M ∗ , whenever Σ is an M -support for p . PECIALIZING TREES WITH SMALL APPROXIMATIONS I 21
Lemma 4.34.
Suppose p is a condition in P T , and that M ∈ E p . Let Σ be an M -supportset for p . Then p ∈ R p ( M, Σ) .Proof. Let us check the items in Definition 4.33. Item 1 is essentially Proposition 4.16.To see Item 2a holds, we observe that if O M ( t ) is guessed in M , then O M ( t ) ∈ M , andhence O M ( t ) = ˆ t , which in turn implies that the interval [ht(ˆ t ) , η M ( t )) is empty. If O M ( t )is not guessed in M , then the by the property of Σ, there is no node in dom( f p ) withheight in the interval [ˆ t, η M ( t )).To see Item 2b holds true, suppose is s ∈ g t is such that f p ( s ) = f p ( t ), then since f p ( t ) ∈ M , and that s is guessed in M , we should have that s ∈ M . Thus s ≤ T O M ( t ),and hence s < T ˆ t ≤ T t , which is a contradiction. Lemma 4.35.
Suppose p is a condition in P T , and that M ∈ E p . Assume that q ∈ M ∩ R p ( M, Σ) . Then letting r := p ∧ q , r ′ = ( M r , f r \ { ( t, f p ( t )) : t / ∈ O ( p, M ) } ) is acondition.Proof. Lemmas 3.11, 4.18 and 4.29 imply that r ′ satisfies Items 1, 3 and 4 of Defini-tion 4.12, respectively. Thus it remains to show that the well-defined function f r ′ is in facta condition in B ω ( T ). To see this, let s ∈ dom( f q ) \ dom( f p ) and t ∈ dom( f r ′ ) \ dom( f q ).Assume that s and t are comparable in T , we shall show that f q ( s ) = f p ( t ). We mayassume that f p ( t ) ∈ M . Thus t < T s is impossible, as then t is guessed in M , and hence t ∈ M , a contradiction! Thus the only possible case is s < T t . In this case, s ≤ T O M ( t ).We claim that s ∈ g t . This is clear if O M ( t ) is guessed in M . On the other hand if O M ( t ) is not guessed in M , then η M ( t ) ∈ M as t / ∈ O ( p, M ). Therefore, by Item 2a ofDefinition 4.33, the height of s avoids the interval [ht(ˆ t ) , η M ( t )), where ˆ t is some nodein M with ˆ t ≤ T O M ( t ) such that g t = b ˆ t . Thus s < T ˆ t . In either case, s ∈ g t , but thenItem 2b of Definition 4.33 implies that f p ( t ) = f q ( s ). Proposition 4.36.
Suppose that p ∈ P T . Let θ ∗ be a sufficiently large regular cardinal.Assume that M ∗ ≺ H θ ∗ is countable and contains T and θ . If M := M ∗ ∩ H θ ∈ M p .Then p is ( M ∗ , P T ) -generic.Proof. Let D ∈ M ∗ be a dense subset of P T . We may assume that p ∈ D . Since M ∗ is fixed in this proof, for a node t ∈ T , by η t everywhere in this proof we mean η M ( t ).By Lemmas 4.32 and 4.34, there is an M -support Σ = { g t : t ∈ D ( p, M ) } for p so that p ∈ R p ( M, Σ). Recall that R p ( M, Σ) ∈ M ∗ . Let ( t i ) i ≤ m enumerate O ( p, M ), and let( η i ) i ≤ m ′ enumerate { η t i : i ≤ m ′ } . To reduce the amount of notation, we may assume that m = m ′ . Set η ∗ i = min( M ∩ ω \ η i ), for every i ≤ m , possibly except η m if it issup( M ∩ ω ), for which we then let η ∗ m = ω . Notice that η ∗ i < η i +1 . Let us call a mapΘ : x → p x from P ω ( T ) into T , a T -assignment if the following properties are satisfiedfor every x ∈ P ω ( T ).(1) p x ∈ R p ( M, π ) ∩ D .(2) | f p x | = | f p | .(3) For every s ∈ dom( p x ), if ht( s ) ∈ [ht(ˆ t i ) , η ∗ i ), thensup(ht( u ) : u ∈ x ∩ T <η ∗ i ) < ht( s ) , where ˆ t i is such that g t i = b ˆ t i .We first show that there are T -assignments in M ∗ . Claim 4.37.
There is a T -assignment in M ∗ .Proof. We observe that all the parameters in the above properties are in M ∗ . By ele-mentarity, it is enough to show that for every x , there is such p x ∈ H θ ∗ . In fact, themapping x p is a T -assignment. The first items is clear by Lemma 4.34 and that thesecond one is trivial. To see the third one holds, on the one hand by the construction of g t i , there is no node whose height lying in [ht(ˆ t i ) , η i ), and on the other hand if x ∈ M ∗ ,then x ∩ T <η ∗ i is bounded below η i , thus if s ∈ dom( f p ) is of height at least ht(ˆ t i ), thenthen ht( s ) ≥ η i , and thus sup(ht( u ) : u ∈ x ∩ T <η ∗ i ) < η i ≤ ht( s ).Therefore, for each x ∈ M ∗ , p witness the above three properties, thus for each x ∈ M ∗ ,there is p x ∈ M ∗ such that x p x satisfies the above properties. It turns out that T -assignments exist in M ∗ . We shall show that there is a set B ∗ ∈ M ∗ unbounded in P ω ( T ) such that for every x ∈ M ∗ ∩ B ∗ , p x and p are compatible. Let n := | dom( f x ) | . For each x ∈ P ω ( T ), fix anenumeration of dom( f p x ), say ( t xj : j ≤ n ). For every B ⊆ P ω ( T ), let B ( i, j ) := { x ∈ B : ht( t xj ) ≥ ht(ˆ t i ) } . Claim 4.38.
Let i ≤ m and j ≤ n . Suppose that B ∈ M ∗ an unbounded subset of P ω ( T ) . Assume that B ( i, j ) is cofinal in B . Then, there is a cofinal subset B i,j of B ( i, j ) in M ∗ such that for every x ∈ M ∗ ∩ B i,j , t xj ≮ T O M ( t i ) .Proof. Let Ψ i be the characteristic function of b M ( t i ) on T . For every x ∈ T , we let ψ xj : x → ψ xj ( s ) = 1 if and only if s < T t xj . Now consider the mapping x ψ xj . Since Ψ i is not guessed in M , Lemma 2.13 implies that there is a set B i,j ∈ M ∗ cofinal in B ( i, j ) such that for every x ∈ B i,j , ψ xj * Ψ i . Now if x ∈ M ∗ ∩ B i,j , then t xj ∈ M . Moreover, if s ∈ x is of height at least η ∗ i , then ψ xj ( s ) = 0 = Ψ i ( s ). Thus PECIALIZING TREES WITH SMALL APPROXIMATIONS I 23 ψ xj * Ψ i implies that there is some s ∈ T <η i ∩ M such that ψ xj ( s ) = Ψ i ( s ), which impliesthat t xj ≮ T O M ( t ). Notice that by the third property above ht( s ) < ht( t xj ). Now let e be a bijection between mn and m × n , say e ( k ) = ( e ( k ) , e ( k )). Letus set B = B − = P ω ( T ), and consider B (0 , P ω ( T ), thenapply Claim 4.38 to find B , , and set B = B , ; if it is not cofinal, then let B = P ω ( T ) \ B (0 , B k +1 ⊆ B k inductively; at stage k +1 ask whether C k := B k ( e ( k +1) , e ( k +1)) is cofinal in B k , if the answer is affirmative,then apply Claim 4.38 to obtain C ke ( k +1) ,e ( k +1) , and set B k +1 = C ke ( k +1) ,e ( k +1) , and ifthe answer is negative, then choose B k \ C k . It is clear that B k +1 ⊆ B k . Set B ∗ = B mn − .Notice that if x ∈ C ke ( k +1) ,e ( k +1) , then t xe ( k +1) ≮ T O M ( t e ( k +1) ). Claim 4.39.
For every x ∈ B ∗ ∩ M ∗ , p x and p are compatible.Proof. Fix x ∈ B ∗ ∩ M ∗ . Let r = p x ∧ p . We claim that r is a condition. By Lemma 4.35,we only need to check if there are comparable s ∈ dom( f p x ) \ dom( f p ) and t ∈ O ( p, M )such that f p x ( s ) = f p ( t ). We shall see that this is impossible. Assume towards a contra-diction that it holds true. Let t = t i and s = t xj , and consider ˆ t i . Now t xj ∈ M as x ∈ M ∗ .Then ht( t xj ) ≮ ht(ˆ t i ) since f p x ( s ) = f p ( t ) and that p x ∈ R p ( M, Σ). Thus ht( t xj ) ≥ ht(ˆ t i ).Let k ≥ e ( k ) = ( i, j ). Since x ∈ B ∗ ⊆ B k ⊆ B k − and that ht( t xj ) ≥ ht(ˆ t i ),we should have that B k = C k − i,j , but then t xj (cid:2) T O M ( t i ) by Claim 4.38, which in turnimplies that t xj (cid:2) T t i as x ∈ M , a contradiction! Corollary 4.40. P T is proper.Proof. Let θ ∗ be a sufficiently large regular cardinal. Assume that M ∗ ≺ H λ is countableand contains H θ , T, E and E . Set M = M ∗ ∩ H θ , and let p ∈ M ∗ be a condition.By Proposition 4.14, p M is a condition such that M ∈ M p M . Now, Proposition 4.36guarantees that p M is ( M ∗ , P T )-generic. Remark . Notice that P T forces | θ | = | T | = ℵ .We use the above strategy and Lemma 2.13 to show that P T has the ω -approximationproperty. Proposition 4.42. P T has the ω -approximation property.Proof. Suppose that ˙ f is a P T -name forced by a condition p to be a countably approx-imated { , } -valued function in V . Suppose that dom( ˙ f ) is forced to be some set in V , say X . Without loss of generality, we may assume either T ⊆ X or X ⊆ T . We consider the former; the other one can be proved similarly. Let θ ∗ be a sufficiently largeregular cardinal. Let M ∗ ≺ H θ ∗ be a countable model containing all relevant objects.Set M = M ∗ ∩ H θ . Let q ≤ p M decides ˙ f ↾ M ∗ , say g : M ∗ ∩ X → V is such that q (cid:13) ˙ f ↾ M = ˇ g . It is not hard to see that if g is guessed in M ∗ , then ˙ f is decided by acondition extending p . Thus ˙ f should be forced by p to be in V . We may thus assumethat g is not guessed in M ∗ , and we will find a contradiction. Fix an M -support set Σfor q . Now in M ∗ , let x → ( q x , q x ) be an assignment on P ω ( X ) such that(1) q x ≤ q ↾ M .(2) q x ∈ R q ( M, Σ).(3) | f q x | = | f q | .(4) For every s ∈ dom( q x ), if ht( s ) ∈ [ht(ˆ t i ) , η ∗ i ), thensup(ht( u ) : u ∈ x ∩ T <η ∗ i ) < ht( s ) , where ˆ t i is such that g t i = b ˆ t i .(5) g x : dom( g x ) → x as a subset.(6) q x (cid:13) g x ↾ x = ˙ f ↾ X .Here, η i and η ∗ i is as in Proposition 4.36. Such an assignment exists in M ∗ as witnessedby x ( q, g ). Now by Lemma 2.13, there is a set B ∈ M ∗ cofinal in P ω ( X ) such thatfor every x ∈ B ∗ , g x * g . Now let C be the restriction of B to T i.e C = { x ∩ T : x ∈ B } .Then C is cofinal in P ω ( T ). Using Axiom of Choice, for each c ∈ C , pick let x c ∈ B such that x c ∩ T = c . Fix such a choice function in M ∗ . Now consider the assignment c q x c . By the above properties, c q c = q x c is a T -assignment in M ∗ , thus as inProposition 4.36 there is some c ∈ C ∩ M ∗ such that q c is compatible with q . Thus thereis some x ∈ B ∩ M ∗ with x c = x . This is a contradiction as g x * g implies that q x is notcompatible with q , a contradiction! Lemma 4.43.
Suppose that p ∈ P T and t ∈ T . Then there is some q ≤ p such that t ∈ dom( f q ) .Proof. Assume that t is not in dom( f p ). If t is not in any model in E p , then pick ν below ω different from the values of f p such that ν > max( M ∩ ω : M ∈ E p ), and then set q = ( M p , f p ∪ { ( t, ν ) } ). Then Item 1 of Definition 4.12 holds true obviously, Item 2 holdstrue by our choice of ν . Items 3 and 4 are obvious of course as f q ( t ) = ν belongs to nomodel in E q = E p . PECIALIZING TREES WITH SMALL APPROXIMATIONS I 25
Now assume that there are some models in E p containing t . Let M be the least modelon the chain M p with t ∈ M . Let ν ∈ M ∩ ω \ Im( f p ) be such that ν > max( N ∩ ω : N ∈ E p ∩ M ) . Set q = ( M , f p ∪ { ( t, ν ) } ). We claim that q is a condition. As in the previous case,Items 1 and 2 of Definition 4.12 hold true, thus we only need to check Items 3 and 4.To see Item 3 holds, assume that N ∈ E p contains t . By the minimality of M , M ∈ ∗ N .We claim that M ⊆ N . Suppose this is not the case, thus there is some P ∈ E p suchthat P ∩ N ∈ ∗ M ∈ P ∈ N , but then t ∈ P ∩ N , which contradicts the minimality of M . Thus M ⊆ N , and hence ν ∈ M ⊆ N . For Item 4, suppose that N ∈ E p is suchthat ν ∈ N and t is guessed in N . We shall show that M ⊆ N , and hence t ∈ N . Wefirst show that N ∈ ∗ M is impossible. To see this, observe that N / ∈ M by our choice of ν . Thus if N ∈ ∗ M , then there is some P ∈ E p ∩ M such that N ∈ [ P ∩ M, P ) p . Now t belongs to P as it is guessed in N , and thus t ∈ P ∩ M , which contradicts the minimalityof M . Therefore, M should be below N in M p . If M * N , there is P ∈ M p such that P ∩ N ∈ ∗ M ∈ P ∈ N . Then since t ∈ P is guessed in N , by Lemma 4.9 it should beguessed in P ∩ N . Notice that ν ∈ P ∩ N , which is a contradiction as P ∩ N ∈ ∗ M , aswe have seen before. Conclusion
Suppose T is a tree of height ω without cofinal branches. We may assume that T is rooted. By Fact 2.3, we may also assume that T is a normal tree. Consider now P T constructed in the previous section. For a V -generic filter G ⊆ P T , set f G = [ { f p : p ∈ G } . Proposition 5.1. T is specializable via P T .Proof. By Corollaries 4.40 and 4.28, P T preserves ℵ and ℵ , respectively. Let G ⊆ P T be V -generic filter. By Lemma 4.43, dom( f G ) = T . It is clear that f G is a special functionon T . Putting all together we have proved the following.
Theorem 5.2.
Assume that GM ∗ ( ω ) holds. Then, every tree of height ω withoutcofinal branches is specializable using a proper forcing with finite conditions. Moreover,the forcing has the ω -approximation property. Since PFA implies GM ∗ ( ω ) by Proposition 2.12, we obtain the following corollary. Corollary 5.3.
Assume
PFA . Suppose T is a tree of height ω without cofinal branches.Then there is a proper ℵ -preserving forcing with the ω -approximation property suchthat P T specializes T in generic extensions. Acknowledgment
The author is grateful to M. Golshani and B. Veliˇckovi´c for the stimulating conversa-tions about the contents of this paper.
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