SPECIALISING TREES WITH SMALL APPROXIMATIONS I

Abstract Assuming 
$\mathrm{PFA}$
 , we shall use internally club 
$\omega _1$
 -guessing models as side conditions to show that for every tree T of height 
$\omega _2$
 without cofinal branches, there is a proper and 
$\aleph _2$
 -preserving forcing notion with finite conditions which specialises T. Moreover, the forcing has the 
$\omega _1$
 -approximation property.

the connection between the specialisation problem and generalised forms of Martin's axiom, and ask if we still need to consider countably closed forcings in this context. If the consistency of a higher analogue of PFA is achievable, it is then natural to speculate whether such a forcing axiom can imply that all trees in an appropriate subclass of trees of height and size 2 are special. As an early application of his method, Neeman [18] attempted to (partially) specialise trees of height 2 with finite conditions. To achieve this, he attaches the partial specialising functions to the sequences of models as side conditions. He then demonstrates that the resulting construction belongs to an iterable class which also includes a forcing notion for adding a nonspecial 2 -Aronszajn tree.
The second factor mentioned above may also lead one to recast the program of finding a generalised MA for the problem of special 2 -Aronszajn trees, as such trees intrinsically involve a particular compactness phenomenon. One can use some forms of the square principle to construct trees without cofinal branches that cannot be special, even in transitive outer models with the same cardinals. The basic idea goes back to Laver (see [21]) who isolated the concept of an ascending path through a tree and showed that an 2 -Aronszajn tree with an ascending path is non-special even in any transitive outer model that computes the relevant cardinals correctly. However, the earliest example of a non-special 2 -Aronszajn tree was constructed by Baumgartner using 1 , which was also independently discovered and generalised by Shelah and Stanley [21]. They showed that implies the existence of nonspecialisable + -Aronszajn trees. The connection between square-like principles and ascending paths through trees or tree-like systems has been studied by several people, just to mention a few: Baumgartner (as mentioned above), Brodsky and Rinot [3], Devlin [6], Cummings [5], Lambie-Hanson [10], Lambie-Hanson and Lücke [11], Laver and Shelah [12], Lücke [13], Neeman [18], Shelah and Stanley [21], and Todorčević [22].
To see why specialising a tree of height beyond 1 is subtly different from that of a tree of height 1 , let us first recall that the standard forcing to specialises a tree T of height κ + uses partial specialising functions of size less than κ, and let us denote this forcing by S κ (T ). For a cardinal ≤ κ, S (T ) is defined naturally. Lücke [13] studied the chain condition of S (T ), and complete the bridge between the notion of an ascending path and the chain condition of S (T ). Under some cardinal arithmetic assumptions, he showed that the nonexistence of a weak form of ascending paths 1 of width less than through T is equivalent to the κ + -chain condition of S (T ). Note that it is easily seen that S (T ) collapses κ + if T has a cofinal branch. Observe that also by Baumgartner-Malitz-Reinhardt [2], if T is of height 1 without cofinal branches, then S (T ) has the countable chain condition, as the existence of a cofinal branch through such tree is equivalent to the existence of a (weak) ascending path of finite length. It is also not hard to see that if κ = 1 and the CH fails, then S 1 (T ) collapses the continuum onto 1 . Thus not only the CH is necessary for preserving ℵ 2 , but also by Lücke's result, the lack of cofinal branches through T is not enough to ensure that S 1 (T ) preserves ℵ 2 . On the other hand, if T is of height 2 and has no cofinal branches, then S (T ) has the ℵ 2 -chain condition, but then the question is how to preserve 1 ? Therefore, the behaviour of the continuum function and the existence of ascending paths of width can prevent us from specialising trees of height 2 merely with countable conditions. Lücke [13] asked the following questions: (1) Assume PFA. Is every tree of height 2 without cofinal branches specialisable? (2) If T is a tree of height κ + , for an uncountable regular cardinal κ without ascending paths of width less than κ, is then T specialisable? Let us end our discussion with a couple of general questions: Do we still need to consider the specialisation of all 2 -Aronszajn trees in the context of generalised Martin's axiom? If looking for a generalised MA, do we want to have some kinds of compactness at ℵ 2 or not?
In this paper, we prove the following theorem.
Theorem 1.1. Assume PFA. Every tree of height 2 without cofinal branches is specialisable via a proper and ℵ 2 -preserving forcing with finite conditions. Moreover, the forcing has the 1 -approximation property.
This theorem answers Lücke's first question in the affirmative. 2 Given a tree T of height 2 with no cofinal branches, we shall use internally club 1 -guessing models to construct a proper forcing notion P T similar to Neeman's in [18], so that forcing with P T specialises T. Notice that the existence of sufficiently many 1 -guessing models of size 1 implies the failure of certain versions of the square principle. It is also worth mentioning that by an observation due to Lücke, the existence of sufficiently many 1 -guessing models of size ℵ 1 (and hence under PFA) no tree of height 2 without cofinal branches contains an ascending path of width . Interestingly, we will not use this fact, as the presence of guessing models in our side conditions suffices. By a theorem due to Viale and Weiß [25], under PFA, there are stationarily many internally club guessing models, and by a theorem due to Cox and Krueger [4], this consequence of PFA is consistent with arbitrarily large continuum. Thus essentially, the fact that 2 ℵ 0 = ℵ 2 holds under PFA does not play a role in our result and proofs.
We shall also answer the second question above consistently in the affirmative, for trees of height κ ++ without cofinal branches, in our forthcoming paper [16], which in particular includes a proof of the following theorem.
Our paper includes four additional sections. We give the preliminaries in Section 2. Section 3 is devoted to the introduction and the basic properties of forcing with pure side conditions. We shall introduce our main forcing and state its basic properties in Section 4. Finally, we establish our main result in Section 5. §2. Preliminaries. We shall follow standard conventions and notation, but let us recall some of the most important ones. In this paper, by p ≤ q in a forcing ordering ≤, we mean p is stronger than q; for a cardinal , H denotes the collection of sets whose hereditary size is less than ; for a set X, we let P(X ) denote the power-set of X, and if κ is a cardinal, we let P κ (X ) := {A ∈ P(X ) : |A| < κ}; recall that a set S ⊆ P κ (H ) is stationary, if for every function F :

Trees.
Let us recall the definition of a tree and some related concepts.
Definition 2.1. A tree is a partially ordered set (T, < T ) such that for every t ∈ T , b t := {s ∈ T : s < T t} is well ordered with respect to < T .
(1) For every t ∈ T , the height of t, denoted by ht T (t), is the order type of b t .
(3) For every α ≤ ht(T ), T α denotes the set of nodes of height α. T ≤α and T <α have the obvious meanings. In particular, T = T <ht(T ) and T ht(T ) = ∅.
is a downward-closed and linearly ordered set. A branch is a cofinal branch if its order type is the height of T. (5) T is called Hausdorff if for every limit ordinal α(α = 0 is allowed), and every Observe that a Hausdorff tree is rooted, i.e., it has a unique minimal point.
Definition 2.4. Suppose that ≤ κ are infinite regular cardinals. Assume that T is a tree of height κ + . Let S (T ) denote the forcing notion consisting of partial specialising functions, of size less than , ordered by reversed inclusion, that is f ∈ S (T ) is a partial function from T to κ such that if s, t ∈ dom(f) are comparable in T, then f(t) = f(s).
Lemma 2.5. In order to specialise a tree T (of height κ + , for some infinite cardinal κ), one may assume, without loss of generality, that T is a Hausdorff tree.
Proof. Recall that a function f : T 1 → T 2 between two trees is called a weak embedding if f respects the strict orders. It is easily seen that if T 1 weakly embeds into T 2 and T 2 is special, then T 1 is special, as the inverse image of an antichain in T 2 under a weak embedding is an antichain in T 1 . Thus to prove the lemma, it is enough to show that there is a weak embedding from T into a Hausdorff tree T * of the same height as T.
Let T * be the set of all non cofinal branches through T. Then, (T * , ⊂) is a tree of the same height as T. Note that ∅ is the root of T * . Moreover, if a ∈ T * , then the order type of (a, < T ) is exactly ht T * (a). Suppose that α is a nonzero limit ordinal and a, a ∈ T * α with b a = b a . We claim that a = a . Let t ∈ a. Since the order type of a is a limit ordinal, there is s ∈ a with t < T s. Let x = {u ∈ T : u < T s}. Now x < T * a. Thus x ∈ b a = b a . Then t ∈ x ⊆ a . So we have a ⊆ a . Similarly, we have a ⊆ a, and therefore, a = a . Now, let f : T → T * be defined by f(t) = b t . If s < t, then b s is a proper subset of b t , and hence f is a weak embedding.

Strong properness and the approximation property.
Recall that if M ≺ H contains a forcing P, then a condition p ∈ P is called (M, P)-generic if for every dense subset D of P in M, D ∩ M is pre-dense below p. Definition 2.6. Assume that P is a forcing, and is a sufficiently large regular cardinal. Suppose S ⊆ P κ (H ) consists of elementary submodels. Then, P is said to be proper for S, if for every M ∈ S and every p ∈ P ∩ M , there is an (M, P)-generic condition q ≤ p . Lemma 2.7. Let κ be a regular cardinal. Assume that P is a forcing, and > κ is a sufficiently large regular cardinal. Suppose S ⊆ P κ (H ) is a stationary set of elementary submodels. If P is proper for S, then P preserves the regularity of κ.
Proof. Let < κ be an ordinal. Assume towards a contraction that some p ∈ P forces thatḟ is an unbounded function from into κ. Pick M ∈ S such that , κ, p,ḟ ∈ M . Let q ≤ p be an (M, P)-generic condition. Note that ⊆ M and M ∩ κ ∈ κ. By our assumption, we can find a condition q ≤ q, and ordinal < and an ordinal ≥ M ∩ κ such that, q "ḟ( ) = ." Set Then D is a dense subset of P and belongs to M. Since q is (M, P)-generic, there is r ∈ D ∩ M such that r||q . Thus r is compatible with p, and hence, by elementarity, there is ∈ M such that r " =ḟ( )." Now if s is a common extension of q and r, we have s " = ." Thus = ∈ M ∩ κ, a contradiction! Let us now recall the following closely related definitions from [9,14], respectively.
Definition 2.8 (Strong properness). Suppose P is a forcing notion.
(1) Let X be a set. A condition p ∈ P is said to be strongly (X, P)-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) For a collection of sets S, we say P is strongly proper for S, if for every X ∈ S and every p ∈ P ∩ X , there is a strongly (X, P)-generic condition extending p.
Remark 2.9. It is easily seen that if p is strongly (X, P)-generic and M ≺ H is such that M ∩ P = X ∩ P, then p is strongly (M, P)-generic, and hence (M, P)generic. It turns out that if a forcing notion is strongly proper for some stationary set S ⊆ P κ (H ), then P is S-proper, and hence it preserves κ, by Lemma 2.7. Definition 2.10 (κ-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, and every A ∈ V [G] with A ⊆ V , the following are equivalent.
(2) For every a ∈ V with |a| V < κ, we have a ∩ A ∈ V .
Note that it is well-known that if a forcing notion is strongly proper for sufficiently many models in P κ (H ), then it has the κ-approximation property (see [15]).

Guessing models.
For a set M, we say that a set x ⊆ M is bounded in M if there is y ∈ M such that x ⊆ y. Recall that an elementary submodel M of H is called an internally club model (or IC-model for short) if it is the union of a continuous ∈-sequence M α : α < 1 of countable elementary submodels of H .
Let κ M be undefined if the above supremum does not exist.
We now recall the definition of a guessing model from [25].
Definition 2.14 (GM * ( 2 )). The principle GM * ( 2 ) states that for every sufficiently large regular cardinal , the set of 1 -guessing elementary IC-submodels of H is stationary in P 2 (H ).
Proof. The proposition above was mentioned without proof in [25]. A sketch of a proof can be found in [24,Theorem 4.4].
The following lemma plays a crucial role in our later proofs.
Proof. For each ∈ Z, and = 0, 1, let Notice that the sequence belongs to M. We are done if there is some ∈ Z such that both A 0 and A 1 are cofinal in B, as then by elementarity one can find such ∈ M ∩ Z, and then pick A 1-f( ) . Therefore, let us assume that for every ∈ Z, there is an ∈ {0, 1}, which Pure side conditions. This section is devoted to the forcing with pure side conditions. Such a forcing notion, as well as a finite-support iteration of proper forcings with side conditions, was introduced by Neeman in [17]. However, we cannot use Neeman forcing directly, since we shall work with non-transitive models. Instead, we follow Veličković's presentation [23] of Neeman forcing with finite ∈-chains of models of two types, where both types of models are non-transitive. We shall sketch some proofs of the necessary facts in this section, and we encourage the reader to consult [23] for more details.
Fix an uncountable regular cardinal , and let x ∈ H be arbitrary. We let E 0 := E 0 (x) denote the collection of all countable elementary submodels of (H , ∈, x), and let E 1 := E 1 (x) denote a collection of elementary IC-submodels of (H , ∈, x). Note that for every N ∈ E 1 and every It is easily seen that if M ∈ * N holds in an ∈-chain M, and that N ∈ E 1 , then M ∈ N . We simply write M ∈ * N , whenever M is clear from the context.  The following is easy and we leave the proof to the reader.
Therefore, the LHS is a subset of RHS. To see the other direction, suppose P does not belong to any interval as described in the above equation. In particular, P ∈ * M . Now, if P / ∈ M , it then means there are some models in E 1 p ∩ (P, M ) p . Let N be the least such model. Then, N ∩ M ∈ * P, since otherwise by the minimality of N, we have It is not hard to see that p M is an ∈-chain. Now, the following is immediate.
Remark 3.10. The above condition is the greatest lower bound of p and q, and denoted by p ∧ q. Notice that Now if r ∈ M ∩ M extends q M , then q is compatible with r by Fact 3.9. Thus q is strongly (M, M)-generic. By Lemma 2.7 and Remark 2.9, P perseveres ℵ 2 .
p . Then every condition q ∈ M extending p M is compatible with q. In fact, the closure of M p ∪ M q is a condition in M, which is also the greatest lower bound of p and q. Remark 3.13. As before we again denote the above common extension by p ∧ q. Notice that The following is similar to Fact 3.11 in light of Lemma 3.12.
Fact 3.14. M is strongly proper for E 0 . §4. The forcing construction. In this section, we first present the phenomenon of overlapping that was introduced by Neeman in his paper [18] regarding (partial) specialisation of trees of height and size 2 . Neeman's strategy is to attach S (T ) to side conditions consisting of models of two types: countable and transitive, where he also requires several constraints describing the interaction of the working parts, which are elements of S (T ), and the models as side conditions. He then analyses this interaction. Our approach is similar to Neeman's, and we still need to require one of the fundamental constraints, though our forcing is simpler than Neeman's. His definition of overlapping reads as follows: A model M overlaps a node t ∈ T \ M , if there is no non-cofinal branch b ∈ M with t ∈ b. Our terminology is different from Neeman's; we say a node t ∈ T is guessed in M if t belongs to some (non-cofinal) branch b ∈ M .
Throughout this section, we fix a Hausdorff tree (T, < T ) of height 2 without cofinal branches. We also fix a regular cardinal such that P(T ) ∈ H . We let E 0 := E 0 (T ) and E 1 := E 1 (T ) consist, respectively, of countable elementary submodels, and 1 -guessing elementary IC-submodels of (H , ∈, T ). We reserve the symbols p, q, r for forcing conditions, and s, t, u for nodes in T.  Thus every t ∈ M is already guessed in M, and that no node t with ht(t) ≥ sup(M ∩ 2 ) is guessed in M, since M has no cofinal branches. We shall often use the following without mentioning.

Overlaps between models and nodes.
Observe that O M (t) is always well-defined as T is a rooted tree belonging to every model in E 0 ∪ E 1 Proof. Of course, the first item follows from the proof of the second one, but we prefer to give independent proofs.
(2) We may assume that M is in E 0 as otherwise it is trivial. One easily observes that M (t) is below sup(M ∩ 2 ) since T does not have cofinal branches. Now * := min(M ∩ 2 \ M (t)) is an ordinal below 2 , but above M (t). Let b ∈ M be a branch containing t. Assume towards a contradiction that ht(t) > * , then there is some node s ∈ b of height * , and thus s < T t. It then follows that M (t) ≥ * > M (t), a contradiction.
The following is too easy, and we leave the proof to the reader.   The following is key for us.
Then exists as t ∈ N and ht(t) ≤ . Note that ∈ M ∩ 2 by elementarity. Observe that if = ht(s), for some s ∈ N ∩ b, then by elementarity, s ∈ N ∩ M . We then have t ∈ b s ∈ N ∩ M . Thus let us assume that the supremum is not obtained by any element of N ∩ b. In particular, ht(t) < and the cofinality of is either or 1 . We consider two cases: Case 1: cof( ) = . By elementarity, there is a strictly < T -increasing sequence s n : n ∈ ∈ M of nodes in b ∩ N such that sup{ht(s n ) : n ∈ } = . Since we assumed ht(t) < , there is n such that t ≤ T s n . Note that s n ∈ N ∩ M , and hence t ∈ b sn ∈ N ∩ M . Therefore, t is guessed in N ∩ M .
Case 2: cof( ) = 1 . We claim that b ∩ T ≤ is guessed in N. To see this, observe that b ∩ T ≤ is 1approximated in N, since if a ∈ N is a countable set, then there is s ∈ N ∩ b ∩ T ≤ such that a ∩ b ∩ T ≤ = a ∩ b s (as the cofinality of is 1 .) But a ∩ b s ∈ N . As N is an 1 -guessing model, we have b ∩ T ≤ is guessed in N. By the elementarity of Assume towards a contradiction that the equality fails. Thus, there is some s ∈ M whose height is above N ∩M (t) such that s ≤ T O M (t) ≤ T t. Then s ∈ N as 1 ∪ {t} ⊆ N . Therefore, s ∈ N ∩ M , and hence ht(s) ≤ N ∩M (t), a contradiction. Since both O N ∩M (t) and O M (t) are below t and of the same height, they are equal.

The forcing construction and its basic properties.
We are now ready to define our forcing notion P T to specialise T in generic extensions.  (2) f p ∈ S (T ).
We say p is stronger than q if and only if the following are satisfied.
Given a condition p in P T and a model M ∈ E 0 ∪ E 1 containing p, we define an extension of p that will turn later to be generic for the relevant models.  Proof. We check Definition 4.11 item by item. Item 1 is essentially Fact 3.5. Item 2 is obvious of course. To see Items 3 and 4 hold true, let N ∈ E 0 p M . We may assume that N / ∈ M p . Therefore, the only interesting case is M ∈ E 0 and N = P ∩ M , for some P ∈ E 1 p . Thus fix such models. Item 3: Let t ∈ dom(f p M ) ∩ N . We have f p (t) ∈ M , as p ∈ M , and also we have f p (t) ∈ P, as 1 ⊆ P. Thus f p (t) ∈ P ∩ M = N .
Finally, by the construction of p M , we have M ∈ M p M , and by Fact 3.5, p M ≤ p.
We now define the restriction of a condition to a model in the side conditions coordinate.   Note that p ∧ q is not necessarily a condition; however we shall use it as a pair of objects. Notice that M p∧q is the closure of M p ∪ M q under intersections, and belongs to M (see Remark 3.10 and Remark 3.13) and that also f p∧q is a well-defined function due to the fact that p satisfies Item 3 of Definition 4.11. Proof. Fix N ∈ E 0 p∧q and t ∈ dom(f p ) ∪ dom(f q ). Assume that t is in N. We shall show that f p∧q (t) ∈ N . We split the proof into two cases.
Observe that it is enough to assume N ∈ M p ∪ M q : if N ∈ M p ∧ M q , then N = P ∩ N , for some P ∈ M p ∪ M q , and some N ∈ M p ∪ M q . By our assumption, f p∧q (t) belongs to N , and hence, f p∧q (t) ∈ P ∩ N = N , as 1 ⊆ P .
As in the previous case, we may assume t ∈ dom(f q ) and N ∈ M p \ M q . Let us first assume that N ∈ * M . Suppose that N is the minimal counter-example with the above properties. Thus there is P ∈ E 1 p ∩ M such that N ∈ [P ∩ M, P) p . Now P ∩ M N , as otherwise f q (t) ∈ N , since t ∈ P ∈ M q and f q (t) ∈ P ∩ M . Therefore, 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 . Now Q ∩ N is also a counter-example to our claim, since t ∈ Q ∩ N ⊆ N , Q ∩ N ∈ M p \ M q (as otherwise, we would have f q (t) ∈ Q ∩ N ⊆ N ), and Q ∩ N ∈ * M . This contradicts our minimality assumption.
Two cases remain. The case N = M is trivial, and thus we only need to assume Remark 3.2). Notice that t ∈ P ∩ N . Thus by the previous paragraph, f q (t) ∈ P ∩ N ⊆ N .

Preserving ℵ 2 .
In this subsection, we prove that P T preserves the regularity of ℵ 2 . With a similar idea, we shall establish the properness of P T in the subsequent subsection. Proof. Set r = p ∧ q. Notice that f r is well-defined as a function. Now fix t ∈ dom(f r ) and N ∈ E 0 ∩ M r so that f r (t) ∈ N . We shall show that if t is guessed in N, then t ∈ N . Notice that by Remark 3.10, we have M r = M p ∪ M q . We shall consider the nontrivial cases: As b is of size ≤ ℵ 1 and 1 ⊆ M , we have t ∈ b ⊆ M . Thus t ∈ M , which in turn implies that t ∈ dom(f q ) and f q (t) = f p (t) = f r (t) ∈ N . But then t ∈ N , as q is a condition.
Assume that t is guessed in N. By Lemma 4.9, t is guessed in M ∩ N . On the one hand, Thus far, we have shown that p ∧ q satisfies all items in Definition 4.11, possibly except Item 2. We shall show that under appropriate circumstances, p ∧ q is indeed a condition. We now prepare the ground for this.    Proof. Fix p ∈ P T . It is enough to define for t ∈ D(p, M ) with O M (t) / ∈ M . Thus fix such a t. Notice that dom(f p ) is finite, and that, by Lemma 4.6, M (t) is a limit ordinal. Thus one may easily find a node (t) with the above properties. (1) q ≤ p M .
(2) For every t ∈ dom( ), the following hold: (a) There is no node in dom(f q ) whose height is the interval ht( (t)), M (t) . (b) For every s ∈ dom(f q ), if s < T (t), then f q (s) = f p (t).
Let R p (M, ) be the set of (M, )-reflections of p with support .  Proof. We check the items in Definition 4.22. Item 1 is essentially Proposition 4.15. Item 2a follows from the definition of . Item 2b follows from the fact that p is a condition, and that (t) < T t. Proof. Since q ≤ p M , f r is well-defined as a function. We shall show that it satisfies the specialising property. To do this, we only discuss the nontrivial case by considering two arbitrary comparable nodes t ∈ dom(f p ) \ dom(f q ) and s ∈ dom(f q ) \ dom(f p ). We claim that f r (t) = f r (s). Observe that s ∈ M . The fact that M ∩ 2 is an ordinal imply that if t ≤ T s, then t ∈ M , which is a contradiction as t / ∈ dom(f q ). Thus, the only possibility is s < T t. Since q ∈ R p (M, ) ∩ M , the height of s is not in the interval ht( (t)), M (t) . Thus s < T (t). Then Item 2b of Definition 4.22 implies that f q (s) = f p (t). Therefore, f r (t) = f r (s).
We have now all the necessary tools to prove the preservation of ℵ 2 by P T .

Properness.
This subsection is devoted to the proof of the properness of P T . We will closely follow our strategy in the previous subsection. Notice that our notation and definition related to models in E 0 are similar to the ones we used for the preservation of ℵ 2 , but hopefully there will be no confusion, since these two parts are completely independent. comparable in T, we shall show that f q (s) = f p (t). We may assume that f p (t) ∈ M . Thus t < T s is impossible, as otherwise t is guessed in M, and hence t ∈ M , which is a contradiction! Consequently, the only possible case is s < T t. In this case, M ). Therefore, by Item 2a of Definition 4.32, the height of s avoids the interval ht(sup( (t))), M (t) . Thus s < T sup( (t)), and hence s ∈ (t). In either case, s ∈ (t), but then Item 2b of Definition 4.32 implies that f p (t) = f q (s). Proof. Assume that p ≤ p. Since M ∈ M p , we may assume without loss of generality that p = p. Let D ∈ M * be a dense subset of P T . We may also assume, without loss of generality, that p ∈ D. Since M * is fixed throughout proof, we simply denote M (t) by t . By Lemmas 4.31 and 4.33, there is an M-support for p so that p ∈ R p (M, ). Observe that R p (M, ) ∈ M * . Let t i : i < m enumerate O(p, M ) so that t i ≤ t i+1 , for every i < m -1. Let i : i < m be the strictly increasing enumeration of { t i : i < m}. To reduce the amount of notation, we may assume that m = m . For every i < m, set * Notice that * i < i+1 , for every i < m -1. For every i < m, we let alsot i denote sup( (t i )). Note thatt i exists, as t i ∈ O(p, M ). Let us call a map x → p x from P 1 (T ) into P T , a T-assignment if the following properties are satisfied for every x ∈ P 1 (T ).
(3) For every s ∈ dom(f px ) and every i < m, if ht(s) ∈ ht(t i ), * i , then sup{ht(u) : u ∈ x ∩ T < * i } < ht(s). We first show that there are T-assignments in M * . Proof. We observe that all the parameters in the above properties are in M * . By elementarity and the Axiom of Choice, it is enough to show that for every x ∈ M * , there is such p x ∈ H * . Thus fix x ∈ M * . We claim that p is such a witness. The first item is clear by Lemma 4.33 and that the second one is trivial. To see the third one holds true, fix i < m and observe that • there is no node in dom(f p ) whose height lies in the interval ht(t i ), i (by the construction of (t i ), see Item 2 of Definition 4.30).
Thus if s ∈ dom(f p ) is of height at least ht(t i ), then ht(s) ≥ i , and thus Fix a T-assignment x → p x in M * . We shall show that there is a set B * ∈ M * cofinal in P 1 (T ) such that for every x ∈ M * ∩ B * , p x and p are compatible. Let n := |dom(f p )|. For each x ∈ P 1 (T ), fix an enumeration of dom(f px ), say t x j : j < n . For every B ⊆ P 1 (T ), let Claim 4.37. Let i < m and j < n. Suppose that B ∈ M * is an unbounded subset of P 1 (T ). Assume that Proof. Let Ψ i be the characteristic function of b M (t i ) on T. Note that Ψ i is not guessed in M. For every x ⊆ T , we let x j : x → 2 be defined by x j (s) = 1 if and only if s < T t x j . Now consider the mapping x → x j . Since Ψ i is not guessed in M, On the other hand, by Item 3 in the definition of a T-assignment, we have ht(s) < ht(t x j ). Thus s < T t x j if and only if s ≮ T O M (t i ), which contradicts t x j < T O M (t i ). Returning to our main proof, let e be a bijection between mn and m × n. For every k < mn, set e(k) := (e 0 (k), e 1 (k)). We build a descending sequence B k : -1≤ k < mn of cofinal subsets of P 1 (T ) with B k ∈ M * as follows. Let also B -1 := P 1 (T ). Suppose that B k , for k ≥ -1, is constructed. Set C k := B k (e 0 (k), e 1 (k)) and ask the following question: Then proceed as follows: • If the answer to the above question is YES, then apply Claim 4.37 to C k , e 0 (k + 1) and e 1 (k + 1) to obtain C k e 0 (k+1),e 1 (k+1) ∈ M * as in the claim, and then set B k+1 := C k e 0 (k+1),e 1 (k+1) . • If the answer to the above question is NO, then let B k+1 = B k \ C k .
It is clear that B k : -1≤ k < mn is descending and each B k is in M * . Set B * := B mn-1 . Note that if x ∈ C k e 0 (k+1),e 1 (k+1) , then t x e 1 (k+1) ≮ T O M (t e 0 (k+1) ), by Claim 4.37. Proof. Fix x ∈ B * ∩ M * . Then p x ∈ M * ∩ D. Let r = p x ∧ p. We claim that r is a condition. By Lemma 4.34, we only need to check if there are comparable s ∈ dom(f px ) \ dom(f p ) and t ∈ O(p, M ) such that f px (s) = f p (t). We shall see that it does not happen. Thus assume towards a contradiction that there are such t and s. Then t = t i and s = t x j , for some i < m and j < n. Note that f px (s), t x j ∈ M , as x ∈ M * . Observe that if t i ≤ T t x j , then t i is guessed in M, and hence it belongs to M by Item 4 of Definition 4.11, which is a contradiction. Thus t x j < T t i , which in turn implies that t Remark 4.39. Note that to find the cofinal set B * in the above proof, we could start with any set which is cofinal in P 1 (T ). Proof. Let * be a sufficiently large regular cardinal. Assume that M * ≺ H * is countable and contains H , T, E 0 , and E 1 . Set M = M * ∩ H , and let p ∈ M * be a condition. Notice that the set of such models is a club in P 1 (H * ). By Proposition 4.13, p M is a condition with p M ≤ p such that M ∈ M p M . Now, Proposition 4.35 guarantees that p M is (M * , P T )-generic. Thus P T is proper.
We shall use the above strategy and Lemma 2.16 to show that P T has the 1approximation property. Proof. Assume towards a contradiction thatȦ is a P T -name such that for some p ∈ P T and some X ∈ V , we have • p "Ȧ ⊆X , " • p "Ȧ / ∈ V, " and • p "Ȧ is countable approximated in V, " i.e., for every countable set a ∈ V , p "Ȧ ∩ǎ ∈ V." Without loss of generality, we may work with a P T -name for the characteristic function ofȦ, sayḟ. We may also, without loss of generality, assume that either T ⊆ X or X ⊆ T . To see this, observe that by passing to an isomorphic copy of T, we may assume that the underlying set of T is |T |. On the other hand, using a bijection between X and |X |, we can assume that the domain ofḟ is forced to be |X |. As |X | and |T | are comparable, we may assume that either T ⊆ X or X ⊆ T .
Let us assume that T ⊆ X , the other case is proved similarly. Let * be a sufficiently large regular cardinal. Let M * ≺ H * be a countable model containing all the relevant objects, including p. Set M = M * ∩ H . We can extend p M to a condition q such that q decidesḟ M * , i.e., for some function g : M * ∩ X → 2 in V, q "ḟ M * =ǧ." Proof. Suppose that g is guessed in M * . Let g * ∈ M * be such that g * ∩ M * = g. Set D = {r ≤ p : ∃x ∈ X r "g * (x) =ḟ(x)"} ∪ {r ∈ P T : r ⊥ p}.
Obviously D ∈ M * . We use elementarity to show that D is dense in P T . Thus let r ∈ M * ∩ P T . We may assume that r is compatible with p. Thus, there is s ∈ M ∩ P T such that s ≤ p, r. Since p "ḟ / ∈ V, " there is x ∈ M * ∩ X and there is s ≤ s in M * such that s "g * (x) =ḟ(x)." Thus s ∈ D ∩ M . On the other hand, by Proposition 4.35, q is (M * , P T )-generic. Thus, there is u ∈ D ∩ M * such that u||q. But then u||p, and thus there is x ∈ M * ∩ X such that u "g * (x) =ḟ(x)." This is impossible, as q g * (x) = g(x) =ḟ(x).
Fix an M-support set for q. As in the proof of Proposition 4.35, we can find, in M * , a function x → (q x , g x ) on P 1 (X ) such that: (1) q x ∈ R p (M, ).
(4) g x : dom(g x ) → 2 is a function with countable domain containing x as a subset. (5) q x g x x =ḟ x .
Here, i , * i , andt i are as in the proof of Proposition 4.35. Note that to find an assignment in M * , observe that if x ∈ M * , then x ⊆ dom(g), and thus we can use (q, g) as a witness. Since we assumed T ⊆ X and by the above claim g is not guessed in M * , we first apply Lemma 2.16 to find a set B ∈ M * , cofinal in P 1 (X ), such that for 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 1 (T ). Using the Axiom of Choice, for each c ∈ C , pick x c ∈ B such that x c ∩ T = c. Fix such a choice function c → x c in M * and consider the assignment c → q xc . By the above properties, c → q c = q xc is a T-assignment in M * . Thus, as in Proposition 4.35, there is some c ∈ C ∩ M * such that q c is compatible with q. There exists x ∈ B ∩ M * with x c = c, but this is a contradiction, as g x g implies that q xc = q c is not compatible with q! 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 belonging to E 0 p , then pick below 1 and different from the values of f p such that > max{M ∩ 1 : M ∈ E 0 p }, and then set q = (M p , f p ∪ {(t, )}). Then Item 1 of Definition 4.11 is easily fulfilled, Item 2 holds true as / ∈ rang(f p ). Item 3 is obvious as t does not belong to any model in M q = M p . Finally, Item 4 is fulfilled, since f q (t) = belongs to no model in E 0 q = E 0 p . Now assume that there are some models in E 0 p containing t. Let M be the least countable model in M p with t ∈ M . Let ∈ M ∩ 1 \ ran(f p ) be such that > max{N ∩ 1 : N ∈ E 0 p ∩ M }.