Uncountable trees and Cohen κ-reals

Transcription

1 Uncountable trees and Cohen -reals Giorgio Laguzzi (U. Freiburg) January 26, 2017 Abstract We investigate some versions of amoeba for tree forcings in the generalized Cantor and Baire spaces. This answers [9, Question 3.20] and generalizes a line of research that in the standard case has been studied in [10], [11] and [7]. Moreover, we also answer questions posed in [4] by Friedman, Khomskii and Kulikov, about the relationships between regularity properties at uncountable cardinals. We show Σ 1 1-counterexamples to some regularity properties related to trees without club splitting. In particular we prove a strong relationship between the Ramsey and the Baire properties, in slight contrast with the standard case. 1 Introduction This paper is concerned with forcings consisting of uncountable trees. In particular we focus on some issues about pure decision and Cohen -reals, attacking some points raised in [9, Question 3.20]. The main part of the paper is centered around the investigation of amoeba forcings, and more generally, the effects of adding uncountable generic trees over models of ZFC. The importance of such a topic is that it has crucial applications in questions concerning cardinal invariants associated with tree-ideals and regularity properties. In the standard case, such a topic has been extensively studied; see [10], [11], [1] and [7], [12] for important results in the context of 2 ω and ω ω. When dealing with trees on ω, even if the most natural versions of amoeba usually do not have pure decision, some refinements can be defined in order to even get the Laver property. This is indeed possible for Sacks, Miller, Laver and Mathias forcing, whereas in [12] Spinas has shown this cannot be done for Silver forcing. Rather surprisingly, we show that the situation with trees on > ω is completely different, and we are going to show that pure decision gets very often lost. In the last section, we also present some results about regularity properties for tree-forcings at, showing Σ 1 1-counterexamples for Mathias and Laver measurability even without club splitting, and we obtain an interesting and rather surprising result connecting the generalizations of Ramsey property and the property of Baire. This contributes to solve some questions raised by Friedman, Khomskii and Kulikov concerning the regularity properties diagram at (see [4]). We remark that this reaserch field has largely spread out in the last years among set theorists. In particular, the problems analysed in this paper are part of those collected in [9], which is the output of a series of workshops which took place in Amsterdam (November 2014), Hamburg (September 2015) and Bonn (September 2016). 1

2 2 Definitons and notation Throughout the paper we assume 2 < = and we use the following notation. The elements in 2 and are called -reals or -sequences. A tree T 2 < is a subset closed under initial segments and its elements are called nodes. We consider < -closed trees T, i.e., for every -increasing sequence of length < of nodes in T, the supremum (i.e., union) of these nodes is still in T. Moreover, t denotes the order type of dom(t). We say that a tree T is perfect iff for every s T there is t s such that t 0 T and t 1 T ; we call such t a splitting node (or splitnode) and set Split(T ) := {t T : t is splitting}. We say that a splitnode t T has order type α (and we write t Split α (T )) iff {s T : s t s Split(T )} = α. t T is a terminal node iff there are no s t such that s T, and we write t Term(T ). Moreover, we say that T end-extends S iff T S and for every t T \ S there exists s Term(S) such that s t; stem(t ) is the longest node in T which is compatible with every node in T ; T t := {s T : s is compatible with t}; for p T, T p := {t T : s Term(p)(s t t s)}. [T ] := {x 2 : α < (x α T )} is called the set of branches (or body) of T. For s C, we put [s] := {x 2 : s x}. For every tree T we define the boundary of T b(t ) as: b(t ) := {t / T : s t(s T )}. (Note that the same notions can be introduced for trees in <.) We say that a poset P is a tree-forcing if the conditions are perfect trees in < or 2 < with the property that if T P and t T, then T t P too. The ordering is the inclusion. The generic -real added by a tree-forcing P is x G := T G stem(t ). Along the paper we are going to introduce several types of tree-forcings: Sacks, Silver, Miller, Cohen, Laver and Mathias. We remark that some popular forcings can be seen as tre-forcings, even if this might not be evident a priori. For instance: -Cohen forcing C := {2 <, } can be seen as a tree-forcing by associating s 2 < with the tree T s := {t 2 < : t s}. Also we will often write [s] := {x 2 : x s} instead of [T s ]. -Mathias forcing R := {(s, x) : s 2 <, x 2, s x x s = s}, ordered by: (t, y) (s, x) s t i < (y(i) = 1 x(i) = 1). Again, one can associate (s, x) R with a tree T (s,x) as follows: T (s,x) := {t 2 < : t s i < t (t(i) = 1 x(i) = 1)}.

3 Each section is indeed devoted to focus on a particular kind of trees. The specific definitions are given at the beginning of the corresponding section. Definition 1. (Tree-ideals and tree-measurability) Let P be a tree-forcing and let X be a set of -reals. We define: X is P-open dense iff T P T T (T P [T ] X). The complement of a P-open dense set is called P-nowhere dense. X is P-meager iff it can be covered by a -size union of P-nowhere dense sets. The ideal of P-meager sets is denoted by I P. X is P-measurable iff for every T P there is T T, T P such that [T ] X I P or [T ] \ X I P. Definition 2. Let P be a tree-forcing. We say that T P is an absolute P-generic tree over V if for every forcing extension N V via a < -closed poset N = T P x [T ](x is P-generic over V ). We say that AP is an amoeba forcing for P whenever it adds an absolute generic tree T P over V. Definition 3. (-axiom A) Let P be a forcing notion. We say that P satisfies -axiom A iff there is a sequence { α : α < } of partial orders satisfying the following properties: 1. 0 = and for every α <, α, and for every α < β, β α ; 2. if {p α : α < } P is such that for every α < β, p β α p α, then there is q P such that for all α <, q α p α (such q is called fusion); 3. if A P is an antichain, p P and α <, then there is q α p such that {p A : p and q are compatible} has size. Definition 4. Let P be a forcing satisfying axiom A and { α : α < } be the corresponding sequence of orders. We say that a tree-forcing P satisfies pure decision iff for every formula ϕ, p P there exists q 0 p such that either q ϕ or q ϕ. 3 -Sacks trees Definition 5. A tree T 2 < is called club -Sacks (T S Club ) iff T is perfect and for every x [T ], {α < : x α Split(t)} is closed unbounded (briefly called club, from now on). This forcing was introduced by Kanamori ([6]) as a suitable generalization of Sacks forcing in order to obtain < -closure and preservation of + under - support iteration. In more recent years, S Club has been investigated by several authors, such as Friedman and Zdomskyy ([3]) and Friedman, Khomskii and Kulikov ([4]). Definition 6. (p, T ) AS Club T S Club ; iff the following holds: p T is a < -closed subtree of size < such that if {t i : i < δ} is a < -sequence of splitnodes of p then lim i<δ t i Split(p) too;

4 Term(p) Split α (T ), for some α <. The order is given by: (p, T ) (p, T ) T T p p. If G is AS Club -generic over V, put T G := {p : T (p, T ) G}. The following result shows that AS Club is really an amoeba forcing for S Club with respect to Definition 2. The proof is similar to the one of [7, Lemma 12]. Proposition 7. Let G be AS Club -generic over V. Then T G is an absolute S Club -generic tree over V. Proof. T G S Club is clear, since any approximation p T G has the property that limits of splitting nodes are splitting. We first check that V [G] = x [T G ](x is S Club -generic over V ) Fix D S Club open dense and put E D := {(p, T ) : t b(p)(t t D)}. We claim E D is open dense. In fact, fix (p, T ) AS Club, and for every t b(p), pick S t T t such that S t D. Then put S := t b(p) S t. Clearly (p, S) E D. Hence, for every D S Club open dense in V, V [G] = x [T G ](H x D ), where H x := {T S Club : x [T ]}. Note that H x is a filter. If not, there are T, T H x, such that T T is not in S Club. Let A = T T ; then D A := {S S Club : [S] [A] = } is dense, as for every S S Club we can find t S / A. But then H x D A, contradicting the fact that x [T ] [T ] = [A]. Note that what we have proven is V [G] = ϕ : F 2 < x 2 (x [T G ] t F (t x (T G ) t D)). But ϕ is a Σ 1 2( ) statement and so for every < -closed forcing extension N V [G], N = ϕ. Since D S Club is arbitrary, we are done. We now assume be inaccessible and we consider the ordering: (p, T ) (p, T ) T T p end-extends p. Now we aim at showing the following. Proposition 8. Let be inaccessible. AS Club satisfies -axiom A. The sequence of orders { α : α < } we choose is the following: first we say that T α T, for T, T S Club iff T T and Split α (T ) = Split α (T ). Then we put (p, T ) α (p, T ) iff p = p T α T. Proposition 8 follows from the following two lemmata. Lemma 9. AS Club (p, T ) AS Club satisfies quasi pure decision, i.e., given D AS Club such that T T, (p, T ) AS Club and there is T S Club (q, S) (p, T )((q, S) D (q, T q) D). Proof. To simplify the notation, we give a proof for p = and leave the general case to the reader. We use the following notation: given a < -size tree p we say that q p is a terminal subtree iff t Term(q), t Term(p) too. In the following construction, we use the following notation: and

5 - T [β] denotes the tree generated by Split β (T ); - for a tree T, p < -size subtree of T, and S T end-extending p, put T p S := {t T : ( t 0 Term(p), t 0 t) t S}. We build a fusion sequence {T α : α < } by induction as follows (with T 0 = T ): Step α + 1. Let {p i α : i < δ α } enumerate all terminal subtrees of T α [α + 1]. Note that δ α 2 2α <, since is inaccessible. Then we proceed by induction on i < δ α as follows: i = 0. If S 0 α T α so that (p 0 α, S 0 α) D, then put T 0 α := T α p0 α S 0 α ; otherwise let T 0 α := T α. i + 1. If Sα i+1 Tα i so that (p i+1 α, Sα i+1 ) D, then put Tα i+1 := Tα i pi+1 α Sα i+1 ; otherwise let Tα i+1 := Tα. i i limit. Put Tα i := j<i T α. j Then put T α+1 := i<δ α T i α. Note that T α+1 α T α. Step α limit. Put T α := ξ<α T ξ. Note that for all ξ < α, T α ξ T ξ. Finally put T := α< T α. We claim that T as the required property. Indeed pick any (q, S) (, T ) such that (q, S) D. Choose α < and i < δ α such that q = p i α. Then the statement S 0 (p i α, S 0 ) D is satisfied, and so (q, T q) (q, Tα q) i (q, S 0 ) D. Lemma 10. Let A AS Club α <. Then there exists T S Club (p, T ) only has -elements compatible in A. be a maximal antichain, (p, T ) AS Club and, T T such that (p, T ) α (p, T ) and Proof. Let A AS Club be a maximal antichain and D A := {(p, T ) : (q, S) A, (p, T ) (q, S)} its associated open dense set. Given any (p, T ) AS Club, α <, let {p i : i < δ} (δ < ) list all terminal subtrees of T [α+1] end-extending p and apply Lemma 9 in order to find T i T p i, for i < δ, satisfying quasi pure decision for (p i, T p i ) with D = D A. Then define T as the limit of the following recursive construction on i < δ (starting with S 0 = T ): successor case i + 1: put S i+1 := S i pi T i ; limit case i: S i := j<i S j. Finally put T := i<δ S i. We get (p, T ) α (p, T ) and {(q, S) A : (q, S) (p, T )} i<δ{(q, S) A : (q, S) (p i, T i )}. But for every i < δ, Lemma 9 implies {(q, S) A : (q, S) (p i, T i )} has size, and so also the δ-size union has size. Remark 11. Looking at the ω-case, without particular care, amoeba forcings tend to add Cohen reals (so they do not have the Laver property). For instance, the naive Sacks amoeba adds the following Cohen real c 2 ω : let T G be the generic Sacks tree added by the amoeba, z its leftmost branch, {t n : n ω} the set of splitting nodes such that z = n ω t n. Define c(n) to be 0 iff min{ s : t n 0 s s Split(T G )} min{ s : t n 1 s s Split(T G )}. This construction straightforwardly generalizes to our context 2. But this kind of Cohen real can be suppressed by considering a finer version of Sacks amoeba, that not only kills

6 this instance of Cohen real, but actually turns out to have pure decision and the Laver property (see [10] and [7]). However, in the generalized framework we are considering, this kind of construction fails, since we do not have an appropriate partition property for perfect trees in S Club. A symptom of this problem is revealed by the existence of another kind of Cohen -real, that seems to be more robust compared to the previous one. To build this Cohen -real we fix a stationary and co-stationary subset S. Then let x T G be the leftmost branch, where T G is the generic tree added by AS Club, and let {t α : α < } enumerate the splitting nodes that are initial segments of x. Then put c(α) = 0 iff t α+1 S. It is easy to check that c is a Cohen -real, since any club contains unboundedly often both elements in S and elements in its complement. In the next section, we are going to prove one of the main and most surprising results of this paper, showing that this kind of Cohen -real cannot be avoided whenever we add an absolute generic S Club tree. 3.1 Coding by stationary sets The main result in this section will be the following. Theorem 12. Let V N be ZFC-models such that in N there is an absolute S Club -generic tree over V and N is a forcing extension of V via a < -closed poset. Then there exists z N 2 that is Cohen over V. We remark that this result highlights a strong difference from the standard Sacks forcing in the ω-case, for which one can construct an amoeba for Sacks satisfying pure decision and the Laver property. Theorem 12 is therefore a rather strong result, which asserts that no matter how we refine our amoeba for S Club, we will never find a version not adding Cohen -reals. To prove Theorem 12 we introduce a way to read off a Cohen -sequence from T S Club in an absolute way. This coding will use stationary subsets of. So fix {S τ : τ 2 < } family of disjoint stationary subsets of. Let {λ α : α < } be an increasing enumeration of all limit ordinals <. The set H α we will refer to is meant in different ways, depending whether we are dealing with a successor or an inaccessible. inaccessible case: put H α := 2 λα. successor case: let W 0 be a well-ordering of all s 2 λ0 and for every α <, t 2 λα, let Wt α+1 be a well-ordering of all s 2 λα+1 extending t. In what follows ot(s) refers to the order type of s 2 λα+1 in the wellordering Wt α+1. Then recursively define: H 0 := {t W 0 : ot(t) < λ 0 } and t H α+1 iff there exists {t ξ : ξ α+1} with the following properties: for every ξ α, one has: t ξ 2 λ ξ, t ξ+1 W ξ+1 t ξ, and ot(t ξ ) < λ α+1, ot(t α+1 ) < λ α+1, t λ = lim ξ<λ t ξ, for λ limit, t = t α+1. H λ := α<λ H α, for λ limit. Note that every H α has size <.

7 Lemma 13 (Coding Lemma). Let T S Club, {D ξ : ξ < } be a -decreasing family of open dense subsets of C. Then there is T S Club, T T such that for every α <, there exists a unique τ α 2 < such that t Split α+1 (T ) s H α ( t S τα s τ α D α ). The first step is to prove the following. Claim 14. Given T S Club, α, τ 2 <, there is T α T, T S Club such that t Split α+1 (T )( t S τ ). Proof of Claim. Fix α < and τ 2 <. For every t Split α (T ), i {0, 1}, pick σ(t, i) Split(T ) such that σ(t, i) t i and σ(t, i) S τ. Note that we can do that, since S τ is stationary and we have club many splitnodes above each t i. Moreover, note that for every τ τ, σ(t, i) / S τ, since the stationary sets are pairwise disjoints. Finally let T := {T σ(t,i) : t Split α (T ), i {0, 1}}. By construction, T S Club, T α T and has the desired property. Proof of Coding Lemma. We build a fusion sequence {T α : α < }, with T α+1 α T α as follows (we start with T 0 = T ): case α + 1: first find τ α 2 < so that s H α, s τ α D α ; note this is possible since H α < and D α is open dense in C. Then apply the previous Claim for T = T α and τ = τ α to obtain T α+1 α T α such that t Split α+1 (T α+1 ) s H α ( t S τα s τ α D α ). case λ limit: put T λ := α<λ T α. Finally put T := α T α. Note that, Split α (T ) = Split α (T α ). Hence, by construction T has the desired properties. Notation: Given T S Club as in the Coding Lemma, pick τ α 2 < be such that any splitting node t Split α+1 (T ) satisfies t S τα (by construction, note that τ α is uniquely determined). We call z := τ 0 τ 1... τα... the Cohen -sequence associated with T. Proof of Theorem 12. Let c be a Cohen -real over N and work in N[c]. Let T G N denote the absolute S Club -generic tree over V, f : 2 < Split(T G ) -isomorphism, f : 2 [T G ] the homeomorphism induced by f, and finally let x = f(c) [T G ]. Remark that T G, f, f N, while f(c) N[c] \ N obviously. Note that x is S Club -generic over V, since T G is absolute generic and N V is a forcing extension via a < -closed poset. For every maximal antichain A S Club in V, pick T A such that s ẋ [T A ], for some s C. For every s C, one can then define, B(s) := {T A : s ẋ [T A ]}. Fact 15. B(s) contains a tree in S Club. Proof of Fact. To reach a contradiction, assume not. Define D s := {t 2 < : [ f(t )] [B(s)] = }. Note D s N. For every t 2 < there is t Split(T G ) such that t f(t) and t / B(s); this is possible as (T G ) f(t) S Club and so there is t (T G ) f(t) \ B(s). Then pick t 2 < so that f(t ) = t is in D s and t t. That implies D s is dense in C. Hence, c D s, i.e., there is i < such that c i D s, which gives [ f(ċ i)] [B(s)] =. But we know s ẋ = f(ċ) [B(s)], by definition. Contradiction.

8 So we can assume for every s 2 < there is T (s) B(s) in S Club. Now let {T i : i } enumerate all such T (s) s, and let {τα i : α < } be the components of the Cohen -sequence associated with T i. Then put z := i< σ i, where the σ i s are recursively defined as follows: σ 0 := σ i+1 := σ i τ α i i+1, where α i+1 is chosen in such a way that H αi+1 σ i σ i := j<i σ j, for i limit ordinal. We aim at showing that z is Cohen over V. Let D C be open dense in V ; we say T satisfies the Coding Lemma for D if the sequence is so that D ξ = D, for every ξ <. Set S(D) := {T S Club : T satisfies Coding Lemma for D}, which is a dense subset of S Club. Pick A S(D) maximal antichain (note A is a maximal antichain in S Club as well, as S(D) is a dense subposet of S Club ). Then there exists i < such that T i T A, for some T A A. For every s H αi+1 we have s τα i i+1 D. By construction, j<i σ j H αi+1, and so σ i+1 D. Hence, for every D V open dense of C, there exists α < such that z α D, which means z is Cohen over V. Corollary 16. Let AS be any amoeba for S Club satisfying < -closure and G be AS-generic over V. Then there is c 2 V [G] which is Cohen over V. Proof. It is simply a direct application of the main theorem. By definition, an amoeba for S Club adds an absolute generic tree T G. Then simply apply the theorem for N = V [G], which satisfies the assumption, as AS is < -closed. Remark 17. Given T S Club satisfying the Coding Lemma, let τ α 2 < be such that for every t Split α+1 (T ), t S τα. We can define E (α, T ) = s H α [s τ α ] and then E (T ) := E (α, T ) and E(T ) := E (T t ). β< α β t Split(T ) By construction, both E (T ) and E(T ) are dense Π 0 2 sets (-intersection of open dense). So there is a way to associate T S Club with a dense Π 0 2 sets. This might be useful to answer the following interesting and natural question. Question 1. Let M be the ideal of meager sets, I S Club is the ideal of S Club meager sets, and T denotes Tukey embedding. Is M T I S Club? - Note that we can prove an analogue of Proposition 12 by replacing -Cohen reals with dominating -reals. Recall that z is dominating over V iff x V α < β α(x(β) < z(β)). Indeed the analogue of the Coding Lemma we need in this case is the following. Lemma 18. Let T S Club, let {t α : α < } denote the increasing sequence of leftmost splitting nodes. Let {x ξ : ξ < } be a family of -reals. Then there is T S Club, T T such that for every α <, t α+1 t α > ξ α x ξ(α). This results about dominating -reals is not so surprising, since the same occur for the standard generic Sacks tree in the ω-case.

9 3.2 Other versions of Sacks amoeba May one generalize the Sacks forcing in order to get an amoeba with pure decision and not adding Cohen -reals? In this section we actually give a partially negative answer to this issue, by showing that even finer versions of amoeba for Sacks without club splitting have problems in killing all Cohen -sequences. A version of amoeba one immediately thinks of is the following one, in order to maintain < -closure. We work in 2, with measurable. For U normal measure on, we define S U := {T S : x [T ]({α < : x α Split(T )} U)}. To avoid the Cohen -real as defined for the ω-case in Remark 11, one can introduce a slightly finer version of S U as follows: given T S U, let h : 2 < Split(T ) be the natural -isomorphism. It is easy to show that for any T S U there is T T in S U with the property that for all s, t, T, if h(s) < h(t) then s t. We refer to such trees T as well-sorted trees in S U and re-define S U := {T S U : T is well-sorted}. The amoeba AS U can be defined as follows: (p, T ) AS U iff T S U and p is a well-sorted initial subtree of T. A similar argument to Lemma 9 shows that AS U has quasi pure decision and satisfies -axiom A. One of the main differences from our point of view between AS Club and AS U is that the latter satisfies the following partition property, whose proof is rather easy and left to the reader. Fact 19 (Basic partition property). Let T S U and C : Split(T ) {0, 1} be a 2-coloring. Then there exists T T, T S U such that C Split(T ) is constant, i.e., there is i {0, 1} such that t Split(T ), C(t) = i. However, even if this is a first important step to prove pure decision, it is not sufficient in general. In fact, one would also need the following generalization. Partition property. Let {T i : i < δ}, with δ < and T i S U. Let C : i<δ Split(T i) {0, 1} be a 2-coloring. Then there exist T i T i, T i SU such that C i<δ Split(T i ) is constant, i.e., there is k {0, 1} such that (t i : i < δ) i<δ Split(T i ), C((t i : i < δ)) = k. Note that such a partition property holds for appropriate Sacks forcing in the ω- case (and might be considered as a particular case of Halpern-Läuchli theorem). We are going to build a counterexample to such a partition property in our generalized context. We remark that our construction is going to hold for AS even without the requirement of splitting in U. Definition 20. Given T 2 < perfect tree, we say that T is ω-perfect iff there is an -isomorphism h : 2 ω T, i.e., for every s, t 2 ω one has s t h(s) h(t) and s t h(s) h(t) (roughly speaking, T is ω-perfect if it is an isomorphic copy of 2 ω inside 2 < ). Then we define Ω := {T 2 < : T is ω-perfect}. Given T Ω, x T denotes the leftmost ω-branch in T. For every T, T Ω we define T T x T = x T t x T (T t = T t). It is easy to check that is an equivalence relation. Pick a representative for each equivalence class. We now define the following coloring C : Ω {0, 1}. For every representative T we put C(T ) = 0. Given T Ω, pick the corresponding

10 representative T T. Then find S T T such that S = T σ = Tσ, where σ = stem(s). Moreover let τ = stem(t ) and τ = stem(t ). For every t, t T, with t t x T, let { (t, t 0 iff {s Split(T ) : t s t } is even, ) := 1 else Then define C(S) := (σ, τ ) and { 0 iff C(S) = (σ, τ) C(T ) := 1 else Claim 21. There is no T S U homogeneous for C w.r.t. ω-perfect trees, i.e., there is no T S U and i {0, 1} such that T T, T Ω one has C(T ) = i. Indeed, given any T Ω, let σ = stem(s) with S = T T (where T is the representative of T ), and pick τ be the first splitnode of T extending σ 0. Then clearly C(T ) C(T τ ). Corollary 22. AS U adds Cohen -reals. Proof. Given T G generic tree added via AS U, we can define z 2 as follows: Let {t α : α <, α limit ordinal} be an incresing subsequence of the leftmost splitnodes in T G and let T α be the ω-perfect tree generated by Split α+ω (T tα+1 ). Then define z(α) = C(T α ), for every α <. To show that z is Cohen we argue as follows: given (p, T ) AS U, and t 2 < arbitrary, let z be the part of z and {t α : α λ} the part of leftmost splitnodes decided by (p, T ) (w.l.o.g. we can assume λ limit ordinal). Pick t λ and let p 0 be the ω-perfect tree generated by Split λ+ω (T tλ+1 ). By definition of C, we can always find q 0 p 0, q 0 Ω, such that C(q 0 ) = t(0). Then replace p 0 by q 0 in T, i.e., put T 1 := {t T : s Term(p 0 )(t s) s 0 Term(q 0 )(t s)}. Then proceed by induction on ξ < t : let p ξ be the ω-perfect tree generated by Split λ+ξ+ω (T ξ t λ+ξ+1 ) and pick an ω-perfect tree q ξ p ξ such that C(q ξ ) = t(ξ) and put T ξ+1 := {t T ξ : s Term(p ξ )(t s) s Term(q ξ )(t s)}. Finally let T := ξ< t T ξ and p be the tree generated by p ξ< t q ξ. By construction, T end-extends p and (p, T ) z z t, and this shows that z is -Cohen. Question 2. Can we prove an analogue of Proposition 12? In other words: can we prove that if N V is a ZFC-model containing an absolute S U -generic tree over V, then there is c 2 N Cohen over V? 4 -Miller and -Silver trees The situation for -Miller and -Silver trees is rather similar to that of -Sacks forcing. Definition 23. A tree T < is club -Miller (T M Club ) iff

11 for every s T there is t s, t Split(T ) and {α : t α T } is club; for every x [T ], {α : x α Split(T )} is club. A tree T 2 < is club -Silver (T V Club ) iff T is perfect and for every s, t T such that s = t one has s i t i, for i {0, 1}; {α < : t T (t Split(T ))} is club. When is measurable, we can similarly define M U and V U by replacing being club with being in normal measure U. Pure decision for -Miller forcing has been studied in detail by Brendle and Montoya. They indeed proved that M Club does not have pure decision and adds Cohen -reals, while M U satisfies the -Laver property (and so it does not add Cohen -reals). Remark 24. The situation occurring for -Silver trees is essentially the same as for -Sacks trees, when is inaccessible; the only innocuous difference when defining the corresponding amoeba AV is that one has to maintain the uniformity of the frozen part as well, and the same care has to be taken when doing the various fusion arguments. Apart from that, the reader can easily realize that all of the definitions and proofs in the previous section work for the -Silver case as well. So we only focus on the -Miller case, which requires some slight modifications, though it is rather similar. When defining the corresponding amoeba we have to require the frozen part to have size <. This will be crucial to have quasi pure decision, and therefore not to collapse +. Definition 25. We say that (p, T ) AM Club T M Club, p T and p < ; iff the following hold: if γ is limit and {t α : α < γ} is a -increasing sequence of splitting nodes in p, then α<γ t α Split(p); if γ is limit and {α j : j < γ} is a set of elements such that t α j p, then t α p, where α := j γ α j. Proposition 26. Let G be AM Club -generic over V and T G := {p : T (p, T ) G}. Then, for every < -closed forcing extension N V [G], N = T G M Club x [T G ](x is M Club -generic over V ). Proof. Similar to the one of amoeba for Sacks. For checking that the set H x be a filter, for T, T H x, we argue by contradiction as follows: if T T / M Club then D := {S : [S] [T T ] = } is dense. Hence we should have H x D, i.e., x [S] for some S D, but also x [T T ]. As for Sacks forcing, we focus on the case inaccessible. Lemma 27. AM Club (p, T ) AM Club has quasi pure decision, i.e., given D AM Club there is T M Club such that T T, (p, T ) AM Club (q, S) (p, T )((q, S) D (q, T q) D). and and

12 Given T M Club let {t T σ : σ < } be the natural enumeration of all splitting nodes of T with the property that for every σ, σ <, if σ lex σ then t σ lex t σ. Given T, T M Club we define T α T iff T T and for every σ α α, t T σ = t T σ. Proof. The proof is analogous to the proof of quasi pure decision for AS Club. The only difference is that in step α + 1, instead of considering all terminal subtrees of T α [α + 1], we consider all terminal subtrees of the tree generated by the set {t Tα σ : σ (α + 1) (α+1) }. Corollary 28. AM Club has quasi pure decision. We remark that the analogous results hold for AM U as well. AM Club adds Cohen -reals, since even M Club itself adds Cohen -reals. On the contrary M U does not add Cohen -reals, but the reader can easily realize that we can consider a construction as in the case of AS U in order to show that AM U adds Cohen -reals. 5 -Mathias and -Laver trees -Mathias forcing. The -Mathias forcing R for uncountable is defined as the poset of pairs (s, A), where s of size < and A of size such that sup(s) < min(a), ordered by (s, A) (t, B) t s A B t \ s A. It is easy to see that R can be viewed as a tree-forcing, i.e., a forcing whose conditions are < -closed homogeneuos perfect trees in 2 < with splitting levels A and stem s. As for the other tree-forcings, it is convenient to assume some further assumptions, in order to obtain a < -closed forcing, having some kind of fusion (such as A Club, or A U, for some normal measure U). We remark that -Mathias satisfies quasi pure decision and so -axiom A as well, for inaccessible. The proof works exactly as in the ω-case, so we can omit it. We just remark that the use of inaccessible is important in the proof; in fact we have to recursively run through all < -size subsets of a given set of splitting levels of order type α, for every α <, and we need this procedure to end in < -many steps, for each α. The situation for successor in not known and it is listed as an open question in the last section. Remark 29. Note that for R Club facts: we have the following three straightforward 1. R Club adds Cohen -reals. Let z be the canonical R Club -generic branch and then define c 2 by: c(α) = 0 iff z(α + 1) S. One can easily check that c is -Cohen. 2. R Club does not have pure decision. In fact, let T R Club and α successor ordinal such that T does not decide ż(α). Consider the formula ϕ = ż(α) S; then ϕ cannot be purely decided by T. 3. Let f : 2 < 2 < be a map defined by: f(t(α)) = 0 iff t(α + 1) S, for t 2 <. Then let f : 2 2 be the extension induced by f. Then f is obviously continuous. Moreover, if X 2 is closed nowhere dense, then f 1 [X] is R Club -null; indeed, for every T R Club, let s = f(stem(t )), and pick s s such that [s ] X =. Note that we can pick t stem(t ) so that f(t ) = s. Then, by putting T = T t, we get [T ] f 1 [X] =.

13 Now we want to show that any version of -Mathias forcing adds Cohen -reals, and in particular has no pure decision. For every a, b [] ω, we define the following equivalence relation: a b a b < ω. We also choose a representative for any equivalence class. We then define a coloring C : [] ω {0, 1} as follows: for b [] ω, let a be the representative of [b]. Then put: { 0 iff a b is even C(b) := 1 else. Let x be the Mathias generic and {α j : j < } enumerate the limit ordinals <. i x (ξ) denotes the ξth elements of x. Define, for j <, { 0 iff C({i x (ξ) x : α j < ξ < α j+1 )})=0 z(j) := 1 else (Note that α j+1 = α j + ω and so C is well defined, since the set {i x (ξ) x : α j < ξ < α j+1 )} [] ω.) We claim z is Cohen. Fix (s, A) R and let z 0 be the < -initial segment of z already decided by (s, A). Let t 2 <. We are going to find A A and s s such that (s, A ) R and (s, A ) z 0 t z. This will imply z be Cohen. W.l.o.g. assume sup s is limit and let λ be such that α λ = sup s. Then let b j := {i A (ξ) A : α λ+j < ξ < α λ+j+1 }, a j be the corresponding representative, and ξ j := min(b j a j ). We then recursively define b j b j, for j < t, as follows: { b b j if C(b j ) = t(j) j := b j \ {ξ j } if C(b j ) t(j) Let Γ := {ξ j : b j b j} and A := A \ Γ. Moreover, let s = s σ, where σ 2 < is the sequence corresponding to the set j< t b j. Hence, for every j < t, (s, A ) z(λ + j) = t(j), since (s, A ) z(λ + j) = C(b j ) = t(j). Note that this construction provides a counterexample to pure decision as well. Indeed, given (s, A) R, pick α λ limit ordinal > s, and put b := {i x (ξ) x : α λ < ξ < α λ+1 }, where x is the Mathias generic. Then the formula ϕ = C(b) = 0 cannot be purely decided by (s, A). Proposition 30. Let Γ be a topologically reasonable family. Then Γ(R ) Γ(Baire). Proof. Let {α j : j < } enumerate all limit ordinals <. Define h : 2 2 as follows: for x 2, let x := {ξ : x(ξ) = 1} and put h(x)(j) := C({i x (ξ) x : α j < ξ < α j+1 }). We denote by h : 2 < 2 < the function induced by h. It is easy to check that h is continuous, surjective and open. Moreover, for every T R one has h[[t ]] = [ h(stem(t ))]. As an immediate consequence, for every X 2, if h 1 [X] is R -open dense, then X is open dense. Fix X Γ and let Y := h 1 [X]. We want to show that X has the Baire property. Note that Y Γ too, and so it is R -measurable. This provides us with two possible cases.

14 Case 1: for every T R, it holds [T ] Y I R. Hence, for every s 2 < one has [s] X is -meager, which means that X is -meager as well (here -meager sets are those obtained as a -size union of nowhere dense sets; following the notation of definition 1 we could have denoted them by C -meager). Case 2: there is T R such that [T ] \ Y I R. We claim that X is comeager in h(stem(t )). Put t := h(stem(t )). We aim at building a sequence of open dense sets {U i : i < } such that i< U i [t] X is dense in [t]. For σ < define T σ R such that: T := T ; j< [T σ j] [T σ ] is R -open dense, for every j, stem(t σ j) > stem(t σ ), and i< U i can be written as α< σ =α [T σ j]. This can be done as follows: for every x Y [T σ ], pick T x T σ such that: x [T x ] stem(t x ) > stem(t σ ). Let Σ := {t 2 < : x Y [T σ ](t = stem(t x ))}. Then let {T σ j : j < } enumerate all trees of the form (T σ ) t, with t Σ. By construction, j< [T σ j] [T σ ] and it is R -open dense in [t]. Note this can be done for every σ <, as [T σ ] \ Y I R. Moreover the T σ j s can be chosen in order to satisfy: for every z there is (unique) x such that i< [T z i] = {x}. Now let t σ := h(stem(t σ )), for all σ <. Find a σ such that: for i, j a σ, [t σ i] [t σ j] = j a σ [t σ j] is dense in t σ. Note this can be done by the choices of T σ j s. Then define by recursion: A 0 = { }; A i+1 = σ A i {σ j : j a σ }; finally put U i := σ A i [t σ ]. Then clearly U := i< U i is dense in [t] := stem(t ). Finally, U X; indeed given y U, the construction of the t σ s provides us with a unique η such that y i< [t η i]. Also the construction of the T σ s gives a unique z i< [T η i], and h(z) = y. But z Y := h 1 [X], and so y X. By picking h 1 [Club] we then obtain the following straightforward consequence. Corollary 31. There is a Σ 1 1 set that is not R -measurable. This is a rather surprising result; indeed, to our knowledge, it is the first example where a tree-measurability fails at Σ 1 1 for trees without fat splitting (e.g., club). About absolute R -generic trees and the amoebas for R, it is well-known that R is an amoeba of itself in the ω-case, and the same situation occurs at uncountable. -Laver forcing. such that: First we consider L Club, which consists of trees T < t stem(t )(t Split(T )); t Split(T )(succ(t) is club); For L Club we have an analogue of Remark 29. Like for the -Mathias forcing, we can consider version without club splitting. For inaccessible, a standard proof shows that -Laver satisfies quasi pure decision and -axiom A. We want to show, for = ω 1, that we can build a Cohen ω 1 -real, and implicitely a sentence

15 that cannot be purely decided. So let L ω1 denote any version of Laver forcing at ω 1 with possibly any extra requirement on the splitting nodes in order to have < ω 1 -closure and fusion. Aiming at that, we first consider the following version of Laver forcing L(ω 1, ω) in ω1 ω. We say T L(ω 1, ω) iff T ω 1 <ω for every t stem(t ), succ(t, T ) = ω. Note that a diagonalization against countable subtrees of ω 1 provides us with a Bernstein-type set X, i.e., X ω1 ω such that for every T L(ω 1, ω) one has X [T ] and X \ [T ]. So we can build the following Cohen ω 1 -real. Let z ω ω1 1 be L ω1 -generic. Let {α j : j < } enumerate all limit ordinals < ω 1 and let Z(j) := z(ξ) : α j ξ < α j+1. Note we can view Z(j) as an element of ω1 ω. Define x 2 ω1 as: x(j) = 1 Z(j) X. We claim x is Cohen. Indeed, given T L ω1, let x T be the initial segment of x already decided by T, and fix t 2 <ω1 arbitrarily. We have to find T T such that T x T t x. We recursively build {σ j : j < t } sequence of elements of ω1 ω as follows: Step 0. σ 0 T such that σ 0 X iff t(0) = 1; Step j. σ j T i<jσ i such that σ j X iff t(j) = 1 (where i<j σ i simply consists of the concatenation of the σ i s, for i < j); Finally put σ := lim j< t σ j and T := T σ. By construction, for every j < t, T Z(j) X iff t(j) = 1, and so T x(j) = t(j), as desired. Like for -Mathias forcing, this idea provides us with a counterexample to pure decision too. Indeed, given T L ω1, pick j ω 1 successor ordinal large enough so that T does not decide x(j). Then the formula ϕ = x(j) = 1 cannot be purely decided by T. Hence, for = ω 1, an analogue of proposition 30 ans 31 holds for Laver measurability as well. Proposition 32. Γ(L ω1 ) implies Γ(Baire), for Γ topologically reasonable family. As a corollary, there is a Σ 1 1 set which is not L ω1 -measurable. It remains open what about the case > ω 1. The issue of finding a Bernsteintype set for this forcing then reduces to the following question: is there a maximal antichain such that every two elements are pairwise disjoint? If the answer were positive, the we could argue as follows. Let {p i : i < } be such an antichain, and for every i < build X i ω such that for every p L(, ω), p p i, [p ] X i and [p ] \ X i. This can be done by using a standard construction as for the standard Laver forcing in ω ω. Then put X := i< X i. Then clearly, given any q L(, ω) there is i < such that q and p i are compatible, and so [q] X i and since the p i s are pairwise disjoint also [q] \ X. Amoeba for Laver AL is in general a different forcing from L and a proof like the one given for Sacks and Miller forcing show that it satisfies -axiom A and that the club version necessarily adds both Cohen and dominating -reals. We conclude this section with some interesting observations about L Club and R Club. The Hechler forcing D can be easily introduced in the context of : we say (s, f) D iff f, s <, with s f. The ordering is: (s, f ) (s, f) iff i <, f(i) f (i) and s s. It is easy to see that D adds both dominating and Cohen -reals. In the ω-case, Mathias and Laver forcing were introduced in order to add dominating reals without adding Cohen reals. What follows shows that from this point of view, L Club generalizations; indeed, they are equivalent to D. and R Club are not very suitable

16 Lemma 33. D is equivalent to L Club and R Club. Proof. We show that for L Club and the reader can easily generalize this argument for R Club as well. Fix {S i : i < } family of disjoint stationary subsets of, with the following property: for every j < limit, for every {α i : i < j} such that α i S i, one has lim i<j α i S j. For every T L, we can easily find T T with the following properties: let x [T ] be the leftmost -branch, {α j : j < } be such that x( stem(t ) + j) = α j and η j such that α j S ηj. Then, for every t stem(t ), with t = stem(t ) + j, one has that if succ(t) := {β i : i < } with i < l < (β i < β l ), then i <, β i S ηj+i. For doing that, simply prune T as follows: For t stem(t ), with t = stem(t ) + j, and succ(t) := {β i : i < }, we proceed by recursion on ξ < as follows: β i0 := β 0 ; step ξ + 1. pick β iξ+1 such that β iξ+1 is the least element in {β i : i > β iξ } S ηj+ξ+1; limit step λ. Let β iλ := sup ξ<λ β iξ. Note that by the choice of S i s, one has β iλ S ηj + λ. Call Θ t the set {β iξ : ξ < } obtained by this recursive construction, for a given t. Then pruning T as follows: first put T 1 := β Θ t0 T t β, 0 where t 0 = stem(t ). Recursively; successor case: for every α <, put T α+1 := t = t 0 +α β Θ t T α t β ; limit case: T λ := α<λ T α. Finally T := α< T α. Then let L be the dense subposet consisting of these special -Laver trees. We now define the following map h : L D ; for every T L, put x [T ] be the leftmost -branch and {α j : j < }, {η j : j < } be such that x( stem(t ) +j) = α j and α j S ηj. Then define f such that f(j) := { stem(t )(j) if j < stem(t ) η j if j stem(t ) By construction, one can easily check that h is a dense embedding. 6 Concluding remarks In [8] it was proven that if one drops the club splitting on the trees then it is possible to obtain a tree-measurability which can be forced for all projective sets (e.g., for Silver and Miller forcing). On the other hand, in this paper we have seen that for -Mathias and ω 1 -Laver measurability this cannot be done, by Proposition 30 and 32. In the following table we summarize the current situation. In the column Σ 1 1-counterexample we list all regularities for which it is provable in ZFC the existence of such a non-measurable set; in the column Forceable we list all cases for which Σ 1 1 or even projective measurability is forceable; the last column obviously exhibits the open questions.

17 Forcing notion Σ 1 1-counterexample Forceable Unknown Sacks Silver Miller Laver Mathias S Club ([4]) S ([8]) S U V Club ([8]) V ([8]) V U M Club ([4]) M ([8]) M U L Club ([4]), L ω1, L U ω 1 (Cor.31) L, L U, > ω 1 R Club Cohen C ([5]) ([4]), R, R U (Prop.32) (Recall C -measurability is simply the Baire property.) About generic trees we recall the main questions that remain open. Question 1. Let M be the ideal of -meager sets, I S Club is the ideal of S Club meager sets, and T denotes Tukey embedding. Is M T I S Club? - Question 2. Can we prove an analogue of Proposition 12 for AS U and the other tree-forcings? In other words: can we prove that if N V is a ZFCmodel containing an absolute S U -generic tree over V, then there is c 2 N Cohen over V? Finally we remark that in all proof about -axiom A, we use that be inaccessible. So the following is still open. Question 3. Can one prove -axiom A for the amoebas and tree-forcings analysed in this paper, for regular successor? References [1] Jörg Brendle, How small can the set of generic be?, Logic Colloquium 98, pp , Lect. Notes Logic, 13, Assoc. Symb. Logic, Urbana, IL, (2000). [2] Jörg Brendle, Benedikt löwe, Solovay-type characterizations for forcingalgebra, Journal of Symbolic Logic, Vol. 64, No. 3, pp (1999). [3] Sy D. Friedman, Lyubomyr Zdomskyy, Measurable cardinals and the cofinality of the symmetric group, Fundamenta Mathematicae 207, pp , (2010). [4] Sy D. Friedman, Yurii Khomskii, Vadim Kulikov, Regularity properties on the generalized reals, Annals of Pure and Applied Logic 167, pp , (2016). [5] Aapo Halko, Saharon Shelah, On strong measure zero set in 2, Fundamenta Mathematicae, Volume 170, n. 3, pp , (2001). [6] Akiro Kanamori, Perfect set forcing for uncountable cardinals, Annals of Mathematical Logic, n. 19, pp , (1980). [7] Giorgio Laguzzi, Some considerations on amoeba forcing notions, Archive for Mathematical Logic, Volume 53, Issue 5-6, pp , (2014).

18 [8] Giorgio Laguzzi, Generalized Silver and Miller measurability, Mathematical Logic Quarterly, Volume 61, Issue 1-2, pp (2015). [9] Yurii Khomskii, Giorgio Laguzzi, Benedikt Löwe, Ilya Sharankou, Questions on generalized Baire spaces, Mathematical Logic Quarterly, Volume 62, Issue 4-5 (2016). [10] A. Louveau, S. Shelah, B. Velickovic, Borel partitions of infinite subtrees of a perfect tree, Ann. Pure App. Logic 63, pp (1993). [11] Otmar Spinas, Generic trees, Journal of Symbolic Logic, Vol. 60, No. 3, pp (1995). [12] Otmar Spinas, Silver trees and Cohen reals, Israel Journal of Mathematics 211, , (2016).