Silver trees and Cohen reals

Silver trees and Cohen reals Otmar Spinas Abstract We prove that the meager ideal M is Tukey reducible to the Mycielski ideal J(Si) which is the ideal associated with Silver forcing Si. This implies add (J(Si)) add (M) and answers a question of Laguzzi and others. 1 Introduction A tree p 2 <ω is called Silver tree if there exist a coinfinite set a ω and a function x : a 2 such that p = {ν 2 <ω : n ν a ν(n) = x(n)}. Then we call a the domain and ω \ a the codomain of p. We call x = x p the Silver function associated with p and p the Silver tree associated with x. By Si we denote the set of all Silver trees ordered by inclusion. Then Si is the well-known Silver forcing. The Silver ideal J(Si) consists of all X 2 ω such that p Si q Si (q p [q] X = ), where [q] = {x 2 ω : n < ω x n p}. It is sometimes called Mycielski ideal and has been studied by many authors (e.g. see [M], [ShSt]). In the same way, to each of the classical tree forcings like Sacks forcing Sa, Laver, Miller forcing etc. a forcing ideal is associated. These as well have been studied extensively (e.g. see [JMSh]). All these can easily be seen to be σ-ideals. Hence their additivity coefficient, i.e. the minimal cardinality of a subfamily whose union does not belong to the ideal, is always uncountable. Clearly it is at most 2 ℵ 0. The natural forcing that can be used to increase add (J(P )) for some tree forcing P, i.e. to construct a model for ℵ 1 < add (J(P )), is an amoeba forcing for P, i.e. a forcing adding a generic tree in P such that each of its branches is P -generic over the ground model. Experience shows that, unless special care is taken, amoeba forcings tend to add Cohen reals, hence to increase cov (M), the covering coefficient of the meager ideal The author is partially supported by DFG grant SP683/3-1. Keywords: Silver tree, Cohen real, Silver ideal, additivity, meager ideal, Tukey reduction. 1

M. However, in the case of Sacks forcing (see [LShV]) as well as Laver, Miller forcing respectively (see [Sp]) amoebas not adding Cohen reals have been constructed, hence for these three a model where the additivity of the respective forcing ideal is 2 ℵ 0 = ℵ 2 and cov (M) = ℵ 1 exists. Laguzzi [L] and others have noticed that for Si a similar construction is not known and they asked whether there is an amoeba for Si not adding Cohen reals. Here we prove add (J(Si)) cov (M) in ZFC. This implies that every reasonable amoeba forcing for Si, i.e. one that can be iterated to produce a model for ℵ 1 < add (J(Si)), adds a Cohen real. We actually prove something stronger than this. We show that M is Tukey reducible to J(Si). Recall that given partially ordered sets P, Q a map ϕ : P Q is called a Tukey function iff for every unbounded X P the pointwise image ϕ[x] is unbounded in Q. If such ϕ exists P is called Tukey reducible to Q and we write P T Q. We shall construct a Tukey function ϕ : M J(Si), where M, J(Si) are partially ordered by inclusion. Note that this easily implies add (J(Si)) add (M). By Cichon s diagram (see [BJ]) add (M) = min{b, cov (M)} holds in ZFC, where b is the bounding number. That add (J(Si)) b holds is a result of [SpW]. Note that sets of the form X(A) = 2 ω \ {[p] : p A}, where A is a maximal antichain of Si, form a basis for the ideal J(Si). Therefore, in [SpW] a lot of work has been done to improve our understanding of such antichains. An important result that is essential for the main theorem of this paper as well is the following: Theorem 1 [SpW, Theorem 2.2] Suppose that A Si is a maximal antichain such that for every p A dom(p) is infinite. Then A has size 2 ℵ 0. To be complete we sketch a proof of this result. Let A Si be an antichain such that A < 2 ℵ 0 and dom(p) is infinite for every p A. For p A let x : dom(p) 2 be the associated Silver function. Let e j : j < ω list all finite subsets of ω and let A j be the set of all p A such that e j = {n dom(p) : x p (n) = 1}. Note that, letting B j = {codom(p) : p A j }, B j is an almost disjoint family on ω. We can easily find a [ω] ω such that for every j < ω we have that either a b for some b B j or else a b < ℵ 0 for every b B j. By our assumption A < 2 ℵ 0 we can choose some infinite a 0 a such that a \ a 0 = ℵ 0 and a 0 {n < ω : x p (n) = 1} is infinite for every p A \ A j such that a {n < ω : x p (n) = 1} is infinite. Moreover let a \ a 0 = a 1 a 2 be a partition into two infinite pieces. If we define a Silver function y such that codom(y) = a 2, y is constantly 0 on a 0 and 2

constantly 1 on a 1, then, whatever y ω \ a will be, we have guaranteed that p y, the Silver tree associated with y, is incompatible with all p A except for those countably many p A j with a codom(p) and those p A\ A j with a {n < ω : x p (n) = 1} < ℵ 0. The definition of y on ω \ a will take care of these. Let p(k) : k < ω list all p A j with a codom(p). We define a perfect tree T on ω\a such that every n ω\a appears at most once in T and for every k we have T (k) := {ν(k) : ν T } dom(p(k)) \ (e j a) where j is such that p(k) A j. By A < 2 ℵ 0 we choose a branch z of T that is not definable from {A, T } A. We let a 4 = ran(z) and a 5 = (ω \ a) \ a 4, and we define y constantly 1 on a 4 and constantly 0 on a 5. Then for no p A \ A j can it be that {n dom(p) : x p (n) = 1} z, and hence we conclude that p y and p are incompatible for every p A. This finishes the proof of Theorem 1. By an idea of [JMSh] one can easily build maximal antichains A Si such that given p Si with [p] {[q] : q A} there exists B A such that B < 2 ℵ 0 and [p] {[q] : q B}. As then B induces a maximal antichain below p, we know by Theorem 1 that some member q of B has codom(q) codom(p). This observation has been crucial for the result add (J(Si)) b in [SpW]. It will again be crucial for the construction of a Tukey function ϕ : M J(Si) below. Moreover we apply a new idea how to encode a Cohen real using the Hamming weights of the splitnodes of a Silver tree. Notation We shall use the following notation to deal with Silver trees. Given p Si and σ p we let p σ = {τ p : τ σ σ τ}. Obviously p σ Si. We call σ a splitnode of p if σ 0 and σ 1 both belong to p. By split (p) we denote the set of all splitnodes of p. By Lev p (n) we denote the n-th splitlevel of p, i.e. the set of all σ split (p) such that if σ 0 σ 1... σ k = σ is the increasing enumeration of all τ split (p) with τ σ, then k = n. The unique element of Lev p (0) is called the stem of p. Given σ 2 n, ρ = σ k for some k n and ξ 2 k, by σ ξ we denote the unique τ ρ 2n with τ k = ξ and τ(i) = σ(i) for every i [k, n). Note that given ρ Lev p (n) and σ Lev p (m) such that n < m and ρ σ, σ uniquely determines Lev q (n + 1) for any q Si with q p, Lev q (n) = Lev p (n) σ Lev q (n + 1), namely Lev q (n + 1) = {σ ξ i : ξ Lev ρ σ( ρ ) p(n) i < 2}. By C we denote Cohen forcing (2 <ω, ). Given D C we let D = {[σ] : σ D} where [σ] = {x 2 ω : σ x}. 3

2 Coding by Hamming weights Definition 2 For n, k < ω we define d(n, k) 2 by letting d(n, k) = 0 iff the unique j < ω such that n [j 2 k, (j +1) 2 k ) is even. Let e(n) be the minimal k < ω with 2 k > n. Let c(n) = d(n, k) : k < ω and c (n) = c(n) e(n). Remark 1 Obviously c(n)(k) = 0 for every k e(n) and e(n) < n for every n < ω. Definition 3 For σ 2 <ω the Hamming weight of σ is defined as HW (σ) = {i < σ : σ(i) = 1}. Remark 2 Note that for every n < ω, σ 2 n and σ i : i < k such that σ 0 σ 1... σ k 1 σ, letting ξ i = c (HW (σ i )), we have that i<k HW (σ i ) n and hence, by the previous remark, ξ 0 ξ 1... ξ k 1 has length < n. Coding Lemma 2 Given D C open dense, I n : n < ω a sequence of intervalls in ω with lim I n = and l, m < ω, for almost every n there is n k I n such that (i) k l I n and (ii) for every ν 2 m, for every i [k l, k] ν c (i) D. Proof: Let t be the number of digits of the binary representation of l. We can find σ 2 <ω such that for every ν 2 m and µ 2 t we have ν µ σ D. Let s = σ. Let n be large enough such that at least three multiples of 2 t+s 1 belong to I n. Then clearly there exists k I n of the form j<s λ j 2 t+j, where λ j ω, such that c (k ) = 0 t σ, where 0 t is the string of t many 0 s, and there exists a multiple of 2 t+s 1 in I n bigger that k. We let k = k + l. Note that k I n and for every i [k l, k] there exists µ 2 t such that c (i) = µ σ. By the choice of σ, k is as desired. Thinning-Out-Lemma 3 Given D n : n < ω a family of open dense sets D n C and p Si, there exists q Si such that q p and for every n < ω and σ Lev q (n + 1), if m = τ for any τ Lev q (n) and k = HW (σ), then ν c (k) D j for every ν 2 m and j n. 4

Remark 3 Note that the set of q Si with the property described in the Thinning-Out-Lemma is open, hence open dense. Proof: It is evident that for every p Si and n < ω we have that I p (n) := {HW (σ) : σ Lev p (n)} is an intervall in ω of lenth n + 1. Moreover, given f ω ω we can find q p such that f(n) < min(i q (n)) and max(i q (n)) < min(i q (n + 1)) for every n < ω. Simply delete enough spitlevels of p by deleting there the nodes that go left. Suppose now that we have constructed q p up to Lev q (n) as desired. Let τ be the lexicographically largest element of Lev q (n) and m = τ. We apply the Coding Lemma with D = D j, I pτ 1 (j) : j < ω, l = n + 1 and m and obtain j and σ Lev pτ 1 (j) such that, letting k = HW (σ), we have k l I pτ 1 (j) and ν c (i) D for every ν 2 m and i [k l, k]. Then σ determines Lev q (n + 1) such that it is its lexicographically largest element and {HW (ξ) : ξ Lev q (n + 1)} = [k n 1, k]. Definition 4 Let p Si, let σ n the leftmost element of Lev p (n), m n = σ n, w n = HW (σ n ) and ξ n = c (w n ). Define G n+1 (p) = {[ν ξ n+1 ] : ν 2 mn }, H n+1 (p) = {G j (p) : j n + 1} and H(p) = H n+1 (p). Finally let H (p) = {H(p τ ) : τ split (p)}. Remark 4 Every H n+1 (p) 2 ω is dense open, hence H(p) and H (p) are dense G δ. Lemma 4 Let D n : n < ω, p and q be as in the Thinning-Out-Lemma and let r Si, r q. Then H(r) Dn. Proof: Let ρ n be the leftmost element of Lev r (n), l n = ρ n, v n = HW (ρ n ) and ζ n = c (v n ). Then ρ n+1 Lev q (k + 1) for some k n. Let σ Lev q (k) such that σ ρ n+1 and let m = σ. Clearly ρ n σ and hence l n m. By construction of q we have ν ζ n+1 D j for every ν 2 m and hence j k ν ζ n+1 D j for every ν 2 ln. We conclude G n+1 (r) Dj and hence H n+1 (r) Dj and H(r) Dn. 3 The Tukey function Theorem 5 M T J(Si). 5

Proof: We have to define a Tukey function ϕ : M J(Si). Let F 2 ω be any meager set of type F σ. Let E n : n < ω be a family of dense open sets E n 2 ω such that F = 2 ω \ E n. Let D n C be dense open such that Dn = E n. In order to define ϕ(f ) we construct A Si such that (1) A is a maximal antichain of Si; (2) for every q Si, if [q] {[p] : p A} there exists B [A] <2ℵ 0 such that [q] {[p] : p B}; (3) for every q A, for every n < ω and every σ Lev q (n + 1), if m = τ for any τ Lev q (n) and k = HW (σ), then ν c (k) D j for every j n and ν 2 m. The construction is a pretty straightforward application of the Thinning- Out-Lemma. Let q β : β < 2 ℵ 0 enumerate all Silver trees. Suppose that for γ < 2 ℵ 0 we have constructed an increasing sequence B β : β < γ of antichains B β Si with B β = β together with a sequence x β : β < γ of elements of 2 ω. If [q γ ] {[p] : p B β } we fix x γ [q γ ] \ {[p] : p B β }. Otherwise x γ 2 ω is arbitrary. If some p B β is compatible with q γ we let B γ = B β. Otherwise, by the Thinning-Out-Lemma, we first find q q γ as in (3) and then choose q q in Si with x β / [q ] for every β γ. Then we let B γ = B β {q }. Then A = β<2 ℵ 0 B β is as desired. Now we define ϕ(f ) as 2 ω \ {[p] : p A}. We have to show that ϕ is Tukey. let X M be such that ϕ[x] J(Si). We have to show X M. By assumption we find q Si such that [q] ϕ[x] =. We claim that H (q) F = for every F X, which will suffice. Let F X be arbitrary. Let A Si be the maximal antichain we have constructed above to define ϕ(x). Let D n : n < ω as in that construction. We know that [q] {[p] : p A}. By (2) there exists B [A] <2ℵ 0 such that [q] {[p] : p B}. Note that any two incompatible Silver trees have only finitely many branches in common. Therefore no p A \ B is compatible with q. Hence B includes a maximal antichain of Si below q. More precisely, B q := {p q : p B p, q are compatible} is like that. By Theorem 1 there exists r B q that is relatively finite in q, i.e. r = p q for some p B with codom(q) codom(p). Therefore there exists τ split(r) such 6

that q τ p. By Lemma 4 and by (3) we conclude H(q τ ) Dn and hence H (q) Dn which means H (q) F = as desired. References [BJ] T. Bartoszynski, H. Judah, Set theory - On the structure of the real line, AK Peters Wellesley (1999). [JMSh] H. Judah, A. Miller, S. Shelah, Sacks forcing, Laver forcing, and Martin s axiom, Arch. Math. Logic (1992) 31, 145-161. [L] G. Laguzzi, Some considerations on Amoeba forcing notions, Arch. Logic, to appear. [LShV] A. Louveau, S. Shelah, B. Velickovic, Borel partitions of infinite subtrees of a perfect tree, Ann. Pure Appl. Logic (1993) 63, 271-281. [M] J. Mycielski, Some new ideals of sets on the real line, Coll. (1969) 20, 71-76. Math. [ShSt] S. Shelah, J. Steprāns, The covering numbers of Mycielski ideals are all equal, J. Symb. Logic (2001) 66, 707-718. [Sp] O. Spinas, Generic trees, J. Symb. Logic (1995) 60, 2767-2782. [SpW] O. Spinas, M. Wyszkowski, Silver antichains, J. Symb. Logic, to appear. Mathematisches Seminar Christian-Albrechts-Universität zu Kiel Ludewig-Meyn-Str. 4, 24118 Kiel, Germany 7