Generic Coding with Help and Amalgamation Failure

Abstract

We show that if M is a countable transitive model of ZF and if a, b are reals not in M , then there is a G generic over M such that b ∈ L[a, G]. We then present several applications such as the following: if J is any countable transitive model of ZFC and M ⊆ J is another countable transitive model of ZFC of the same ordinal height α, then there is a forcing extension N of J such that M ∪ N is not included in any transitive model of ZFC of height α. Also, assuming 0 # exists, letting S be the set of reals generic over L, although S is disjoint from the Turing cone above 0 # , we have that for any non-constructible real a, {a ⊕ s : s ∈ S} is cofinal in the Turing degrees. 2 SY-DAVID FRIEDMAN AND DAN HATHAWAY 1) Fix P Corollary 5.5 of [4]). 3) Miha Habic (unpublished) and the first author (see [3] just after "nodes of compatibility") have independently shown that every real unbounded over M is (C, M)-helpful. 4) The first author has shown that every real Sacks generic over M is (C, M)-helpful (unpublished). 5) The central result of this paper (Theorem 1.3) is that every real not in M is (H, M)-helpful, where H is "Tree-Hechler" forcing. 6) The question of whether every real not in M is (C, M)-helpful remains open.Definition 1.2. The forcing H, called Tree-Hechler forcing, consists of all trees T ⊆ <ω ω such that for all t ⊒ Stem(T ) in T ,The ordering is by inclusion.That is, a condition in Tree-Hechler forcing has cofinite splitting beyond its stem.Consider a tree T ⊆ <ω ω and a node t ∈ T . By a successor of t we always mean some t ⌢ z ∈ T for z ∈ ω. By T ↾ t we mean the set of all s ∈ T that are comparable to t. Stem(T ) is the longest element of T that is comparable with all other elements of T .Let M be a transitive model of ZF and suppose G isThat is, g : ω → ω is the union of all the stems of the trees T ∈ G. The set G can be recovered from g (and M). We will treat g : ω → ω as the object which is encoding information.The poset H is σ-centered, because any two conditions with the same stem are comparable. Thus, H is c.c.c. Combining this with the fact that |H| = 2 ω , we have the following: there are only 2 ω maximal antichains in H. So, if M is a transitive model of ZFC and (2 ω ) M is countable, then there is an H M -generic over M.The forcing H is discussed in [9], along with other versions of Hechler forcing, where it is called D tree . A key ingredient for us is that H admits a "rank analysis" of its dense sets (see Definition 5.9 and Lemma 5.10). In [2], Jörg Brendle and Benedikt Löwe carry out a rank analysis of H. The original rank analysis of a Hechler-like forcing