Disjoint Borel functions

Hathaway, Dan

doi:10.1016/j.apal.2017.02.004

Cited by 5 publications

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“…As described in the introduction, here is the original kind of result for which the Main Lemma was created. A proof can be found in [6].…”

Section: 5mentioning

confidence: 99%

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Generic Coding with Help and Amalgamation Failure

Friedman¹,

Hathaway²

2018

Preprint

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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is cofinal in the Turing degrees. 2 SY-DAVID FRIEDMAN AND DAN HATHAWAY 1) Fix P Corollary 5.5 of [4]). 3) Miha Habic (unpublished) and the first author (see [3] just after "nodes of compatibility") have independently shown that every real unbounded over M is (C, M)-helpful. 4) The first author has shown that every real Sacks generic over M is (C, M)-helpful (unpublished). 5) The central result of this paper (Theorem 1.3) is that every real not in M is (H, M)-helpful, where H is "Tree-Hechler" forcing. 6) The question of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

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“…As described in the introduction, here is the original kind of result for which the Main Lemma was created. A proof can be found in [6].…”

Section: 5mentioning

confidence: 99%

“…The goal of this game was to prove results like Proposition 5.12. In [5] and [6] such results are proved, and the current version of the Main Lemma appears in [5]. We want to emphasize that the Main Lemma may have applications other than Theorem 1.3 and Proposition 5.12.…”

Section: Introductionmentioning

confidence: 95%

Generic Coding with Help and Amalgamation Failure

Friedman¹,

Hathaway²

2018

Preprint

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Applying generic coding with help to uniformizations

Hathaway

2023

Annals of Pure and Applied Logic

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Generic Coding With Help and Amalgamation Failure

Friedman¹,

Hathaway²

2020

J. symb. log.

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We show that if M is a countable transitive model of $\text {ZF}$ and if $a,b$ are reals not in M, then there is a G generic over M such that $b \in L[a,G]$ . We then present several applications such as the following: if J is any countable transitive model of $\text {ZFC}$ and $M \not \subseteq J$ is another countable transitive model of $\text {ZFC}$ of the same ordinal height $\alpha $ , then there is a forcing extension N of J such that $M \cup N$ is not included in any transitive model of $\text {ZFC}$ of height $\alpha $ . 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, $\{ a \oplus s : s \in S \}$ is cofinal in the Turing degrees.

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Disjoint Borel functions

Abstract: Abstract. For each a ∈ ω ω, we define a Baire class one function f a : ω ω → ω ω which encodes a in a certain sense. We show that for each Borel g :1 (c) where c is any code for g. We generalize this theorem for g in larger pointclasses Γ., then a ∈ M 1+n (c).

Cited by 5 publications

References 10 publications

Generic Coding with Help and Amalgamation Failure

Generic Coding with Help and Amalgamation Failure

Applying generic coding with help to uniformizations

Generic Coding With Help and Amalgamation Failure

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