Forcing With Bushy Trees

Khan, Mushfeq; Miller, Joseph S.

doi:10.1017/bsl.2017.12

Cited by 24 publications

(43 citation statements)

References 13 publications

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“…This technique is excessive for this result (cf. the short proof recently given in [5]), but helps illustrate both the method used in the later part of the paper and the clear line of development from previous forcing results, especially [1], and the more recent iterated forcing techniques as in [7,10,11].We also prove two related positive implications. First, since RWKL and WWKL represent distinct weakenings of WKL, it is natural to ask about combining them, into a "RWWKL".…”

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confidence: 53%

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Separating principles below

Flood

Towsner

2016

Math. Log. Quart.

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In this paper, we study the Ramsey‐type weak Kőnig's Lemma, written sans-serifRWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an ω‐model satisfying one principle T1 but not another T2. The technique often allows one to translate a “one step” construction (building an instance of T2 along with a collection of solutions to each computable instance of T1) into an ω‐model separation (building a computable instance of T2 together with a Turing ideal where T1 holds but this instance has no solution). We illustrate this translation by separating d-DNR from sans-serifDNR (reproving a result of Ambos‐Spies, Kjos‐Hanssen, Lempp, and Slaman), and then apply this technique to separate sans-serifRWKL from sans-serifDNR (which has been shown separately by Bienvenu, Patey, and Shafer).

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supporting

confidence: 53%

mentioning

confidence: 87%

Separating principles below

Flood

Towsner

2016

Math. Log. Quart.

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“…Theorem 6 (Khan and Miller [10,Theorem 4.3]). For each recursive numbering ψ and for each order function h, there is an f ∈ DNR ψ h such that f computes no Kurtz random real.…”

Section: Solutions To Open Problemsmentioning

confidence: 99%

Covering the recursive sets

Kjos-Hanssen

Stephan

Terwijn

2017

Annals of Pure and Applied Logic

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We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales.At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover.

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“…More precisely, perhaps there is a slow enough recursive bound g such that all g-bounded DNR functions compute EBI sets. Khan and Miller have shown [9] that by varying g, one can obtain a proper hierarchy of mass problems of recursively bounded DNR functions. Our main result in this section settles these questions.…”

Section: Slow-growing Dnr Functionsmentioning

confidence: 99%

“…Proofs of the following lemmas can be found in [5] and [9]. Lemma 3.7 (Smallness preservation property).…”

Section: Definitions and Combinatorial Lemmasmentioning

confidence: 99%

Effective Bi-immunity and Randomness

Beros

Khan

Kjos-Hanssen

2016

Computability and Complexity

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Abstract. We study the relationship between randomness and effective biimmunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing DNR functions relative to X that compute no MartinLöf random set. We show that the same holds when Martin-Löf randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set.We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every Martin-Löf random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.

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Forcing With Bushy Trees

Cited by 24 publications

References 13 publications

Separating principles below

Separating principles below

Covering the recursive sets

Effective Bi-immunity and Randomness

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