Defining the Turing Jump

Shore, Richard A.; Slaman, Theodore A.

doi:10.4310/mrl.1999.v6.n6.a10

Cited by 54 publications

(65 citation statements)

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“…The Kleene-Post theorem and Spector's exact pair theorem also had numerous extensions, heading towards definability results in the degrees, as well as combinations to extensions of embeddings, embeddings with jumps etc. Some noteworthy results here include Slaman-Woodin's proof of the definability from parameters of countable relations in the degrees, this leading to the (parameter-free) definability of the jump operator in the partial ordering of the degrees by Shore and Slaman [131] (Also see Slaman [133]). Still open here is the longstanding question of Rogers: are the Turing degrees rigid?…”

Section: The Global Degreesmentioning

confidence: 99%

Computability theory, algorithmic randomness and Turing's anticipation

Downey

2014

Turing's Legacy

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Section: The Global Degreesmentioning

confidence: 99%

Computability theory, algorithmic randomness and Turing's anticipation

Downey

2014

Turing's Legacy

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“…We have the following bounds on the complexity of each of these notions. , and 5 is implicit in Lemmas 2.10 and 2.11 of [7].…”

Section: Extensions To Other Forcing Notionsmentioning

confidence: 99%

“…Posner and Robinson [4] proved that if S ⊆ ω is non-computable, then there exists a G ⊆ ω such that S ⊕ G ≥ T G ′ . Shore and Slaman [7] extended this result to all n ∈ ω, by showing that if S T ∅ (n−1) then there exists a G such that S ⊕ G ≥ T G (n) . Their argument employs KumabeSlaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing in any way.…”

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Limits to joining with generics and randoms

Day

Dzhafarov

2013

Proceedings of the 12th Asian Logic Conference

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Abstract. Posner and Robinson [4] proved that if S ⊆ ω is non-computable, then there exists a G ⊆ ω such that S ⊕ G ≥ T G ′ . Shore and Slaman [7] extended this result to all n ∈ ω, by showing that if S T ∅ (n−1) then there exists a G such that S ⊕ G ≥ T G (n) . Their argument employs KumabeSlaman forcing, and so the set they obtain, unlike that of the Posner-Robinson theorem, is not generic for Cohen forcing in any way. We answer the question of whether this is a necessary complication by showing that for all n ≥ 1, the set G of the Shore-Slaman theorem cannot be chosen to be even weakly 2-generic. Our result applies to several other effective forcing notions commonly used in computability theory, and we also prove that the set G cannot be chosen to be 2-random.

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“…In the past year, Cooper's original definition of the jump in the Turing degrees [9,10] has been shown to be false -the proposed property does not define the jump (Shore and Slaman [53]). Taking an approach quite different from Cooper's, Shore and Slaman [54] then proved that the jump is definable. Their approach uses results of Slaman and Woodin [60] that strongly employ set theoretic and metamathematical arguments.…”

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The Prospects for Mathematical Logic in the Twenty-First Century

Buss

Kechris²,

Pillay³

et al. 2001

Bull. symb. log.

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Abstract. The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.We can only see a short distance ahead, but we can see plenty there that needs to be done.A. Turing, 1950. §1. Introduction. The annual meeting of the Association for Symbolic Logic held in Urbana-Champaign, June 2000, included a panel discussion on "The Prospects for Mathematical Logic in the Twenty-First Century". The panel discussions included independent presentations by the four panel members, followed by approximately one hour of lively discussion with members of the audience.The main themes of the discussions concerned the directions mathematical logic should or could pursue in the future. Some members of the audience strongly felt that logic needs to find more applications to mathematics; however, there was disagreement as to what kinds of applications were most likely to be possible and important. Many people also felt that applications to computer science will be of great importance. On the other hand, quite a few people, while acknowledging the importance of applications of logic, felt that the most important progress in logic comes from internal developments.It seems safe to presume that the future of mathematical logic will include a multitude of directions and a blend of these various elements. Indeed, it speaks well for the strength of the field that there are multiple compelling directions for future progress. It is to be hoped that logic will be driven both by internal developments and by external applications, Supported in part by NSF grant DMS-9803515.

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Defining the Turing Jump

Cited by 54 publications

References 13 publications

Computability theory, algorithmic randomness and Turing's anticipation

Computability theory, algorithmic randomness and Turing's anticipation

Limits to joining with generics and randoms

The Prospects for Mathematical Logic in the Twenty-First Century

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