Martin's Conjecture: A Classification of the Naturally Occurring Turing Degrees

Montalbán, Antonio

doi:10.1090/noti1940

Cited by 8 publications

(4 citation statements)

References 11 publications

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“…set whose Turing degree lies strictly between computable ones and the halting problem; see e.g. [31]. As one way to solve this problem of the lack of natural intermediate c.e.…”

Section: Game Rulesmentioning

confidence: 99%

Untitled

2023

Trans. Amer. Math. Soc. Ser. B

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In this article, we study the lattice of Lawvere-Tierney topologies on Hyland's effective topos. For this purpose, we introduce a new computabilitytheoretic reducibility notion, which is a common extension of the notions of Turing reducibility and generalized Weihrauch reducibility. Based on the work by Lee and van Oosten [Ann. Pure Appl. Logic 164 (2013), pp. 866-883], we utilize this reducibility notion for providing a concrete description of the lattice of the Lawvere-Tierney topologies on the effective topos. As an application, we solve several open problems proposed by Lee and van Oosten. For instance, we show that there exists no minimal Lawvere-Tierney topology which is strictly above the identity topology on the effective topos.

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“…set whose Turing degree lies strictly between computable ones and the halting problem; see e.g. [31]. As one way to solve this problem of the lack of natural intermediate c.e.…”

Section: Game Rulesmentioning

confidence: 99%

Untitled

2023

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

show abstract

“…set whose Turing degree lies strictly between computable ones and the halting problem; see e.g. [31]. As one way to solve this problem of the lack of natural intermediate c.e.…”

Section: Game Rulesmentioning

confidence: 99%

Lawvere-Tierney topologies for computability theorists

Kihara

2023

Trans. Amer. Math. Soc. Ser. B

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In this article, we study the lattice of Lawvere-Tierney topologies on Hyland’s effective topos. For this purpose, we introduce a new computability-theoretic reducibility notion, which is a common extension of the notions of Turing reducibility and generalized Weihrauch reducibility. Based on the work by Lee and van Oosten [Ann. Pure Appl. Logic 164 (2013), pp. 866-883], we utilize this reducibility notion for providing a concrete description of the lattice of the Lawvere-Tierney topologies on the effective topos. As an application, we solve several open problems proposed by Lee and van Oosten. For instance, we show that there exists no minimal Lawvere-Tierney topology which is strictly above the identity topology on the effective topos.

show abstract

“…set whose Turing degree lies strictly between computable ones and the halting problem; see e.g. [27]. As one way to solve this problem of the lack of natural intermediate c.e.…”

Section: Game Rulesmentioning

confidence: 99%

Lawvere-Tierney topologies for computability theorists

Kihara

2021

Preprint

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In this article, we introduce certain kinds of computable reduction games with imperfect information. One can view such a game as an extension of the notion of Turing reduction, and generalized Weihrauch reduction as well. Based on the work by Lee and van Oosten, we utilize these games for providing a concrete description of the lattice of the Lawvere-Tierney topologies on the effective topos (equivalently, the subtoposes of the effective topos preordered by geometric inclusion). As an application, for instance, we show that there exists no minimal Lawvere-Tierney topology which is strictly above the identity topology on the effective topos.

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Martin's Conjecture: A Classification of the Naturally Occurring Turing Degrees

Cited by 8 publications

References 11 publications

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Lawvere-Tierney topologies for computability theorists

Lawvere-Tierney topologies for computability theorists

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