Bishop-Style Constructive Reverse Mathematics

Diener, Hannes; Ishihara, Hajime

doi:10.1007/978-3-030-59234-9_10

Cited by 9 publications

(12 citation statements)

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“…There are a huge number of mathematical principles which are equivalent to LLPO; see Diener [11] and also Brattka-Gherardi-Pauly [6]. The principle LLPO may also be called de Morgan's law for Σ 0 1 formulas.…”

Section: Game Rulesmentioning

confidence: 99%

“…Coincidentally, it turns out that our notion is heavily related to another notion called extended Weihrauch reducibility, which is introduced in Bauer [2]. By viewing topologies in this way, it is possible, for example, to position the study of the structure of Lawvere-Tierney topologies as an extension of the Weihrauch-style analogue [6] of reverse mathematics (note, however, that this is by no means an extension of the standard reverse mathematics [38,11] at all; reverse mathematics has more to do with the internal logic, and more finitary aspects). By bringing the arguments on topologies into pure computability theory in this way, we solve some problems proposed in [23,24].…”

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confidence: 99%

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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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Section: Game Rulesmentioning

confidence: 99%

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confidence: 99%

Lawvere-Tierney topologies for computability theorists

Kihara

2021

Preprint

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“…We have the following theorem, which sprouts from the section on metastability in Diener's forthcoming Habilitationsschrift ( [24]).…”

Section: 14mentioning

confidence: 99%

The unreasonable effectiveness of Nonstandard Analysis

Sanders

2020

Journal of Logic and Computation

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As suggested by the title, the aim of this paper is to uncover the vast computational content of classical Nonstandard Analysis. To this end, we formulate a template CI which converts a theorem of 'pure' Nonstandard Analysis, i.e. formulated solely with the nonstandard definitions (of continuity, integration, differentiability, convergence, compactness, et cetera), into the associated effective theorem. The latter constitutes a theorem of computable mathematics no longer involving Nonstandard Analysis. To establish the vast scope of CI, we apply this template to representative theorems from the Big Five categories from Reverse Mathematics. The latter foundational program provides a classification of the majority of theorems from 'ordinary', that is non-set theoretical, mathematics into the aforementioned five categories. The Reverse Mathematics zoo gathers exceptions to this classification, and is studied in [70,71] using CI. Hence, the template CI is seen to apply to essentially all of ordinary mathematics, thanks to the Big Five classification (and associated zoo) from Reverse Mathematics. Finally, we establish that certain 'highly constructive' theorems, called Herbrandisations, imply the original theorem of Nonstandard Analysis from which they were obtained via CI.

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“…The notion of BD-N as well as Proposition 9 are due to [13]. For an overview of constructive reverse mathematics, see [11,14,15].…”

Section: Notesmentioning

confidence: 99%