From Nonstandard Analysis to Various Flavours of Computability Theory

Sanders, Sam

doi:10.1007/978-3-319-55911-7_40

Cited by 7 publications

(2 citation statements)

References 12 publications

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“…In a wide sense, the work on this article can also be seen as a contribution to bridge (the antipodes) constructive analysis and nonstandard analysis. This problem has been extensively (and intensively) discussed in the past few years (see for example [40,41,42,8,43,31]).…”

Section: Then By Writingmentioning

confidence: 99%

Naïve Infinitesimal Analysis: Its Construction and Its Properties

Nugraha¹,

McKubre-Jordens²,

Diener³

2020

Preprint

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This paper aims to build a new understanding of the nonstandard mathematical analysis. The main contribution of this paper is the construction of a new set of numbers, R Z< , which includes infinities and infinitesimals. The construction of this new set is done naïvely in the sense that it does not require any heavy mathematical machinery, and so it will be much less problematic in a long term. Despite its naïvety character, the set R Z< is still a robust and rewarding set to work in. We further develop some analysis and topological properties of it, where not only we recover most of the basic theories that we have classically, but we also introduce some new enthralling notions in them. The computability issue of this set is also explored. The works presented here can be seen as a contribution to bridge constructive analysis and nonstandard analysis, which has been extensively (and intensively) discussed in the past few years.

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Section: Then By Writingmentioning

confidence: 99%

Naïve Infinitesimal Analysis: Its Construction and Its Properties

Nugraha¹,

McKubre-Jordens²,

Diener³

2020

Preprint

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“…Nelson's framework has been formulated for higher-order Peano and Heyting arithmetic, and given a computational interpretation via Herbrand realizability and the associated nonstandard Dialectica interpretation [8]-we refer to the later here as the Herbrand functional interpretation. More recently, this Herbrand functional interpretation has been successfully applied in the characterisation of the computational content of nonstandard analysis [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

On the Herbrand functional interpretation

Oliva¹,

2020

Mathematical Logic Qtrly

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We show that the types of the witnesses in the Herbrand functional interpretation can be simplified, avoiding the use of “sets of functionals” in the interpretation of implication and universal quantification. This is done by presenting an alternative formulation of the Herbrand functional interpretation, which we show to be equivalent to the original presentation. As a result of this investigation we also strengthen the monotonicity property of the original presentation, and prove a monotonicity property for our alternative definition.

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