Interpreting weak Kőnig's lemma in theories of nonstandard arithmetic

We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles.The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar's functional interpretation and has similarities with Van den Berg, Briseid and Safarik's functional interpretation but replacing finiteness by majorisability.We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments.1. We add the standard predicates st σ (t σ ) (for each finite type σ).2. We add the standardness axioms:(for each finite types σ and τ ).3. We add the external induction rule Φ(0) ∀x 0 (st 0 (x)→(Φ(x)→Φ(x+1))) ∀x 0 (st 0 (x)→Φ(x))

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“…3. we are in line with the argument that nonstandard methods can be used to analyse compactness arguments [8,7,6,5].…”

Section: Introductionsupporting

confidence: 86%

Intuitionistic nonstandard bounded modified realisability and functional interpretation

Dinis

Gaspar

2018

Annals of Pure and Applied Logic

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“…the previous theorem does not lead to a contradiction. Moreover, the first part of the theorem, involving a classical system, has been proved in [4], and the proof in the latter seems to go through in our (semi-)intuitionistic setting.…”

Section: 2mentioning

confidence: 87%

A note on non-classical nonstandard arithmetic

Sanders

2019

Annals of Pure and Applied Logic

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Recently, a number of formal systems for Nonstandard Analysis restricted to the language of finite types, i.e. nonstandard arithmetic, have been proposed. We single out one particular system by Dinis-Gaspar, which is categorised by the authors as being part of intuitionistic nonstandard arithmetic. Their system is indeed inconsistent with the Transfer axiom of Nonstandard Analysis, and the latter axiom is classical in nature as it implies (higher-order) comprehension. Inspired by this observation, the main aim of this paper is to provide answers to the following questions: (Q1) In the spirit of Reverse Mathematics, what is the minimal fragment ofTransfer that is inconsistent with the Dinis-Gaspar system? (Q2) What other axioms are inconsistent with the Dinis-Gaspar system? Our answer to the first question suggests that the aforementioned inconsistency actually derives from the axiom of extensionality relative to the standard world, and that other (much stronger) consequences of Transfer are actually harmless. Perhaps surprisingly, our answer to the second question shows that the Dinis-Gaspar system is inconsistent with a number of (non-classical) continuity theorems which one would -in our opinion-categorise as intuitionistic in the sense of Brouwer. Finally, we show that the Dinis-Gaspar system involves a standard part map, suggesting this system also pushes the boundary of what still counts as 'Nonstandard Analysis' or 'internal set theory'.

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“…Weak Kőnig's Lemma. As shown in [DF17], WKL 0 (one of the Big Five systems from Reverse Mathematics) is interpretable, over a nonstandard version of primitive recursive arithmetic with extensionality, using a version of the Axiom of Choice and Idealization. It relies on distinguishing two sorts: the number sort is interpreted by the standard numbers, and the set sort is interpreted by bounded type 1 functionals (or by number codes, both standard and nonstandard, of finite sets of numbers, again both standard and nonstandard).…”

Section: Llpomentioning

confidence: 99%

Stateful Realizers for Nonstandard Analysis

Dinis¹,

Miquey²

2023

Logical Methods in Computer Science

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In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of (semi)intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the $\lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $\cal{M}^{\mathbb{N}}$ the computation is being done. We pay attention to the nonstandard principles (and their computational content) obtainable in this setting. In particular, we give non-trivial realizers to Idealization and a non-standard version of the LLPO principle. We then discuss how to quotient this product to mimic the Lightstone-Robinson construction.

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Interpreting weak Kőnig's lemma in theories of nonstandard arithmetic

Cited by 6 publications

References 14 publications

Intuitionistic nonstandard bounded modified realisability and functional interpretation

Intuitionistic nonstandard bounded modified realisability and functional interpretation

A note on non-classical nonstandard arithmetic

Stateful Realizers for Nonstandard Analysis

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