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A MARRIAGE OF BROUWER’S INTUITIONISM AND HILBERT’S FINITISM I: ARITHMETIC

Part of: Proof theory and constructive mathematics

Published online by Cambridge University Press:  07 January 2021

TAKAKO NEMOTO  and
SATO KENTARO
TAKAKO NEMOTO
Affiliation:
DEPARTMENT OF ARCHITECTURAL DESIGN FACULTY OF ENVIRONMENTAL STUDY HIROSHIMA INSTITUTE OF TECHNOLOGY SAEKI-KU MIYAKE 2-1-1 HIROSHIMA 731-5193, JAPAN E-mail: [email protected]
SATO KENTARO
Affiliation:
INSTITUTE OF COMPUTER SCIENCE UNIVERSITÄT BERN NEUBRÜCKSTRASSE 10 BERN 3012, SWITZERLAND E-mail: [email protected]
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Abstract

We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics (i.e., which part is equiconsistent with $\textbf {PRA}$ or consistent provably in $\textbf {PRA}$ ) already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: (i) fan theorem for decidable fans but arbitrary bars; (ii) continuity principle and the axiom of choice both for arbitrary formulae; and (iii) $\Sigma _2$ induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO (except the choice along functions); but that limited principle of omniscience LPO makes the situation completely classical.

Keywords

Hilbert’s programmeintuitionistic mathematicsRussian recursive mathematicsinterpretabilityconsistency strengthsemi-classical principlefunction-based second order arithmeticLifschitz’s realizabilityKleene’s second modelCoquand–Hofmann forcing

MSC classification

Primary: 03F25: Relative consistency and interpretations
Secondary: 03F35: Second- and higher-order arithmetic and fragments 03F50: Metamathematics of constructive systems 03F55: Intuitionistic mathematics 03F60: Constructive and recursive analysis
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 437 - 497
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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