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FREGE’S THEORY OF REAL NUMBERS: A CONSISTENT RENDERING

Part of: General and miscellaneous specific topics Proof theory and constructive mathematics Mathematical Logic and Foundations General logic Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  22 February 2021

FRANCESCA BOCCUNI  and
MARCO PANZA
FRANCESCA BOCCUNI
Affiliation:
VITA-SALUTE SAN RAFFAELE UNIVERSITY MILAN, ITALY E-mail: boccuni.francesca@unisr.it
MARCO PANZA
Affiliation:
IHPST (CNRS AND UNIVERSITÉ DE PARIS 1, PANTHÉONE SORBONNE) PARIS, FRANCE E-mail: marco.panza@univ-paris1.fr CHAPMAN UNIVERSITY ORANGE, CA, USA E-mail: panza@chapman.edu
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Abstract

Frege’s definition of the real numbers, as envisaged in the second volume of Grundgesetze der Arithmetik, is fatally flawed by the inconsistency of Frege’s ill-fated Basic Law V. We restate Frege’s definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege’s own indications is possible at all.

Keywords

FregeFrege’s theory of real numbersfoundation of real analysisLogicismEuclid’s Principle

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 03A05: Philosophical and critical 03-03: Historical (must also be assigned at least one classification number from Section 01) 03B30: Foundations of classical theories (including reverse mathematics) 03F35: Second- and higher-order arithmetic and fragments
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 3 , September 2022 , pp. 624 - 667
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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