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THE POTENTIAL IN FREGE’S THEOREM

Part of: Mathematical Logic and Foundations General logic General and miscellaneous specific topics

Published online by Cambridge University Press:  25 August 2020

WILL STAFFORD
WILL STAFFORD*
Affiliation:
DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE UNIVERSITY OF CALIFORNIA, IRVINE IRVINE, 92617 CA, USA
*
E-mail: [email protected]
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Abstract

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Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.

Keywords

logicismpotential infinitymodal logic

MSC classification

Primary: 03-02: Research exposition (monographs, survey articles)
Secondary: 00A30: Philosophy of mathematics 03B45: Modal logic (including the logic of norms)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 2 , June 2023 , pp. 553 - 577
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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