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MODAL QUANTIFIERS, POTENTIAL INFINITY, AND YABLO SEQUENCES

Part of: Proof theory and constructive mathematics General logic Model theory Nonstandard models Philosophical aspects of logic and foundations General and miscellaneous specific topics

Published online by Cambridge University Press:  17 March 2022

MICHAŁ TOMASZ GODZISZEWSKI  and
RAFAŁ URBANIAK
MICHAŁ TOMASZ GODZISZEWSKI*
Affiliation:
UNIVERSITY OF LODZ POLAND
RAFAŁ URBANIAK
Affiliation:
UNIVERSITY OF GDAŃSK POLAND E-mail: rfl.urbaniak@gmail.com
*
E-mail: mtgodziszewski@gmail.com
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Abstract

When properly arithmetized, Yablo’s paradox results in a set of formulas which (with local disquotation in the background) turns out to be consistent, but $\omega $-inconsistent. Adding either uniform disquotation or the $\omega $-rule results in inconsistency. Since the paradox involves an infinite sequence of sentences, one might think that it doesn’t arise in finitary contexts. We study whether it does. It turns out that the issue depends on how the finitistic approach is formalized. On one of them, proposed by M. Mostowski, all the paradoxical sentences simply fail to hold. This happens at a price: the underlying finitistic arithmetic itself is $\omega $-inconsistent. Finally, when studied in the context of a finitistic approach which preserves the truth of standard arithmetic (developed by one of the authors), the paradox strikes back—it does so with double force, for now the inconsistency can be obtained without the use of uniform disquotation or the $\omega $-rule.

Keywords

potentialismYablo’s paradoxaxiomatic theories of truthfinitismpotential infinitymodalitiesmodal logicmodal quantifiers

MSC classification

Primary: 00A30: Philosophy of mathematics 03A05: Philosophical and critical 03B45: Modal logic (including the logic of norms) 03C13: Finite structures 03H15: Nonstandard models of arithmetic
Secondary: 03F30: First-order arithmetic and fragments

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 19 , Issue 1 , March 2026 , pp. 1 - 22
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

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