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THE MODAL LOGIC OF SET-THEORETIC POTENTIALISM AND THE POTENTIALIST MAXIMALITY PRINCIPLES

Part of: Model theory General logic

Published online by Cambridge University Press:  04 October 2019

JOEL DAVID HAMKINS  and
ØYSTEIN LINNEBO
JOEL DAVID HAMKINS
Affiliation:
FACULTY OF PHILOSOPHY UNIVERSITY OF OXFORDOXFORD, UK and UNIVERSITY COLLEGEOXFORD, UKE-mail: [email protected]: http://jdh.hamkins.org
ØYSTEIN LINNEBO
Affiliation:
DEPARTMENT OF PHILOSOPHY, IFIKK UNIVERSITY OF OSLO, POSTBOKS 1020 BLINDERN 0315OSLO, NORWAYE-mail: [email protected]: http://oysteinlinnebo.org
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Abstract

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_\beta $ ), Grothendieck–Zermelo potentialism (true in all larger $V_\kappa $ for inaccessible cardinals $\kappa $ ), transitive-set potentialism (true in all larger transitive sets), forcing potentialism (true in all forcing extensions), countable-transitive-model potentialism (true in all larger countable transitive models of ZFC), countable-model potentialism (true in all larger countable models of ZFC), and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.

Keywords

potentialismmodal logicmodels of set theory

MSC classification

Secondary: 03C62: Models of arithmetic and set theory 03B45: Modal logic (including the logic of norms)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 1 , March 2022 , pp. 1 - 35
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Copyright
© Association for Symbolic Logic, 2022

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