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LEVEL THEORY, PART 2: AXIOMATIZING THE BARE IDEA OF A POTENTIAL HIERARCHY

Part of: Philosophical aspects of logic and foundations Set theory

Published online by Cambridge University Press:  29 April 2021

TIM BUTTON
TIM BUTTON*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY COLLEGE LONDON GOWER STREET, LONDON, WC1E 6BT, UK E-mail: tim.button@ucl.ac.uk URL: http://www.nottub.com
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Abstract

Potentialists think that the concept of set is importantly modal. Using tensed language as a heuristic, the following bare-bones story introduces the idea of a potential hierarchy of sets: ‘Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets’. Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with Level Theory, as developed in Part 1.

Keywords

set theorylevel theoryrankpotentialismactualism

MSC classification

Primary: 03A05: Philosophical and critical
Secondary: 03E30: Axiomatics of classical set theory and its fragments

Information

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 4 , December 2021 , pp. 461 - 484
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Copyright
The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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References

Boolos, G., The iterative conception of set . The Journal of Philosophy , vol. 68 (1971), no. 8, pp. 215–231.10.2307/2025204CrossRefGoogle Scholar
Boolos, G., To be is to be a value of a variable (or to be some values of some variables) . Journal of Philosophy , vol. 81 (1984), no. 8, pp. 430–449.10.2307/2026308CrossRefGoogle Scholar
Button, T. and Walsh, S., Philosophy and Model Theory, Oxford University Press, Oxford, 2018.CrossRefGoogle Scholar
Fine, K., Relatively unrestricted quantification , Absolute Generality (A. Rayo and G. Uzquiano, editors), Oxford University Press, Oxford, 2006, pp. 20–44.Google Scholar
Goldblatt, R.. Logics of Time and Computation second ed., CSLI Publications, Stanford, CA, 1992.Google Scholar
Linnebo, Ø., Pluralities and sets . Journal of Philosophy , vol. 107 (2010), no. 3, pp. 144–164.10.5840/jphil2010107311CrossRefGoogle Scholar
Linnebo, Ø., The potential hierarchy of sets . Review of Symbolic Logic , vol. 6 (2013), no. 2, pp. 205–228.CrossRefGoogle Scholar
Linnebo, Ø., Putnam on mathematics as modal logic , Hilary Putnam on Logic and Mathematics (G. Hellman and R.T. Cook, editors), Springer, New York, 2018a, pp. 249–267.10.1007/978-3-319-96274-0_14CrossRefGoogle Scholar
Linnebo, Ø., Thin Objects , Oxford University Press, Oxford, 2018b.10.1093/oso/9780199641314.001.0001CrossRefGoogle Scholar
Parsons, C., What is the iterative conception of set? Mathematics in Philosophy (C. Parsons), 1977, pp. 268–297. 10.1007/978-94-010-1138-9_18CrossRefGoogle Scholar
Parsons, C., Mathematics in Philosophy, Cornell University Press, Ithaca, NY, 1983.Google Scholar
Parsons, C., Sets and modality, Mathematics in Philosophy (C. Parsons), 1983, pp. 298–341.Google Scholar
Putnam, H., Mathematics without foundations . Journal of Philosophy, vol. 64 (1967), no. 1, pp. 5–22.CrossRefGoogle Scholar
Putnam, H., Three blog posts: ‘The modal logical interpretation and “equivalent descriptions”’ (11.Dec.2014); ‘Continuing’ (12.Dec.2014); and ‘Mathematical “existence”’ (13.Dec.2014), 2014.Google Scholar
Scott, D., Axiomatizing set theory , Axiomatic Set Theory II (T. Jech, editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, RI, 1974, pp. 207–214.10.1090/pspum/013.2/0392570CrossRefGoogle Scholar
Soysal, Z., Why is the universe of sets not a set? Synthese, vol. 197 (2020), no. 2, pp. 575–597.10.1007/s11229-017-1513-xCrossRefGoogle Scholar
Studd, J.P., The iterative conception of set: A (bi)modal axiomatization . Journal of Philosophical Logic , vol. 42 (2013), pp. 697–725.10.1007/s10992-012-9245-3CrossRefGoogle Scholar
Studd, J.P., Everything, More or Less: A Defence of Generality Relativism, Oxford University Press, Oxford, 2019, pp. 165, 260.10.1093/oso/9780198719649.001.0001CrossRefGoogle Scholar
Williamson, T., Modal Logic as Metaphysics , Oxford University Press, Oxford, 2013.CrossRefGoogle Scholar