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LEVEL THEORY, PART 3: A BOOLEAN ALGEBRA OF SETS ARRANGED IN WELL-ORDERED LEVELS

Part of: Set theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  28 April 2021

TIM BUTTON
TIM BUTTON*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY COLLEGE LONDON GOWER STREET, LONDON, WC1E 6BT, UK E-mail: [email protected] URL: http://www.nottub.com
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Abstract

On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural extension of BLT is definitionally equivalent with ZF.

Keywords

set theorylevel theoryuniversal set

MSC classification

Primary: 03A05: Philosophical and critical
Secondary: 03E30: Axiomatics of classical set theory and its fragments
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 1 , March 2022 , pp. 1 - 26
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The 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.CrossRefGoogle Scholar
Boolos, G., Iteration again. Philosophical Topics, vol. 17 (1989), no. 2, pp. 5–21.Google Scholar
Button, T. and Walsh, S., Philosophy and Model Theory, Oxford University Press, Oxford, UK, 2018.CrossRefGoogle Scholar
Church, A., Set theory with a universal set , Proceedings of the Tarski Symposium (L. Henkin, editor), American Mathematical Society, Providence, RI, 1974, pp. 297–308.CrossRefGoogle Scholar
Conway, J. H., On Numbers and Games, Academic Press, London, UK, 1976.Google Scholar
Cox, M. and Kaye, R., Amphi-ZF: Axioms for Conway games . Archive for Mathematical Logic, vol. 51 (2012), pp. 353–371.CrossRefGoogle Scholar
Forster, T., Church's set theory with a universal set , Logic, Meaning and Computation (C. A. Anderson and M. Zelëny, editors), Springer, Dordrecht, Netherlands, 2001, pp. 109–138.CrossRefGoogle Scholar
Forster, T., The iterative conception of set. Review of Symbolic Logic, vol. 1 (2008), no. 1, pp. 97–110.CrossRefGoogle Scholar
Friedman, H. M. and Visser, A., When bi-interpretability implies synonymy . Logic Group Preprint Series, vol. 320 (2014), pp. 1–19.Google Scholar
Kaye, R. and Wong, T. L., On interpretations of arithmetic in set theory . Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 4, pp. 497–510.CrossRefGoogle Scholar
Le Guin, U. K., The Dispossessed, Harper & Row, 1974.Google Scholar
Löwe, B., Set theory with and without urelements and categories of interpretation . Notre Dame Journal of Formal Logic, vol. 47 (2006), no. 1, pp. 83–91.CrossRefGoogle Scholar
Mitchell, E., A model of set theory with a universal set , Ph.D. thesis, Madison, Wisconsin, 1976.Google Scholar
Montague, R., Set theory and higher-order logic , Formal Systems and Recursive Functions (J. Crossley and M. Dummett, editors), Proceedings of the Eight Logic Colloquium, July, 1963, North-Holland, Amsterdam, Netherlands, 1965, pp. 131–148.CrossRefGoogle Scholar
Montague, R., Scott, D., and Tarski, A., An Axiomatic Approach to Set Theory , Archive copy from the Bancroft Library (BANC MSS 84/69 c, carton 4, folder 29-30.Google Scholar
Oswald, U., Fragmente von “New Foundations” und Typentheorie , Ph.D. thesis, ETH Zürich, 1976.Google Scholar
Potter, M., Sets: An Introduction, Oxford University Press, Oxford, UK, 1990.Google Scholar
Potter, M., Set Theory and Its Philosophy, Oxford University Press, Oxford, UK, 2004.CrossRefGoogle Scholar
Schleicher, D. and Stoll, M., An introduction to Conway's games and numbers . Moscow Mathematical Journal, vol. 6 (2006), no. 2, pp. 359–388.Google Scholar
Scott, D., Definitions by abstraction in set theory . Bulletin of the American Mathematical Society, vol. 61 (1955), no. 5, p. 442.Google Scholar
Scott, D., The notion of rank in set-theory , Summaries of talks Presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, Institute for Defence Analysis, Princeton, NJ, 1960, pp. 267–269.Google Scholar
Scott, D., Axiomatizing set theory , Axiomatic Set Theory II (T. Jech, editor), American Mathematical Society, Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society, July–August 1967, American Mathematical Society, Providence, RI, 1974, pp. 207–14.CrossRefGoogle Scholar
Sheridan, F., A variant of Church's set theory with a universal set in which the singleton function is a set . Logique et Analyse, vol. 59 (2016), no. 233, pp. 81–131.Google Scholar