Level theory, part 1: Axiomatizing the bare idea of a cumulative hierarchy of sets

Bulletin of Symbolic Logic 27 (4):436-460 (2021)
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Abstract

The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, Montague, Derrick, and Potter.

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10.1017/bsl.2021.13

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Tim Button
University College London

Citations of this work

Against Cumulative Type Theory.Tim Button & Robert Trueman - 2022 - Review of Symbolic Logic 15 (4):907-49.
Symmetric relations, symmetric theories, and Pythagrapheanism.Tim Button - 2022 - Philosophy and Phenomenological Research (3):583-612.

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References found in this work

The iterative conception of set.George Boolos - 1971 - Journal of Philosophy 68 (8):215-231.
How we learn mathematical language.Vann McGee - 1997 - Philosophical Review 106 (1):35-68.
Iteration Again.George Boolos - 1989 - Philosophical Topics 17 (2):5-21.
Set Theory: An Introduction to Large Cardinals.F. R. Drake & T. J. Jech - 1976 - British Journal for the Philosophy of Science 27 (2):187-191.

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