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Philosophy and Model Theory$
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Tim Button and Sean Walsh

Print publication date: 2018

Print ISBN-13: 9780198790396

Published to Oxford Scholarship Online: May 2018

DOI: 10.1093/oso/9780198790396.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 17 October 2021

Internal categoricity and the sets

Internal categoricity and the sets

(p.251) 11 Internal categoricity and the sets
Philosophy and Model Theory

Tim Button

Sean Walsh

Oxford University Press

As the previous chapter discussed the internalist perspective on the categoricity of arithmetic, this chapter presents the internalist perspective on sets. In particular, we show both how to internalise Scott-Potter set theory its quasi-categoricity theorem, and how to internalise Zermelo’s Quasi-Categoricity Theorem. As in the case of arithmetic, this gives a non-semantic way to draw the boundary between algebraic and univocal theories. A particularly compelling case of the quasi-univocity of set theory revolves around the continuum hypothesis. Furthermore, by additionally postulating that the size of the pure sets is the same as the size of the universe, these famous quasi-categoricity results can actually be turned into internal categoricity results simpliciter, so that one has full univocity instead of mere quasi-univocity. In the appendices we prove these results, and we discuss how they relate to important work by McGee and Martin.

Keywords:   Internal quasi-categoricity, quasi-intolerance, The Continuum Hypothesis, indefinite extensibility, McGee’s Theorem, Martin on quasi-categoricity, Scott-Potter Set Theory

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