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UNIVERSISM AND EXTENSIONS OF V

Part of: General and miscellaneous specific topics Philosophical aspects of logic and foundations Set theory

Published online by Cambridge University Press:  21 July 2020

CAROLIN ANTOS ,
NEIL BARTON  and
SY-DAVID FRIEDMAN
CAROLIN ANTOS
Affiliation:
FACHBEREICH PHILOSOPHIE/ZUKUNFTSKOLLEG UNIVERSITY OF KONSTANZ KONSTANZ, GERMANYE-mail: [email protected]
NEIL BARTON
Affiliation:
FACHBEREICH PHILOSOPHIE UNIVERSITY OF KONSTANZ KONSTANZ, GERMANYE-mail: [email protected]
SY-DAVID FRIEDMAN
Affiliation:
KURT GÖDEL RESEARCH CENTER UNIVERSITY OF VIENNA, VIENNA, AUSTRIAE-mail: [email protected]
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Abstract

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favor of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in ‘ideal’ outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist.

Keywords

extensionsmultiversismuniversismset theoryadmissible set

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 03A99: None of the above, but in this section 03E70: Nonclassical and second-order set theories
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 1 , March 2021 , pp. 112 - 154
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Copyright
© Association for Symbolic Logic, 2020

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