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CANTORIAN SET THEORY

Published online by Cambridge University Press:  02 January 2019

ALEX OLIVER  and
TIMOTHY SMILEY
ALEX OLIVER
Affiliation:
GONVILLE AND CAIUS COLLEGE UNIVERSITY OF CAMBRIDGE CAMBRIDGE CB2 1TA, UKE-mail: [email protected]
TIMOTHY SMILEY
Affiliation:
CLARE COLLEGE UNIVERSITY OF CAMBRIDGE CAMBRIDGE CB2 1TL, UKE-mail: [email protected]
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Abstract

Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals.

Keywords

03A0503B2003E3003E7003E75arithmetization of analysisAxiom of Pluralityempty setlevelordered pairsingletonSingular LogicCantorScott
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 24 , Issue 4 , December 2018 , pp. 393 - 451
Copyright
Copyright © The Association for Symbolic Logic 2018 

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