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NON-CLASSICAL FOUNDATIONS OF SET THEORY

Part of: Set theory Model theory

Published online by Cambridge University Press:  02 December 2021

SOURAV TARAFDER
SOURAV TARAFDER*
Affiliation:
BUSINESS MATHEMATICS AND STATISTICS ST. XAVIER’S COLLEGE 30 MOTHER TERESA SARANI KOLKATA, WEST BENGAL700016, INDIA and INSTITUTO DE FILOSOFIA E CIÊNCIAS HUMANAS UNIVERSITY OF CAMPINAS (UNICAMP) BARÃO GERALDO, SP 13083-896, BRAZILE-mail:souravt09@gmail.com
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Abstract

In this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories.

Keywords

algebra-valued modelsnon-classical set theoriescardinalityAxiom of Choicegeneralized continuum hypothesis

MSC classification

Primary: 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)
Secondary: 03E10: Ordinal and cardinal numbers 03E25: Axiom of choice and related propositions 03E50: Continuum hypothesis and Martin's axiom 03E70: Nonclassical and second-order set theories
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 347 - 376
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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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