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INDEPENDENCE PROOFS IN NON-CLASSICAL SET THEORIES

Part of: Set theory General logic

Published online by Cambridge University Press:  22 March 2021

SOURAV TARAFDER  and
GIORGIO VENTURI
SOURAV TARAFDER
Affiliation:
BUSINESS MATHEMATICS AND STATISTICS DEPARTMENT OF COMMERCE ST. XAVIER’S COLLEGE 30 MOTHER TERESA SARANI KOLKATA 700016 INDIA INSTITUTE OF PHILOSOPHY AND HUMAN SCIENCES UNIVERSITY OF CAMPINAS (UNICAMP) BARÃO GERALDO, R. CORA CORALINA 100-CIDADE UNIVERSITÁRIA CAMPINAS - SP, 13083-896 BRAZIL E-mail: souravt09@gmail.com
GIORGIO VENTURI
Affiliation:
INSTITUTE OF PHILOSOPHY AND HUMAN SCIENCES UNIVERSITY OF CAMPINAS (UNICAMP) BARÃO GERALDO, R. CORA CORALINA 100-CIDADE UNIVERSITÁRIA CAMPINAS, SP 13083-896 BRAZIL E-mail: gio.venturi@gmail.com
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Abstract

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of $\mathsf {CH}$); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse.

Keywords

set theoryBoolean-valued modelsnon-classical logicsindependenceparaconsistencyZF

MSC classification

Primary: 03E40: Other aspects of forcing and Boolean-valued models
Secondary: 03E35: Consistency and independence results 03B53: Paraconsistent logics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 4 , December 2023 , pp. 979 - 1010
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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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