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LARGE CARDINALS AS PRINCIPLES OF STRUCTURAL REFLECTION

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

Published online by Cambridge University Press:  23 January 2023

JOAN BAGARIA Open the ORCID record for JOAN BAGARIA [Opens in a new window]
JOAN BAGARIA*
Affiliation:
INSTITUCIÓ CATALANA DE RECERCA I ESTUDIS AVANÇATS (ICREA) BARCELONA, CATALONIA, SPAIN and DEPARTAMENT DE MATEMÀTIQUES I INFORMÀTICA UNIVERSITAT DE BARCELONA GRAN VIA DE LES CORTS CATALANES, 585 08007 BARCELONA, CATALONIA, SPAIN E-mail: joan.bagaria@icrea.cat E-mail: bagaria@ub.edu
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Abstract

After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960s, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflection implicit in Cantor’s original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a comprehensive survey of results showing that different forms of the new principle of Structural Reflection are equivalent to well-known large cardinal axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter.

Keywords

large cardinalsstructural reflectionreflection principles

MSC classification

Primary: 03-02: Research exposition (monographs, survey articles) 03E55: Large cardinals 03E65: Other hypotheses and axioms
Secondary: 00A30: Philosophy of mathematics 03A05: Philosophical and critical

Information

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 29 , Issue 1 , March 2023 , pp. 19 - 70
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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