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DOI: 10.4230/LIPIcs.CSL.2024.28

Realizability Models for Large Cardinals

Authors: Laura Fontanella, Guillaume Geoffroy, and Richard Matthews

Published in: LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Abstract

Realizabilty is a branch of logic that aims at extracting the computational content of mathematical proofs by establishing a correspondence between proofs and programs. Invented by S.C. Kleene in the 1945 to develop a connection between intuitionism and Turing computable functions, realizability has evolved to include not only classical logic but even set theory, thanks to the work of J-L. Krivine. Krivine’s work made possible to build realizability models for Zermelo-Frænkel set theory, ZF, assuming its consistency. Nevertheless, a large part of set theoretic research involves investigating further axioms that are known as large cardinals axioms; in this paper we focus on four large cardinals axioms: the axioms of (strongly) inaccessible cardinal, Mahlo cardinals, measurable cardinals and Reinhardt cardinals. We show how to build realizability models for each of these four axioms assuming their consistency relative to ZFC or ZF.

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Laura Fontanella, Guillaume Geoffroy, and Richard Matthews. Realizability Models for Large Cardinals. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fontanella_et_al:LIPIcs.CSL.2024.28,
  author =	{Fontanella, Laura and Geoffroy, Guillaume and Matthews, Richard},
  title =	{{Realizability Models for Large Cardinals}},
  booktitle =	{32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)},
  pages =	{28:1--28:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-310-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{288},
  editor =	{Murano, Aniello and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.28},
  URN =		{urn:nbn:de:0030-drops-196715},
  doi =		{10.4230/LIPIcs.CSL.2024.28},
  annote =	{Keywords: Logic, Classical Realizability, Set Theory, Large Cardinals}
}

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Realizability Models for Large Cardinals

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