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IDENTITY CRISIS BETWEEN SUPERCOMPACTNESS AND VǑPENKA’S PRINCIPLE

Part of: Set theory

Published online by Cambridge University Press:  07 September 2020

YAIR HAYUT ,
MENACHEM MAGIDOR  and
ALEJANDRO POVEDA
YAIR HAYUT
Affiliation:
KURT GÖDEL RESEARCH CENTER INSTITUT FÜR MATHEMATIK UNIVERSITÄT WIEN, 1090 WIEN, AUSTRIAE-mail:[email protected]
MENACHEM MAGIDOR
Affiliation:
INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEMJERUSALEM91904, ISRAELE-mail:[email protected]
ALEJANDRO POVEDA
Affiliation:
DEPARTAMENT DE MATEMÀTIQUES I INFORMÀTICA UNIVERSITAT DE BARCELONA, GRAN VIA DE LES CORST CATALANES 585, BARCELONA08007, CATALONIAE-mail:[email protected]
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Abstract

In this paper we study the notion of $C^{(n)}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{(1)}$ -supercompact but also that the least supercompact is $C^{(1)}$ -supercompact (and even $C^{(n)}$ -supercompact). Furthermore, we prove that under suitable hypothesis the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.

Keywords

forcingPrikry-type forcinglarge cardinalsC(n)-cardinals

MSC classification

Secondary: 03E50: Continuum hypothesis and Martin's axiom 03E57: Generic absoluteness and forcing axioms
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 2 , June 2022 , pp. 626 - 648
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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