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MORE ON THE PRESERVATION OF LARGE CARDINALS UNDER CLASS FORCING

Part of: Set theory

Published online by Cambridge University Press:  13 September 2021

JOAN BAGARIA Open the ORCID record for JOAN BAGARIA [Opens in a new window]  and
ALEJANDRO POVEDA
JOAN BAGARIA
Affiliation:
INSTITUCIÓ CATALANA DE RECERCA I ESTUDIS AVANÇATS (ICREA) BARCELONA, SPAIN and DEPARTAMENT DE MATEMÀTIQUES I INFORMÀTICA UNIVERSITAT DE BARCELONA, GRAN VIA DE LES CORTS CATALANES BARCELONA 585, 08007, SPAIN E-mail: joan.bagaria@icrea.cat
ALEJANDRO POVEDA*
Affiliation:
DEPARTAMENT DE MATEMÀTIQUES I INFORMÀTICA UNIVERSITAT DE BARCELONA, GRAN VIA DE LES CORTS CATALANES BARCELONA 585, 08007, SPAIN Current address: EINSTEIN INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEM JERUSALEM 91904, ISRAEL
*
E-mail: alejandro.poveda@mail.huji.ac.il
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Abstract

We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka’s Principle and $C^{(n)}$-extendible cardinals under Jensen’s iteration for forcing the GCH [17], previously obtained in [8, 27], respectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $\Delta _2$-definable behaviour of the power-set function on regular cardinals. We show that one can force proper class-many disagreements between the universe and HOD with respect to the calculation of successors of regular cardinals, while preserving $C^{(n)}$-extendible cardinals. We also show, assuming the GCH, that the class forcing iteration of Cummings–Foreman–Magidor for forcing $\diamondsuit _{\kappa ^+}^+$ at every $\kappa $ [10] preserves $C^{(n)}$-extendible cardinals. We give an optimal result on the consistency of weak square principles and $C^{(n)}$-extendible cardinals. In the last section prove another preservation result for $C^{(n)}$-extendible cardinals under very general (not necessarily definable or weakly homogeneous) class forcing iterations. As applications we prove the consistency of $C^{(n)}$-extendible cardinals with $\mathrm {{V}}=\mathrm {{HOD}}$, and also with $\mathrm {GA}$ (the Ground Axiom) plus $\mathrm {V}\neq \mathrm {HOD}$, the latter being a strengthening of a result from [14].

Keywords

large cardinalsclass forcing iterationsextendible cardinalsVopěnka’s Principle

MSC classification

Secondary: 03E35: Consistency and independence results 03E55: Large cardinals

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Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 1 , March 2023 , pp. 290 - 323
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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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