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INDESTRUCTIBILITY WHEN THE FIRST TWO MEASURABLE CARDINALS ARE STRONGLY COMPACT

Part of: Set theory

Published online by Cambridge University Press:  23 November 2021

ARTHUR W. APTER
ARTHUR W. APTER*
Affiliation:
DEPARTMENT OF MATHEMATICS BARUCH COLLEGE OF CUNYNEW YORK, NY10010, USA and DEPARTMENT OF MATHEMATICS THE CUNY GRADUATE CENTER 365 FIFTH AVENUE NEW YORK, NY10016, USAE-mail:[email protected]: http://faculty.baruch.cuny.edu/aapter
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Abstract

We prove two theorems concerning indestructibility properties of the first two strongly compact cardinals when these cardinals are in addition the first two measurable cardinals. Starting from two supercompact cardinals $\kappa _1 < \kappa _2$ , we force and construct a model in which $\kappa _1$ and $\kappa _2$ are both the first two strongly compact and first two measurable cardinals, $\kappa _1$ ’s strong compactness is fully indestructible (i.e., $\kappa _1$ ’s strong compactness is indestructible under arbitrary $\kappa _1$ -directed closed forcing), and $\kappa _2$ ’s strong compactness is indestructible under $\mathrm {Add}(\kappa _2, \delta )$ for any ordinal $\delta $ . This provides an answer to a strengthened version of a question of Sargsyan found in [17, Question 5]. We also investigate indestructibility properties that may occur when the first two strongly compact cardinals are not only the first two measurable cardinals, but also exhibit nontrivial degrees of supercompactness.

Keywords

supercompact cardinalstrongly compact cardinalindestructibilitynonreflecting stationary set of ordinalslottery sumEaston support iteration of partial orderings

MSC classification

Primary: 03E35: Consistency and independence results
Secondary: 03E55: Large cardinals
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 214 - 227
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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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