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N-BERKELEY CARDINALS AND WEAK EXTENDER MODELS

Part of: Set theory

Published online by Cambridge University Press:  21 July 2020

RAFFAELLA CUTOLO
RAFFAELLA CUTOLO*
Affiliation:
DEPARTMENT OF MATHEMATICS AND APPLICATIONS UNIVERSITY OF NAPLES “FEDERICO II” VIA CINTIA 21, 80126NAPLES, ITALYE-mail: [email protected]
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Abstract

For a given inner model N of ZFC, one can consider the relativized version of Berkeley cardinals in the context of ZFC, and ask if there can exist an “N-Berkeley cardinal.” In this article we provide a positive answer to this question. Indeed, under the assumption of a supercompact cardinal $\delta $ , we show that there exists a ZFC inner model N such that there is a cardinal which is N-Berkeley, even in a strong sense. Further, the involved model N is a weak extender model of $\delta $ is supercompact. Finally, we prove that the strong version of N-Berkeley cardinals turns out to be inconsistent whenever N satisfies closure under $\omega $ -sequences.

Keywords

N-Berkeley cardinalsweak extender modelsHOD dichotomy

MSC classification

Primary: 03E45: Inner models, including constructibility, ordinal definability, and core models
Secondary: 03E55: Large cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 809 - 816
Copyright
© The Association for Symbolic Logic 2020

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References

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