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N-BERKELEY CARDINALS AND WEAK EXTENDER MODELS
Published online by Cambridge University Press: 21 July 2020
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.
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- © The Association for Symbolic Logic 2020