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N-BERKELEY CARDINALS AND WEAK EXTENDER MODELS
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 2 / June 2020
- Published online by Cambridge University Press:
- 21 July 2020, pp. 809-816
- Print publication:
- June 2020
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- Article
- Export citation
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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.
