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N-BERKELEY CARDINALS AND WEAK EXTENDER MODELS
The Journal of Symbolic Logic ( IF 0.6 ) Pub Date : 2020-07-21 , DOI: 10.1017/jsl.2020.21
RAFFAELLA CUTOLO

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.

中文翻译:

N-伯克利红雀和弱扩展模型

对于给定的内部模型ñ对于 ZFC,可以考虑在 ZFC 的上下文中的相对化版本的伯克利红雀,并询问是否可以存在“ñ——伯克利红衣主教。” 在这篇文章中,我们为这个问题提供了一个肯定的答案。事实上,在超紧致红衣主教的假设下$\三角洲$,我们证明存在一个 ZFC 内部模型ñ这样有一个红衣主教是ñ- 伯克利,即使在强烈的意义上。此外,所涉及的模型ñ是一个弱扩展模型$\三角洲$是超紧凑的。最后,我们证明了强版本ñ- 伯克利红雀队在任何时候都证明是不一致的ñ满足下关闭$\欧米茄$-序列。
更新日期:2020-07-21
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