When cardinals determine the power set: inner models and Härtig quantifier logic

Pub Date : 2023-09-11 DOI:10.1002/malq.202200030

Jouko Väänänen, Philip D. Welch

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Abstract

We show that the predicate “x is the power set of y” is $Σ_{1} (Card)$ -definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here $Card$ is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to $V_{I}$ , the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has a cardinal strong up to one of its ℵ-fixed points, and $ℓ_{I}$ , the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) $ℓ_{I}$ is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.

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当基数确定权力集时:内部模型和Härtig量词逻辑

我们证明了谓词“x是y的幂集”是Σ 1(Card)$ \Sigma _1(\operatorname{Card})$ -可定义的，如果V = L[E]是由扩充器的相干序列构造的扩充器模型，只要没有带有伍丁枢机的内部模型。这里Card $\operatorname{Card}$是一个谓词，仅对无限基数为真。由此得出二阶逻辑的有效性可约化为vi $V_I$，即Härtig量词逻辑的有效性集。进一步证明了如果没有L[E]模型的基数强到它的一个不动点，并且l_1 $\ell _{I}$，这个逻辑的Löwenheim个数小于最小弱不可达的δ，则(i) _I$是K的可测基数的极限，(ii)弱覆盖引理在δ处成立。

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