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

IF 0.4 4区数学 Q4 LOGIC

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

Jouko Väänänen, Philip D. Welch

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引用次数: 0

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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来源期刊

Mathematical Logic Quarterly 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.