We show that the predicate “x is the power set of y” is ${\mathrm{\Sigma }}_1(Card)$-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 ${\ell }_I$, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) ${\ell }_I$ is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.

中文翻译：

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当基数确定幂集时：内部模型和 Härtig 量词逻辑

我们证明谓词“ x是 y的幂集 ”是${\mathrm{\Sigma }}_1（卡片）$- 可定义，如果 V = L[E] 是由连贯的扩展器序列构造的扩展器模型，前提是不存在具有伍丁基数的内部模型。这里$卡片$是一个仅适用于无限基数的谓词。由此我们得出结论：二阶逻辑的有效性可简化为$V_{我}$，Härtig 量词逻辑的有效性集。进一步我们表明，如果没有 L[E] 模型具有强到其 ℵ-不动点之一的基数，并且${\ell }_{我}$，该逻辑的 Löwenheim 数，小于最小弱不可访问 δ，则 (i)${\ell }_{我}$是 K 可测量基数的极限，并且 (ii) 弱覆盖引理在 δ 处成立。