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- Title
When cardinals determine the power set: inner models and Härtig quantifier logic.
- Authors
Väänänen, Jouko; Welch, Philip D.
- Abstract
We show that the predicate "xis the power set ofy" is Σ1(Card)$\Sigma _1(\operatorname{Card})$‐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$\operatorname{Card}$ is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to VI$V_I$, 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 ℓI$\ell _{I}$, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) ℓI$\ell _I$ is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.
- Subjects
LOGIC
- Publication
Mathematical Logic Quarterly, 2023, Vol 69, Issue 4, p460
- ISSN
0942-5616
- Publication type
Article
- DOI
10.1002/malq.202200030