Journal of Symbolic Logic 86 (2):448-470 (2021)

Abstract
We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma _{1}$ -definability at uncountable regular cardinals. In particular we give its exact consistency strength first in terms of the second uniform indiscernible for bounded subsets of $\kappa $ : $u_2$, and secondly to give the consistency strength of a property of Lücke’s.TheoremThe following are equiconsistent:There exists $\kappa $ which is stably measurable;for some cardinal $\kappa $, $u_2=\sigma $ ;The $\boldsymbol {\Sigma }_{1}$ -club property holds at a cardinal $\kappa $.Here $\sigma $ is the height of the smallest $M \prec _{\Sigma _{1}} H $ containing $\kappa +1$ and all of $H $. Let $\Phi $ be the assertion: TheoremAssume $\kappa $ is stably measurable. Then $\Phi $.And a form of converse:TheoremSuppose there is no sharp for an inner model with a strong cardinal. Then in the core model K we have: $\mbox {``}\exists \kappa \Phi \mbox {''}$ is -generically absolute ${\,\longleftrightarrow \,}$ There are arbitrarily large stably measurable cardinals.When $u_2 < \sigma $ we give some results on inner model reflection.
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DOI 10.1017/jsl.2019.82
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References found in this work BETA

The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
The Fine Structure of the Constructible Hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229.
Measurable Cardinals and Good Σ1-Wellorderings.Philipp Lücke & Philipp Schlicht - 2018 - Mathematical Logic Quarterly 64 (3):207-217.

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