STABLY MEASURABLE CARDINALS

Abstract

AbstractWe 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 $ : $u_2(\kappa )$ , and secondly to give the consistency strength of a property of Lücke’s.TheoremThe following are equiconsistent:(i)There exists $\kappa $ which is stably measurable;(ii)for some cardinal $\kappa $ , $u_2(\kappa )=\sigma (\kappa )$ ;(iii)The $\boldsymbol {\Sigma }_{1}$ -club property holds at a cardinal $\kappa $ .Here $\sigma (\kappa )$ is the height of the smallest $M \prec _{\Sigma _{1}} H ( \kappa ^{+} )$ containing $\kappa +1$ and all of $H ( \kappa )$ . Let $\Phi (\kappa )$ be the assertion: TheoremAssume $\kappa $ is stably measurable. Then $\Phi (\kappa )$ .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: $\mbox {``}\exists \kappa \Phi (\kappa ) \mbox {''}$ is (set)-generically absolute ${\,\longleftrightarrow \,}$ There are arbitrarily large stably measurable cardinals.When $u_2(\kappa ) < \sigma (\kappa )$ we give some results on inner model reflection.