Stably Measurable Cardinals

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 firstly 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 L\"ucke's. Theorem: The following are equiconsistent: (i) There exists $\kappa$ which is stably measurable; (ii) for some cardinal $\kappa$, $u_2(\kappa)=\sigma(\kappa)$; (iii) The {\boldmath $\Sigma_1$}-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 $H(\kappa)$.