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 Lücke'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 $H(\kappa )$.