STABLY MEASURABLE CARDINALS

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)(kappa), and secondly to give the consistency strength of a property of Lucke's. THEOREM The following are equiconsistent: (i) There exists. which is stably measurable; (ii) for some cardinal kappa, u(2)(k) = sigma(kappa); (iii) The Sigma(1)-club property holds at a cardinal kappa. Here sigma(kappa) is the height of the smallest M <(Sigma 1) H(kappa(+)) containing kappa + 1 and all of H(kappa). Let Phi(kappa) be the assertion: for all X subset of R for all r is an element of R[X is Sigma(1)(kappa, r)-definable <-> X is an element of Sigma(1)(3) (r)]. THEOREM Assume kappa is stably measurable. Then Phi(kappa). And a form of converse: THEOREM Suppose there is no sharp for an inner model with a strong cardinal. Then in the core model K we have: "there exists kappa Phi(kappa)" is (set)-generically absolute <-> There are arbitrarily large stably measurable cardinals. When u(2)(kappa) < sigma(kappa) we give some results on inner model reflection.