The Largest Countable Inductive Set is a Mouse Set

Abstract

Let kappa be the least ordinal alpha such that L_{alpha}(R) is admissible. Let A be the set of reals x such that x is ordinal definable in L_{\alpha}(R), for some alpha<kappa. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the following theory: ZFC - Replacement + "There exists $\omega$ Woodin cardinals which are cofinal in the ordinals." T has consistency strength weaker than that of the theory ZFC + "There exists omega Woodin cardinals", but stronger than that of the theory ZFC + "There exists n Woodin Cardinals", for each n. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A is equal to the set of reals in M. Since M is a "mouse", we say that A is a "mouse set." As an application, we use our characterization of A to give an inner-model-theoretic proof of Martin's theorem that A is equal to the set of reals which are Sigma^*_n for some n.