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_α(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 + &#34;There exists $ω$ Woodin cardinals which are cofinal in the ordinals.&#34; T has consistency strength weaker than that of the theory ZFC + &#34;There exists omega Woodin cardinals&#34;, but stronger than that of the theory ZFC + &#34;There exists n Woodin Cardinals&#34;, 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 &#34;mouse&#34;, we say that A is a &#34;mouse set.&#34; As an application, we use our characterization of A to give an inner-model-theoretic proof of Martin&#39;s theorem that A is equal to the set of reals which are Sigma^*_n for some n.