Abstract

Let $\kappa^\mathbb{R}$ be the least ordinal $\kappa$ such that L$_\kappa$ is admissible. Let $A = \{x \in \mathbb{R} \mid $ such that x is ordinal definable in $L_\alpha \}$. It is well known that A is the largest countable inductive set of reals. Let T be the 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 $n \in \omega$. Let $\mathcal{M}$ be the canonical, minimal inner model for the theory T. In this paper we show that A = $\mathbb{R} \cap \mathcal{M}$. Since $\mathcal{M}$ 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 a theorem of Martin which states that for all n, every $\Sigma^*_n$ real is in A.