AbstractLet ?Rbe the least ordinal ? such that L?(R) is admissible. Let A= {x? R | (?a < ?R) such that xis ordinal definable in La(R)}. It is well known that (assuming determinacy) Ais the largest countable inductive set of reals. Let Tbe the theory: ZFC - Replacement + “There exists ? Woodin cardinals which are cofinal in the ordinals.” Thas consistency strength weaker than that of the theory ZFC + “There exists ? Woodin cardinals”, but stronger than that of the theory ZFC + “There exists nWoodin Cardinals”, for each n? ?. Let Mbe the canonical, minimal inner model for the theory T. In this paper we show that A= R n M. Since Mis a mouse, we say that Ais a mouse set. As an application, we use our characterization of Ato give an inner-model-theoretic proof of a theorem of Martin which states that for all n, every real is in A.