  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.
