PROJECTIVELY WELL-ORDERED INNER MODELS

We show that the reals in the minimal iterable inner model having n Woodin cardinals are precisely those which are Delta(n+2)(1) definable from some countable ordinal. (One direction here is due to Hugh Woodin.) It follows that this model satisfies ''There is a Delta(n+2)(1) well-order of the reals''. We also describe some other connections between the descriptive set theory of projective sets and inner models with finitely many Woodin cardinals.