Projectively well-ordered inner models Original Research Article

We show that the reals in the minimal iterable inner model having n Woodin cardinals are precisely those which are Δn + 21 definable from some countable ordinal. (One direction here is due to Hugh Woodin.) It follows that this model satisfies “There is a Δn + 21 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.