Complexity of reals in inner models of set theory

AbstractWe consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either ℵ1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly ℵ1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry