Complexity of reals in inner models of set theory

We 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 aleph(1)(M) 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 F-sigma set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly aleph(1), 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. (C) 1998 Published by Elsevier Science B.V. All rights reserved.