0# and Inner Models

this paper we examine the cardinal structure of inner models that satisfy GCH but do not contain 0 # . We show, assuming that 0 # exists, that such models necessarily contain Mahlo cardinals of high order, but without further assumptions need not contain a cardinal # which is #-Mahlo. The principal tools are the Covering Theorem for L and the technique of reverse Easton iteration. Let I denote the class of Silver indiscernibles for L and #i # | # # ORD# its increasing enumeration. Also fix an inner model M of GCH not containing 0 # and let # # denote the # # of the model M[0 # ], the least inner model containing M as a submodel and 0 # as an element.<F11