0# and Inner Models

In 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 κ which is κ-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 〈iα | α ∈ ORD〉 its increasing enumeration. Also fix an inner model M of GCH not containing 0 and let ωα denote the ωα of the model M [0 ], the least inner model containing M as a submodel and 0 as an element.