Fine structure for tame inner models

Abstract

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3]. Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that (1) every level of is a sound, tame mouse, and (2) ⊨ “ There is a strong cardinal that is a limit of Woodin cardinals” . Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10]. Suppose that fi is a normal measure on Ω and that all premice are tame. Then K c , the background certified core model, exists and is a premouse of height Ω. Moreover , for μ-almost every α < Ω. Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application. If there is a cardinal κ such that κ is κ + -strongly compact, then there is a premouse that is not tame .