Deconstructing Inner Model Theory

In this paper we shall repair some errors and fill some gaps in the inner model theory of [2]. The problems we shall address affect some quite basic definitions and proofs. We shall be concerned with condensation properties of canonical inner models constructed from coherent sequences ~ E of extenders as in [2]. Condensation results have the general form: if x is definable in a certain way over a level J ~ E α , then either x ∈ J ~ E α , or else from x we can reconstruct J ~ E α in a simple way. The first condensation property considered in [2] is the initial segment condition, or ISC. In section 1 we show that the version of this condition described in [2] is too strong, in that no coherent ~ E in which the extenders are indexed in the manner of [2], and which is such that L[ ~ E] satisfies the mild large cardinal hypothesis that there is a cardinal which is strong past a measurable, can satisfy the full ISC of [2]. It follows that the coherent sequences constructed in [2] do not satisfy the ISC of [2]. We shall describe the weaker ISC which these sequences do satisfy, and indicate the small changes in the arguments of [2] this new condition requires. In section 2, we fill a gap in the proof that the standard parameters of a sufficiently iterable premouse are solid. This is Theorem 8.1 of [2], one of its central fine structural results. In section 3, we fill a gap in the proof that the Dodd parameter of a sufficiently iterable premouse is Dodd-solid. This is Theorem 3.2 of [4], and is an important ingredient in the proofs of square in L[ ~ E] and of weak covering for K. The difficulties we overcome in sections