Coding over Core Models

contributions to two important areas of set theory, core model theory (cf. [10]) and coding (cf. [1]), respectively. In this article we aim to survey some of the work that has been done which combines these two themes, extending Jensen’s original Coding Theorem from L to core models witnessing large cardinal properties. The original result of Jensen can be stated as follows. Theorem 1. (Jensen, cf. [1]) Suppose that (V, A) is a transitive model of ZFC + GCH (i.e., V is a transitive model of ZFC + GCH and replacement holds in V for formulas mentioning A as an additional unary predicate). Then there is a (V, A)-definable, cofinality-preserving class forcing P such that if G is P-generic over (V, A) we have: (a) For some real R, (V [G], A) | = ZFC + the universe is L[R] and A is definable with parameter R. (b) The typical large cardinals properties consistent with V=L are preserved from V to V [R]: inaccessible, Mahlo, weak compact, Π 1 n indescribable, subtle, ineffable, α-Erdős for countable α. Corollary 2. It is consistent to have a real R such that L and L[R] have the same cofinalities but R belongs to no set-generic extension of L. The theme of this article is to consider the following question: To what extent is it possible to establish an analagous result when L is replaced by a core model K and the large cardinal properties in (b) are strengthened to those consistent with V=K (measurable, hypermeasurable, strong, Woodin)