K without the measurable

We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definable core model that is close to V in various ways. §1. The main theorem. If the universe V of sets does not have within it very complicated canonical inner models for large cardinal hypotheses, then it has a canonical inner model K that in some sense is as large as possible. K is absolutely definable, its internal structure can be analyzed in fine-structural detail, and yet it is close to the full universe V in various ways. If 0 does not exist, then K = L. Set forcing cannot add 0 or change L, so K = K [G ] = LwheneverG is set-generic overV . The fine-structure theory of [7] produces a detailed picture of the first order theory ofL. Jensen’s Covering Theorem [1] describes oneof themost importantwaysL is close toV : any uncountableX ⊆ L has a superset Y of the same cardinality such that Y ∈ L. If 0 does exist, then L is quite far from V , and so K must be larger than L. Dodd and Jensen developed a theory of K under the weaker hypothesis that there is no proper class inner model with a measurable cardinal in [2], [3], and [4]. This hypothesis is compatible with the existence of 0, and if 0 exists, then 0 in K , and henceK is properly larger thanL. Under this weaker anti-large-cardinalhypothesis, K is again absolutely definable, admits a fine structure theory like that of L, and is close to V , in that every uncountable X ⊆ K has a superset Y of the same cardinality such that Y ∈ K . Several authors have extended the Dodd–Jensen work over the years. We shall recount some of the most relevant history in the next section. In this paper, we shall prove a theorem which represents its ultimate extension in one direction. Our discussion of the history will be clearer if we state that theorem now. Theorem 1.1. There are Σ2 formulae øK(v) and øΣ(v) such that, if there is no transitive proper class model satisfying ZFC plus “there is a Woodin cardinal”, then (1) K = {v | øK(v)} is a transitive proper class premouse satisfying ZFC, (2) {v | øΣ(v)} is an iteration strategy for K for set-sized iteration trees, and moreover the unique such strategy, Received March 21, 2010. 1One can think of 0♯ as a weak approximation to a canonical inner model with a measurable cardinal. c © 2013, Association for Symbolic Logic 0022-4812/13/7803-0002/$3.70 DOI:10.2178/jsl.7803020