The core model up to a woodin cardinal

Abstract

Publisher Summary This chapter provides an exposition of the current state of core model theory, and in particular, of the core model theory for large cardinals up to a Woodin cardinal. The existence of a Woodin cardinal is equiconsistent with the axiom of determinacy for Σ 1 2 formulas, and that the existence of infinitely many Woodin cardinals is equiconsistent with the full axiom of determinacy. This work has led to major advances in core model theory. It has provided an additional impetus to the study of cardinals small enough that a core model theory is practical at the present time, while on the other hand, it has provided new tools together with a new understanding of some of the older tools. “The true core model” is a model, denoted by K, which contains all of the large cardinal structure existing in the universe, but which is, at the same time, as much as possible like the constructible sets L. The existence of this model is speculative, but K is known to exist under appropriate assumptions restricting the size of large cardinals existing in the universe. The results discussed in the chapter imply that it exists under the assumption that there is no Woodin cardinal together with a further technical assumption.