Fine Structure and Iteration Trees

This monograph lays the foundations for the theory of canonical inner models of set theory which are large enough to satisfy the statement "There is a Woodin cardinal". It does so by combining Jensen's fine structure models, already useful in the study of smaller inner models, with the theory of iteration trees and Woodin cardinals developed recently by Martin and Steel. The resulting theory is a powerful tool in studying the structure of models of set theory. The main result in this monograph is the construction, given the existence of a Woodin cardinal, of an L-like inner model containing a Woodin cardinal and satisfying the generalized continuum hypothesis, but its real significance is as an indispensable tool for further work with large cardinals in set theory.