We extend the theory of “Fine structure and iteration trees” to models having more than one Woodin cardinal.

In this paper we explore a connection between descriptive set theory and inner model theory. From descriptive set theory, we will take a countable, definable set of reals, A. We will then show that , where is a canonical model from inner model theory. In technical terms, is a “mouse”. Consequently, we say that A is a mouse set. For a concrete example of the type of set A we are working with, let ODnω1 be the set of reals which (...) are ∑n definable over the model Lω1 , from an ordinal parameter. In this paper we will show that for all n 1, ODnω1 is a mouse set. Our work extends some similar results due to D.A. Martin, J.R. Steel, and H. Woodin. Several interesting questions in this area remain open. (shrink)