Mouse sets

Abstract

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 A = R ∩ M , where M is a canonical model from inner model theory. In technical terms, M 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 OD n ω 1 be the set of reals which are ∑ n definable over the model L ω 1 (R), from an ordinal parameter. In this paper we will show that for all n ⩾ 1, OD n ω 1 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.