Abstract

Before one can construct scales of minimal complexity in the Real Core Model, K, one needs to develop the fine-structure theory of K. In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L. In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K and the strength of the theory ZF + AD + $\neg DC_\mathbb{R}$.