## Journal of Symbolic Logic

### The Fine Structure of Real Mice

Daniel W. Cunningham

#### Abstract

Before one can construct scales of minimal complexity in the Real Core Model, K($\mathbb{R}$), one needs to develop the fine-structure theory of K($\mathbb{R}$). 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($\mathbb{R}$). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(R) and the strength of the theory ZF + AD + $\neg DC_\mathbb{R}$.

#### Article information

Source
J. Symbolic Logic, Volume 63, Issue 3 (1998), 937-994.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745575

Mathematical Reviews number (MathSciNet)
MR1649070

Zentralblatt MATH identifier
0921.03047

JSTOR