## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 64, Issue 2 (1999), 443-459.

### The Largest Countable Inductive Set is a Mouse Set

#### Abstract

Let $\kappa^\mathbb{R}$ be the least ordinal $\kappa$ such that L$_\kappa(\mathbb{R})$ is admissible. Let $A = \{x \in \mathbb{R} \mid (\exists\alpha < \kappa^\mathbb{R})$ such that x is ordinal definable in $L_\alpha (\mathbb{R})\}$. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the theory: ZFC - Replacement + "There exists $\omega$ Woodin cardinals which are cofinal in the ordinals." T has consistency strength weaker than that of the theory ZFC + "There exists $\omega$ Woodin cardinals", but stronger than that of the theory ZFC + "There exists n Woodin Cardinals", for each $n \in \omega$. Let $\mathcal{M}$ be the canonical, minimal inner model for the theory T. In this paper we show that A = $\mathbb{R} \cap \mathcal{M}$. Since $\mathcal{M}$ is a mouse, we say that A is a mouse set. As an application, we use our characterization of A to give an inner-model-theoretic proof of a theorem of Martin which states that for all n, every $\Sigma^*_n$ real is in A.

#### Article information

**Source**

J. Symbolic Logic, Volume 64, Issue 2 (1999), 443-459.

**Dates**

First available in Project Euclid: 6 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1183745786

**Mathematical Reviews number (MathSciNet)**

MR1777763

**Zentralblatt MATH identifier**

0930.03059

**JSTOR**

links.jstor.org

**Keywords**

Large Cardinals Descriptive Set Theory Inner Model Theory

#### Citation

Rudominer, Mitch. The Largest Countable Inductive Set is a Mouse Set. J. Symbolic Logic 64 (1999), no. 2, 443--459. https://projecteuclid.org/euclid.jsl/1183745786