Journal of Symbolic Logic

The Largest Countable Inductive Set is a Mouse Set

Mitch Rudominer

Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR.


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

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

First available in Project Euclid: 6 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Large Cardinals Descriptive Set Theory Inner Model Theory


Rudominer, Mitch. The Largest Countable Inductive Set is a Mouse Set. J. Symbolic Logic 64 (1999), no. 2, 443--459.

Export citation