Journal of Symbolic Logic 64 (2):443-459 (1999)

Abstract
Let κ R be the least ordinal κ such that L κ (R) is admissible. Let $A = \{x \in \mathbb{R} \mid (\exists\alpha such that x is ordinal definable in L α (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 ω Woodin cardinals which are cofinal in the ordinals." T has consistency strength weaker than that of the theory ZFC + "There exists ω Woodin cardinals", but stronger than that of the theory ZFC + "There exists n Woodin Cardinals", for each n ∈ ω. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A = R ∩ M. Since 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 Σ * n real is in A
Keywords Large Cardinals   Descriptive Set Theory   Inner Model Theory
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DOI 10.2307/2586477
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References found in this work BETA

Inner Models with Many Woodin Cardinals.J. R. Steel - 1993 - Annals of Pure and Applied Logic 65 (2):185-209.
Projectively Well-Ordered Inner Models.J. R. Steel - 1995 - Annals of Pure and Applied Logic 74 (1):77-104.
Mouse Sets.Mitch Rudominer - 1997 - Annals of Pure and Applied Logic 87 (1):1-100.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
Descriptive Set Theory.Yiannis Nicholas Moschovakis - 1982 - Studia Logica 41 (4):429-430.

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Citations of this work BETA

Descriptive Inner Model Theory.Grigor Sargsyan - 2013 - Bulletin of Symbolic Logic 19 (1):1-55.
The Envelope of a Pointclass Under a Local Determinacy Hypothesis.Trevor M. Wilson - 2015 - Annals of Pure and Applied Logic 166 (10):991-1018.
Inner Model Operators in L.Mitch Rudominer - 2000 - Annals of Pure and Applied Logic 101 (2-3):147-184.

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