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The largest countable inductive set is a mouse set

Published online by Cambridge University Press:  12 March 2014

Mitch Rudominer
Mitch Rudominer*
Affiliation:
Department of Mathematics, Florida International University, Miami. FL 33199, U.S.A. E-mail: rudomine@flu.edu
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Abstract

Let κℝ be the least ordinal κ such that L κ (ℝ) is admissible. Let A = {x ϵ ℝ ∣ (∃α < κℝ) 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 ϵ 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 real is in A.

Keywords

large cardinalsdescriptive set theoryinner model theory

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999, pp. 443 - 459
Copyright
Copyright © Association for Symbolic Logic 1999

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