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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: [email protected]
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Abstract

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