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STRUCTURE THEORY OF L(ℝ, μ) AND ITS APPLICATIONS

Published online by Cambridge University Press:  13 March 2015

NAM TRANG
NAM TRANG*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES, CARNEGIE MELLON UNIVERSITY, 5000 FORBES AVE, PITTSBURGH, PA 15213, USAE-mail: namtrang@andrew.cmu.edu
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Abstract

In this paper, we explore the structure theory of L(ℝ, μ) under the hypothesis L(ℝ, μ) ⊧ “AD + μ is a normal fine measure on ” and give some applications. First we show that “ ZFC + there exist ω2 Woodin cardinals”1 has the same consistency strength as “ AD + ω1 is ℝ-supercompact”. During this process we show that if L(ℝ, μ) ⊧ AD then in fact L(ℝ, μ) ⊧ AD+. Next we prove important properties of L(ℝ, μ) including Σ1 -reflection and the uniqueness of μ in L(ℝ, μ). Then we give the computation of full HOD in L(ℝ, μ). Finally, we use Σ1 -reflection and ℙmax forcing to construct a certain ideal on (or equivalently on in this situation) that has the same consistency strength as “ZFC+ there exist ω2 Woodin cardinals.”

Keywords

Descriptive inner model theorySolovay measureHODderived modelPrikry forcingℙmax forcing
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 1 , March 2015 , pp. 29 - 55
Copyright
Copyright © The Association for Symbolic Logic 2015 

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