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STRUCTURE THEORY OF L(ℝ, μ) AND ITS APPLICATIONS
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- Journal:
- The Journal of Symbolic Logic / Volume 80 / Issue 1 / March 2015
- Published online by Cambridge University Press:
- 13 March 2015, pp. 29-55
- Print publication:
- March 2015
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- Article
- Export citation
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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.”
” and give some applications. First we show that “ ZFC + there exist 
(or equivalently on 
in this situation) that has the same consistency strength as “ZFC+ there exist 