Skip to main content Accessibility help


  • NAM TRANG (a1)


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.”



Hide All
[1]Cummings, J., Iterated forcing and elementary embeddings, Handbook of Set Theory, pp. 775883, 2010.
[2]Ketchersid, R., Toward ADfrom the Continuum Hypothesis and anω 1-dense ideal, Ph. D. thesis, Berkeley, 2000.
[3]Koellner, P. and Woodin, W. H., Large cardinals from determinacy, Handbook of Set Theory, pp. 19512119, 2010.
[4]Larson, Paul B., The stationary tower: Notes on a course by W. Hugh Woodin, vol. 32, University Lecture Series, American Mathematical Society, Providence, RI, 2004.
[5]Larson, Paul B., Forcing over models of determinacy, Handbook of Set Theory, pp. 21212177, 2010.
[6]Sargsyan, Grigor, A tale of hybrid mice, available at∼gs481/.
[7]Schindler, Ralf and Steel, John R., The core model induction, available∼steel.
[8]Solovay, R., The independence of DC from AD, Cabal Seminar 76–77, pp. 171183. Springer, New York, 1978.
[9]Steel, J. R., The derived model theorem, available∼steel.
[10]Steel, J. R., Derived models associated to mice. Computational prospects of infinity. Part I. Tutorials, vol. 14, Lecture Notes Series, Institure of Mathematical Science, National Institure of Singapore, pp. 105–193, World Scientific, Hackensack, NJ, 2008.
[11]Steel, J. R., Scales in K(ℝ). Games, scales, and Suslin cardinals. The Cabal Seminar. Vol. I, Lecture Notes in Logic, pp. 176–208, vol. 31, Association of Symbolic Logic, Chicago, IL, 2008.
[12]Steel, J. R., An outline of inner model theory, Handbook of Set Theory, pp. 15951684, 2010.
[13]Steel, John R. and Woodin, Hugh W., HOD as a core model, 2012.
[14]Steel, John and Zoble, Stuart, Determinacy from strong reflection. Transactions of the American Mathematical Society, vol. 366 (2014), no. 8, pp. 44434490.
[15]Steel, J. R. and Trang, N., AD+, derived models, and σ1 -reflection, available at∼namtrang. 2011.
[16]Trang, N., A hierarchy of measures from AD, available at∼namtrang. 2013.
[17]Trang, N., Determinacy in L(ℝ, μ). Journal of Mathematical Logic, vol. 14 (2014), no. 01.
[18]Wilson, T. M., Contributions to Descriptive Inner Model Theory, PhD thesis, University of California, 2012.
[19]Hugh Woodin, W., The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1999.
[20]Zhu, Y., The derived model theorem II, available at∼g0700513/ der.pdf. 2010.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed