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3 results

  • Chapter 8 - Models of LSA as derived models

  • Grigor Sargsyan, Polish Academy of Sciences, Nam Trang, University of North Texas
  • Book:
    The Largest Suslin Axiom
    Published online:
    07 June 2024
    Print publication:
    27 June 2024, pp 213-234

  • Summary

    In this chapter, we show that under AD^+, the derived model of certain hod pairs satisfies the LSA. We also prove results that are important elsewhere. In particular, we show the derived model of an active \omega.2 lsa Woodin mouse satisfies LSA. This result will be important in Chapter 12, where we obtain the consistency of LSA from PFA.

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

  • NAM TRANG
  • 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

  • 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.”