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2 results

  • Chapter 10 - Applications

  • 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 273-308

  • Summary

    This chapter gives various applications of the theory developed in the previous chapters. The first application is a proof of generation of mouse full pointclasses assuming Strong Mouse Capturing. The second application is a proof that Strong Mouse Capturing holds in the minimal model of LSA; so the Mouse Set Conjecture is true in all models of AD^+ up to the minimal model of LSA. The third application is a proof of consistency of LSA from the existence of a Woodin limit of Woodin cardinals.

  • DIVERGENT MODELS WITH THE FAILURE OF THE CONTINUUM HYPOTHESIS

  • Part of
  • NAM TRANG
  • Journal:
    The Journal of Symbolic Logic , First View
    Published online by Cambridge University Press:
    06 December 2023, pp. 1-11
    • Article
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  • We construct divergent models of $\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis ($\mathsf {CH}$) under various assumptions. Divergent models of $\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in $\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of $\mathsf {AD}^+$. Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings of the reals and principles implying $\neg \mathsf {CH}$, like $\mathsf {MM}$, that divergent models shed light on, see Question 5.1.