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2 results

  • Chapter 9 - Condensing sets

  • 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 235-272

  • Summary

    This chapter develops the theory of condensing sets. Condensing sets give rise to iteration strategies with nice condensation properties. We show the existence of condensing sets under various hypotheses: AD^+ and PFA. We will use the existence of condensing sets in AD^+ in the proof of generation of pointclasses in Chapter 10. We will use the existence of condensing sets in Chapter 12 to construct a model of LSA under PFA.

  • 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.