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Inner Models from Extended Logics: Part 2

Juliette Kennedy, Menachem Magidor, Jouko Väänänen

Abstract

We introduce a new inner model $C(aa)$ arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively $MM^{++}$, the regular uncountable cardinals of $V$ are measurable in the inner model $C(aa)$, the theory of $C(aa)$ is (set) forcing absolute, and $C(aa)$ satisfies CH. We introduce an auxiliary concept that we call club determinacy, which simplifies the construction of $C(aa)$ greatly but may have also independent interest. Based on club determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model $C(aa)$.

Inner Models from Extended Logics: Part 2

Abstract

We introduce a new inner model arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively , the regular uncountable cardinals of are measurable in the inner model , the theory of is (set) forcing absolute, and satisfies CH. We introduce an auxiliary concept that we call club determinacy, which simplifies the construction of greatly but may have also independent interest. Based on club determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model .

Paper Structure

This paper contains 15 sections, 54 theorems, 125 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Inner model $C(\space\mathop{{\hbox{\tt a a}}}\space)$
  4. Club determinacy
  5. Club determinacy from Woodin cardinals
  6. Applications of Club Determinacy
  7. Large cardinals
  8. Forcing absoluteness
  9. The Continuum Hypothesis
  10. aa-premice
  11. The aa-ultrapower
  12. The Continuum Hypothesis in $C(\space\mathop{{\hbox{\tt a a}}}\space)$
  13. Variants of stationary logic
  14. Open problems
  15. Appendix: A counter-example to AC in $C_o(L^*)$

Key Result

Lemma 3.2

$C(\space\mathop{{\hbox{\tt a a}}}\space)$ is a model of $ZFC$. The model $C(\space\mathop{{\hbox{\tt a a}}}\space)$ has a canonical (first order) definable well-order $\prec$.

Figures (2)

  • Figure 1: The iteration.
  • Figure 2: The levels.

Theorems & Definitions (129)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3: MR309729MR3243739
  • Definition 3.4: kmv
  • Proposition 3.5
  • proof
  • Theorem 3.6
  • ...and 119 more