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Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" From Large Cardinals

Andreas Lietz

Abstract

We answer a question of Woodin by showing that assuming an inaccessible cardinal $κ$ which is a limit of ${<}κ$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb P}\models``\mathrm{NS}_{ω_1}\text{ is }ω_1\text{-dense}"$. We also introduce a new forcing axiom $\mathrm{QM}$, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies $\mathbb{Q}_{\mathrm{max}}\text{-}(*)$. Consequently, $\mathrm{QM}$ implies ``$\mathrm{NS}_{ω_1}$ is $ω_1$-dense".

Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" From Large Cardinals

Abstract

We answer a question of Woodin by showing that assuming an inaccessible cardinal which is a limit of -supercompact cardinals exists, there is a stationary set preserving forcing so that . We also introduce a new forcing axiom , show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies . Consequently, implies `` is -dense".
Paper Structure (17 sections, 44 theorems, 205 equations)

This paper contains 17 sections, 44 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. History of "$\mathrm{NS}_{\omega_1}$ is $\omega_1$-dense"
  3. Notation
  4. $\diamondsuit(\omega_1^{{<}\omega})$ and $\diamondsuit^{+}(\omega_1^{{<}\omega})$
  5. A Forcing Axiom That Implies "$\mathrm{NS}_{\omega_1}$ Is $\omega_1$-Dense"
  6. $\mathrm{Q}$-Maximum
  7. $\mathrm{Q}$-iterations
  8. Blueprints for Instances of "$\mathrm{MM}^{++}\Rightarrow(\ast)$"
  9. $\mathbb P_{\mathrm{max}}\text{-variations}$ and the $\mathbb{V}_{\mathrm{max}}$-multiverse view
  10. Asperó-Schindler $(\ast)$-forcing
  11. $\diamondsuit$-iterations
  12. $\diamondsuit\text{-}(\ast)$-forcing
  13. The first blueprint
  14. The second blueprint
  15. The $\mathbb Q_{\mathrm{max}}$-variation $\mathbb Q_{\mathrm{max}}^{-}$
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $\kappa$ is an uncountable cardinal and there is a $\sigma$-additive real-valued measure on $\kappa$ which Then there is a weakly inaccessible cardinal $\leq\kappa$.

Theorems & Definitions (156)

  • Theorem 1: Ulam
  • Theorem 2: Ulam
  • Theorem 4: Taylor
  • Theorem 5: Woodin, unpublished
  • Theorem 6: Shelah, shelahdense
  • Theorem 7: Woodin, woodinbook
  • Definition 8
  • Definition 9
  • Theorem 11
  • Definition 2.1
  • ...and 146 more