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Varsovian models $ω$

Farmer Schlutzenberg

Abstract

For $n<ω$, let $N_n$ be the minimal iterable proper class mouse $M$ such that $M\models$ "there are ordinals $δ_0<κ_0<\ldots<δ_{n-1}<κ_{n-1}$ such that each $δ_i$ is a Woodin cardinal and each $κ_i$ is a strong cardinal", and let $N_ω$ be likewise, but with $M\models$ "there is an ordinal $λ$ which is a limit of Woodin cardinals and a limit of strong cardinals". Under appropriate large cardinal hypotheses, Sargsyan and Schindler introduced and analysed in "Varsovian models I" the Varsovian model of $N_1$, and Sargsyan, Schindler and the author introduced and analysed in "Varsovian models II" the Varsovian model of $N_2$. We extend this to $N_ω$, assuming that $*$-translation integrates routinely with the P-constructions of this paper (the write-up of which is yet to be completed). We show, under this assumption, that $N_ω$ has a proper class inner model $\mathscr{V}_ω$ which is a fully iterable strategy mouse with $ω$ Woodin cardinals, closed under its strategy, and that the universe of $\mathscr{V}_ω$ is the eventual generic HOD, and the mantle, of $N_ω$. We also show, under the same assumption, that the core model $K$ of $N_ω$ (which can be defined in a natural manner) is an iterate of $N_ω$, is an inner model of $\mathscr{V}_ω$, and is fully iterable in $M$ and in $\mathscr{V}_ω$.

Varsovian models $ω$

Abstract

For , let be the minimal iterable proper class mouse such that "there are ordinals such that each is a Woodin cardinal and each is a strong cardinal", and let be likewise, but with "there is an ordinal which is a limit of Woodin cardinals and a limit of strong cardinals". Under appropriate large cardinal hypotheses, Sargsyan and Schindler introduced and analysed in "Varsovian models I" the Varsovian model of , and Sargsyan, Schindler and the author introduced and analysed in "Varsovian models II" the Varsovian model of . We extend this to , assuming that -translation integrates routinely with the P-constructions of this paper (the write-up of which is yet to be completed). We show, under this assumption, that has a proper class inner model which is a fully iterable strategy mouse with Woodin cardinals, closed under its strategy, and that the universe of is the eventual generic HOD, and the mantle, of . We also show, under the same assumption, that the core model of (which can be defined in a natural manner) is an iterate of , is an inner model of , and is fully iterable in and in .
Paper Structure (20 sections, 81 theorems, 114 equations, 1 figure)

This paper contains 20 sections, 81 theorems, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Terminology and notation
  4. Properties of $M_{\mathrm{sw}\omega}$ and $\Sigma_{M_{\mathrm{sw}\omega}}$
  5. The first Varsovian models $\mathscr{V}_1$ and $\mathscr{W}_1$
  6. From $\mathscr{W}_{n+1}$ to $\mathscr{W}_{n+2}$
  7. The structures and strategies
  8. Inductive hypotheses
  9. Inference of $\Sigma_{\mathscr{W}_{n+1{\downarrow} i}}$ from $\Sigma_{\mathscr{W}_{n+1}}$
  10. $\delta_{n+1}$-short tree strategies
  11. Uniform grounds and $\mathscr{M}_{\infty,n+1}[*_{n+1}]$
  12. Action of iteration maps on $\mathscr{M}_{\infty,n+1}$
  13. $\mathscr{V}_{n+2}$ and $\mathscr{W}_{n+2}$
  14. Self-coherence of $\Sigma_{\mathscr{W}_{n+2}}$
  15. Conclusion
  16. ...and 5 more sections

Key Result

Theorem 2

Assume ZFC + "$M_{\mathrm{sw}\omega}^\#$ exists and is $(\omega,\mathrm{OR})$-iterable". Assume that the appropriate adaptation of $*$-translation goes through routinely (see Remark rem:*-trans). Let $M=M_{\mathrm{sw}\omega}$. Then $M$ has an inner model $\mathscr{V}_\omega$, and $\mathscr{V}_\omega

Figures (1)

  • Figure 1: Layout in case $n=2$. Note that $M=\mathscr{V}_0=\mathscr{W}_0$, and $\mathscr{V}_{i+1}=\mathscr{V}_{i+1}^{\mathscr{W}_0}$, and $\mathscr{M}_{\infty i}=\mathscr{V}_{i+1}{\downarrow} i$. If $P$ appears directly above $Q$, then $P\subsetneq Q$, and $Q$ is a set-generic extension of $P$. If $P$ appears directly left of $Q$ then $P\subsetneq Q$, and $Q$ is not a set-generic extension of $P$. The diagonal arrows are ultrapower maps; those in the bottom row are ultrapowers via $e_0=e_0^{\mathscr{W}_3}=e_0^{\mathscr{W}_2}=e_0^{\mathscr{W}_1}$, those in the next row are via $e_1=e_1^{\mathscr{V}_2^{\mathscr{W}_1}}=e_1^{\mathscr{V}_3^{\mathscr{W}_2}}=i^{\mathscr{W}_3}_{e_0}(e_1^{\mathscr{W}_3})$; etc.

Theorems & Definitions (255)

  • Definition 1.1
  • Remark 1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Remark 3
  • Definition 2.3
  • Lemma 4
  • proof
  • Definition 3.1
  • ...and 245 more