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Mice with Woodin cardinals from a Reinhardt

Farmer Schlutzenberg

Abstract

Suppose there is a Reinhardt cardinal. Then (1) $M_n(X)$ exists and is fully iterable (above $X$) for every transitive set $X$ and every $n<ω$ (here $M_n(X)$ denotes the canonical minimal proper class inner model containing $X$ and having $n$ Woodin cardinals above the rank of $X$); and (2) Projective Determinacy holds in every set generic extension.

Mice with Woodin cardinals from a Reinhardt

Abstract

Suppose there is a Reinhardt cardinal. Then (1) exists and is fully iterable (above ) for every transitive set and every (here denotes the canonical minimal proper class inner model containing and having Woodin cardinals above the rank of ); and (2) Projective Determinacy holds in every set generic extension.
Paper Structure (5 sections, 8 theorems, 28 equations)

This paper contains 5 sections, 8 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Some background
  4. $M_n^\#(X)$
  5. ZFR and mice with some Woodin cardinals

Key Result

Theorem 1.1

Assume $\mathsf{ZF}(j)$ and that there is a Reinhardt cardinal as witnessed by $j$. Then $M_n^\#(X)$ exists and is fully iterable (above $X$) for every transitive set $X$ and every $n<\omega$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 28 more