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Maximal Prikry Sequences

Ernest Schimmerling, Jiaming Zhang

Abstract

In this paper we investigate the covering machinery of the Mitchell-Steel core model $K$, under the hypothesis that there is no inner model with a Woodin cardinal. In an earlier work, Mitchell and the first author showed that if $ν>ω_2$ is a regular cardinal in $K$ but a singular ordinal in $V$, then $ν$ is a measurable cardinal in $K$. In this article, we further show that under certain circumstances, there exists a maximal Prikry sequence $C$ for a measure on $ν$ in $K$. The first author shows that the anti-large cardinal hypothesis is necessary. In a more restrictive setting, we prove that every subset of $ν$ with size $<|ν|$ can be covered by a set in $K[C]$ with size $<|ν|$. Benhemou and the first author show that the result is optimal.

Maximal Prikry Sequences

Abstract

In this paper we investigate the covering machinery of the Mitchell-Steel core model , under the hypothesis that there is no inner model with a Woodin cardinal. In an earlier work, Mitchell and the first author showed that if is a regular cardinal in but a singular ordinal in , then is a measurable cardinal in . In this article, we further show that under certain circumstances, there exists a maximal Prikry sequence for a measure on in . The first author shows that the anti-large cardinal hypothesis is necessary. In a more restrictive setting, we prove that every subset of with size can be covered by a set in with size . Benhemou and the first author show that the result is optimal.
Paper Structure (13 sections, 52 theorems, 266 equations)

This paper contains 13 sections, 52 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. A Brief Review of MS2
  3. Preliminaries
  4. Analysis of $\mathcal{Q}_\alpha$ and $\mathcal{S}_\alpha$
  5. The $\mathcal{S}_{XY}$-structure
  6. Proof of Theorem \ref{['T2-1']}.
  7. $o^K(\nu) = 1$ with Closure
  8. Stationary tower example
  9. Tree Prikry example
  10. $o^K(\nu) = 1$ without Closure
  11. $o^K(\nu)>1$ with closure
  12. First correction
  13. Second correction

Key Result

Theorem 1

Assume that $0^\dagger$ does not exist. Let $\nu$ be a singular cardinal. Suppose that $\nu$ is regular in $K$. Then 1) there is a normal measure $U$ over $\nu$ in $K$, 2) there is a maximal Prikry sequence $C$ for $U$, and 3) for every $A \subseteq \nu$ such that $|A| < \nu$, there exists $B \in K[

Theorems & Definitions (100)

  • Theorem : Dodd-Jensen
  • Theorem : Mitchell-Schimmerling
  • Theorem 1.1
  • Definition 1
  • Theorem 1.2: Zhang
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2: MSS, MS2
  • Theorem 2.3: MSS, MS2
  • Lemma 2.4: Lemma 2.1, MS2
  • ...and 90 more