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The number of measures on very large measurable cardinals

Arthur W. Apter, Eyal Kaplan, Alejandro Poveda

Abstract

We study the possible number of normal measures on a measurable cardinal in settings where inner model techniques are unavailable. Instead, we exploit consequences of the Ultrapower Axiom to obtain our theorems. We show that the classical Kimchi-Magidor result -that the first $n$ measurable cardinals can be strongly compact- can be combined with an arbitrary prescribed pattern for the number of normal measures they carry. We also prove that the first measurable cardinal above a supercompact cardinal can carry any given number of normal measures; the same conclusion is established for the first measurable limit of supercompact cardinals. As further applications of our techniques, we strengthen an unpublished theorem of Goldberg--Woodin and a theorem of Goldberg, Osinski, and Poveda. Our analysis circumvents both the reliance of Friedman--Magidor on core model methods and the limitations of the Prikry-type forcing iterations of Gitik--Kaplan.

The number of measures on very large measurable cardinals

Abstract

We study the possible number of normal measures on a measurable cardinal in settings where inner model techniques are unavailable. Instead, we exploit consequences of the Ultrapower Axiom to obtain our theorems. We show that the classical Kimchi-Magidor result -that the first measurable cardinals can be strongly compact- can be combined with an arbitrary prescribed pattern for the number of normal measures they carry. We also prove that the first measurable cardinal above a supercompact cardinal can carry any given number of normal measures; the same conclusion is established for the first measurable limit of supercompact cardinals. As further applications of our techniques, we strengthen an unpublished theorem of Goldberg--Woodin and a theorem of Goldberg, Osinski, and Poveda. Our analysis circumvents both the reliance of Friedman--Magidor on core model methods and the limitations of the Prikry-type forcing iterations of Gitik--Kaplan.
Paper Structure (12 sections, 34 theorems, 74 equations)

This paper contains 12 sections, 34 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalities on nonstationary support iterations
  4. The splitting forcing
  5. Laver's indestructibility via nonstationary support iterations
  6. Forcing nonreflecting stationary sets
  7. The main theorems
  8. The first finitely many measurable cardinals
  9. The first measurable above a supercompact cardinal
  10. The first measurable limit of supercompact cardinals
  11. A generalization of the Goldberg--Woodin Theorem
  12. A comment on Woodin's HOD hypothesis

Key Result

Theorem \ref{thm: IdentityCrises + Normal measures}

Assume the $\mathrm{GCH}$ holds, there are $n<\omega$ supercompact cardinals $\langle \kappa_i: i<n\rangle$, there are no measurable cardinals above $\kappa_{n-1}$, and each $\kappa_i$ has a unique normal measure of Mitchell order $0$. For every $i<n$, let $\tau_i\leq\kappa^{++}_i$ be a cardinal. Th

Theorems & Definitions (85)

  • Theorem \ref{thm: IdentityCrises + Normal measures}
  • Theorem \ref{thm: above supercompact}
  • Theorem \ref{thm: first limit of supercompacts}
  • Theorem \ref{thm: Goldberg-Woodin}
  • Theorem \ref{thm: Hod hypothesis}
  • Definition 2.1: $\alpha$-strategic closure
  • Lemma 2.2: The fusion lemma
  • proof
  • Claim 2.2.1
  • proof : Proof of claim
  • ...and 75 more