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A Note on a Theorem of Apter

Rahman Mohammadpour, Otto Rajala, Sebastiano Thei

Abstract

We show that the consistency of $\mathrm{ZF} + \mathrm{AD}_{\mathbb{R}} + ``Θ$ is measurable$"$ implies the consistency of $\mathrm{ZF} +``Θ$ is the least strongly regular cardinal and the least measurable cardinal$"$ + $``$all uncountable cardinals below $Θ$ are of countable cofinality$"$.

A Note on a Theorem of Apter

Abstract

We show that the consistency of is measurable implies the consistency of is the least strongly regular cardinal and the least measurable cardinal + all uncountable cardinals below are of countable cofinality.
Paper Structure (4 sections, 12 theorems, 14 equations)

This paper contains 4 sections, 12 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Basics
  3. The main theorem
  4. Open Questions

Key Result

Theorem 1.1

Assume $\textsf{ZF}+ \Theta_{\sf meas}$. There is an extension satisfying $\textsf{ZF}$, in which $\Theta$ is strongly regular cardinal, and simultaneously, the least uncountable regular cardinal and the least measurable cardinal.

Theorems & Definitions (24)

  • Theorem 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Theorem 2.2: Apter Apter_AD_pattern, ZF
  • proof
  • Definition 2.3
  • Lemma 2.4: Apter Apter_AD_pattern, ZF
  • proof
  • Lemma 3.1
  • proof
  • ...and 14 more