Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Critical embeddings

Asaf Karagila, Jiachen Yuan

Abstract

Hayut and first author isolated the notion of a critical cardinal in [1]. In this work we answer several questions raised in the original paper. We show that it is consistent for a critical cardinals to not have any ultrapower elementary embeddings, as well as that it is consistent that no target model is closed. We also prove that if $κ$ is a critical point by any ultrapower embedding, then it is the critical point by a normal ultrapower embedding. The paper contains several open questions of interest in the study of critical cardinals.

Critical embeddings

Abstract

Hayut and first author isolated the notion of a critical cardinal in [1]. In this work we answer several questions raised in the original paper. We show that it is consistent for a critical cardinals to not have any ultrapower elementary embeddings, as well as that it is consistent that no target model is closed. We also prove that if is a critical point by any ultrapower embedding, then it is the critical point by a normal ultrapower embedding. The paper contains several open questions of interest in the study of critical cardinals.
Paper Structure (9 sections, 10 theorems, 2 equations)

This paper contains 9 sections, 10 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric extensions
  4. Lifting theorem
  5. Choice in measure
  6. Main theorems
  7. No ultrapower maps are elementary
  8. No closed target models
  9. Questions

Key Result

Lemma 1

Let $p\in\mathbb P$, $\pi\in\mathop{\mathrm{Aut}}\nolimits(\mathbb P)$, and $\dot x$ a $\mathbb P$-name. $p\mathrel{\Vdash}\varphi(\dot x)\iff\pi p\mathrel{\Vdash}\varphi(\pi\dot x).$∎

Theorems & Definitions (16)

  • Lemma
  • Theorem
  • Theorem 2.1
  • Theorem 2.2: Simplified lifting theorem
  • Definition 2.3
  • Theorem 2.4: Spector Spector:1988
  • Corollary 2.5
  • Theorem 2.6
  • proof
  • Corollary 2.7
  • ...and 6 more