Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The uniqueness of elementary embeddings

Gabriel Goldberg

Abstract

Much of the theory of large cardinals beyond a measurable cardinal concerns the structure of elementary embeddings of the universe of sets into inner models. This paper seeks to answer the question of whether the inner model uniquely determines the elementary embedding.

The uniqueness of elementary embeddings

Abstract

Much of the theory of large cardinals beyond a measurable cardinal concerns the structure of elementary embeddings of the universe of sets into inner models. This paper seeks to answer the question of whether the inner model uniquely determines the elementary embedding.

Paper Structure

This paper contains 14 sections, 62 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ultrapower embeddings and extender embeddings
  4. The uniqueness of embeddings on the ordinals
  5. Woodin's results
  6. Uniqueness of embeddings on the ordinals
  7. The uniqueness of embeddings above large cardinals
  8. Completely definable sets
  9. Embeddings above an extendible cardinal
  10. Application: the Kunen inconsistency for ultrapowers
  11. Weaker hypotheses
  12. Ultrapower axioms
  13. The Ultrapower Axiom
  14. The Weak Ultrapower Axiom

Key Result

Lemma 2.4

Suppose $i : P\to Q$ is an elementary embedding, $a$ is a point in $Q$, and $\lambda = \lambda_i(a)$. If $\lambda > 1$, then $i(\lambda) \neq \sup i[\lambda]$.∎

Theorems & Definitions (123)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Theorem 3.1: Woodin
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3: Woodin
  • ...and 113 more