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No cardinal correct inner model elementarily embeds into the universe

Gabriel Goldberg, Sebastiano Thei

Abstract

An elementary embedding $j:M\rightarrow N$ between two inner models of ZFC is cardinal preserving if $M$ and $N$ correctly compute the class of cardinals. We look at the case $N=V$ and show that there is no nontrivial cardinal preserving elementary embedding from $M$ into $V$, answering a question of Caicedo.

No cardinal correct inner model elementarily embeds into the universe

Abstract

An elementary embedding between two inner models of ZFC is cardinal preserving if and correctly compute the class of cardinals. We look at the case and show that there is no nontrivial cardinal preserving elementary embedding from into , answering a question of Caicedo.

Paper Structure

This paper contains 10 sections, 20 theorems.

Table of Contents

  1. Introduction
  2. Forcing axioms
  3. Singular cardinal combinatorics
  4. Weak forms of square.
  5. Jónsson cardinals.
  6. Discontinuities
  7. The main result
  8. Factor embeddings.
  9. Good scales at $\lambda$.
  10. Embeddings into $V$.

Key Result

Theorem 1.1

There is no nontrivial elementary embedding from the universe to itself.

Theorems & Definitions (41)

  • Theorem 1.1: Kunen, MR0311478
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.5: goldberg2021note, Theorem 2.10
  • Theorem 1.6: Ca, Theorem 2.11
  • Conjecture 2.1: Caicedo, Veličković
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4: MR2385636, Corollary 28
  • ...and 31 more