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Strong Rigidity and Elementary Embeddings

Marwan Salam Mohammd

TL;DR

This work investigates the strongly rigid relation principle (SRR) and its relation to choice principles. It develops a method to obtain nontrivial elementary embeddings from graph homomorphisms and extends this approach to subsets of $\mathbb{R}\times \mathrm{OR}$, using these embeddings to show SRR is independent of $\mathrm{ZF}$ and does not imply $\mathrm{AC}$. It then constructs a model of $\mathrm{ZF}+\neg\mathrm{AC}+\mathrm{SRR}$ via a symmetric Cohen forcing argument, confirming the independence result, and finally characterizes proto Berkeley cardinals through a homomorphism/embedding lens, linking large-cardinal notions to rigidity in graphs. Together, these results clarify the strength of SRR as a weak choice principle and illuminate its connections to elementary embeddings and large-cardinal concepts, with practical implications for understanding the landscape between rigidity phenomena and choice axioms.

Abstract

We present a method for producing elementary embeddings from homomorphisms. This method is utilized in the study of the "strongly rigid relation principle" as defined by Hamkins and Palumbo in their paper "The Rigid Relation Principle, a New Weak Choice Principle." We establish that the strongly rigid relation principle is also a weak choice principle that is independent of ZF. Finally, we characterize proto Berkeley cardinals in terms of a strong failure of the strongly rigid relation principle.

Strong Rigidity and Elementary Embeddings

TL;DR

This work investigates the strongly rigid relation principle (SRR) and its relation to choice principles. It develops a method to obtain nontrivial elementary embeddings from graph homomorphisms and extends this approach to subsets of , using these embeddings to show SRR is independent of and does not imply . It then constructs a model of via a symmetric Cohen forcing argument, confirming the independence result, and finally characterizes proto Berkeley cardinals through a homomorphism/embedding lens, linking large-cardinal notions to rigidity in graphs. Together, these results clarify the strength of SRR as a weak choice principle and illuminate its connections to elementary embeddings and large-cardinal concepts, with practical implications for understanding the landscape between rigidity phenomena and choice axioms.

Abstract

We present a method for producing elementary embeddings from homomorphisms. This method is utilized in the study of the "strongly rigid relation principle" as defined by Hamkins and Palumbo in their paper "The Rigid Relation Principle, a New Weak Choice Principle." We establish that the strongly rigid relation principle is also a weak choice principle that is independent of ZF. Finally, we characterize proto Berkeley cardinals in terms of a strong failure of the strongly rigid relation principle.
Paper Structure (5 sections, 10 theorems, 8 equations, 8 figures)

This paper contains 5 sections, 10 theorems, 8 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Countable Sets and Ordinal Numbers
  3. Subsets of $\mathbb{R}\times\mathrm{OR}$
  4. A Model for $\neg \mathrm{AC}+\mathrm{SRR}$
  5. Proto Berkeley Cardinals

Key Result

Theorem 1.3

$\mathrm{RR}$ is independent from $\mathrm{ZF},$ follows from $\mathrm{AC},$ but is not equivalent to $\mathrm{AC}.$

Figures (8)

  • Figure 1: A strongly rigid graph on a countable set
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 3 more figures

Theorems & Definitions (22)

  • Definition 1.1
  • Definition 1.2: hamkins
  • Theorem 1.3: vopenka and hamkins
  • Theorem 1.4
  • Definition 1.5: lcbc
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • ...and 12 more