Strong Rigidity and Elementary Embeddings

TL;DR

This paper introduces a method to generate elementary embeddings from homomorphisms, explores the strongly rigid relation principle as a weak choice principle independent of ZF, and characterizes proto Berkeley cardinals through its failure.

Contribution

It presents a novel method for producing elementary embeddings and establishes the independence of the strongly rigid relation principle from ZF, linking it to proto Berkeley cardinals.

Findings

The strongly rigid relation principle is a weak choice principle independent of ZF.

A new method for producing elementary embeddings from homomorphisms.

Characterization of proto Berkeley cardinals via the failure of the principle.

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.