Reinhardt Cardinals and Eventually Dominating Functions

TL;DR

This paper investigates Reinhardt embeddings and eventually dominating functions, establishing new results on their properties, conditions for their existence, and implications for large cardinals and the Kunen inconsistency.

Contribution

It introduces novel results on Reinhardt embeddings and their relation to large cardinals, providing new proofs and insights into their structure and limitations.

Findings

Existence of Reinhardt embeddings with specific properties

Connections between large cardinals and Reinhardt embeddings

Alternative proof of the Kunen inconsistency

Abstract

We prove a result concerning elementary embeddings of the set-theoretic universe into itself (Reinhardt embeddings) and functions on ordinals that "eventually dominate" such embeddings. We apply that result to show the existence of elementary embeddings satisfying some strict conditions and that are also reminiscent of extendibility in a more local setting. Building further on these concepts, we make precise the nature of some large cardinals whose existence under Reinhardt embeddings was proven by Gabriel Goldberg in his paper "Measurable Cardinals and Choiceless Axioms." Finally, these ideas are used to present another proof of the Kunen inconsistency.