Reinhardt Cardinals and Eventually Dominating Functions

TL;DR

This work analyzes Reinhardt embeddings $j:V\to V$ in a $ ext{ZF}(j)$ (no AC) setting, exploring how $j$ interacts with functions that eventually dominate it on regular cardinals. The authors develop root and closure constructions to uncover an extendibility-like behavior inside $V_\delta$ and prove a no-dominating-function theorem for regular $\delta>\lambda$ with $j(\delta)=\delta$, while showing no $(j,\delta)$-small sets exist. They connect these local extendibility phenomena to large-cardinal concepts such as almost supercompact and almost extendible, following Goldberg’s results, and derive consequences for the regular-cardinal landscape above $\lambda$, including a club of regular cardinals. Finally, they provide an AC-based route to Kunen inconsistency by producing small sets, yielding an alternative proof that $ ext{ZFC}(j)$ is inconsistent. Overall, the paper links local embedding behavior to global large-cardinal structure in the absence of Choice and furnishes a new perspective on Kunen-type results in this setting.

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.

Figures (2)

• Figure 1: Almost supercompactness
• Figure 2: Almost extendibility

Theorems & Definitions (43)

• Definition 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof