On a cofinal Reinhardt embedding without Powerset

TL;DR

This paper proves the consistency of a non-trivial cofinal Reinhardt embedding within ZF^- set theory, building on models with large cardinal assumptions, specifically I_0, and providing a positive answer to a longstanding question.

Contribution

It demonstrates the consistency of a cofinal Reinhardt embedding in ZF^- without relying on the Powerset axiom, using models with large cardinal assumptions like I_0.

Findings

Consistency of cofinal Reinhardt embeddings in ZF^- established

Model construction based on I_0 large cardinal assumption

Extension of Reinhardt embedding theory without Powerset

Abstract

In this paper, we provide a positive answer to the question of Matthews whether $\mathsf{ZF}^-$ is consistent with a non-trivial cofinal Reinhardt elementary embedding $j\colon V\to V$. The consistency follows from $\mathsf{ZFC} + I_0$, and more precisely, it is witnessed by Schlutzenberg's model of $\mathsf{ZF}$ with an elementary embedding $k\colon V_{\lambda+2}\to V_{\lambda+2}$.