Inconsistency of Reinhardt cardinals with $\mathsf{ZF}$

TL;DR

This paper proves that the existence of a certain type of non-trivial elementary embedding in set theory is inconsistent with ZF, clarifying the limitations of large cardinal hypotheses.

Contribution

It establishes the inconsistency of specific Reinhardt cardinals with ZF and introduces new large cardinal properties with intermediate consistency strength.

Findings

Non-trivial $oldsymbol{ ext{Sigma}_1}$-elementary embeddings are inconsistent with ZF.

Introduces new large cardinal properties between $oldsymbol{ ext{I}_3}$ and $oldsymbol{ ext{I}_2}$.

Proves the main inconsistency result within ZF framework.

Abstract

A proof will be presented that the existence of a non-trivial $\Sigma_1$-elementary embedding $j: V_{\lambda+3} \prec V_{\lambda+3}$ is inconsistent with $\textsf{ZF}$. Sections 1 and 2 shall review various important contributions from the literature, notably including \cite{Goldberg2020}, \cite{Schlutzenberg2020}, and \cite{Woodin2010}, the latter reference being where the crucial forcing construction is presented. Section 3 shall introduce some new large cardinal properties, of consistency strength intermediate between $\mathsf{I_3}$ and $\mathsf{I_2}$, and greater than $\mathsf{I_1}$, respectively. The proof of the inconsistency with $\mathsf{ZF}$ of the existence of a non-trivial $\Sigma_1$-elementary embedding $j:V_{\lambda+3} \prec V_{\lambda+3}$ shall be given in Section 4. The claims of Sections 2 and 4 are provable in $\textsf{ZF}$; those of Section 3, with the exception of the last two theorems, in $\textsf{ZFC}$.