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

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 $\text{ZF}$. 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 $\text{ZF}$; those of Section 3, with the exception of the last two theorems, in $\text{ZFC}$.

arXiv commentAn issue has been identified with the claim about continuity of Skolem functions, in the final section of the paper. Although I will attempt to repair this problem, I thought it best to indicate that an error has been found