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

Rupert McCallum

TL;DR

The paper proves that the existence of a non-trivial $Σ_1$-elementary embedding $j:V_{λ+3} prec V_{λ+3}$ is inconsistent with $ extsf{ZF}$, using Woodin-style forcing and inverse-limit reflection to force a model where $ extsf{ZFC+I_{0}}$ holds below a critical sequence and then deriving a contradiction via a $oldsymbol{ω}$-Jónsson argument. It introduces new large-cardinal notions—$oldsymbol{α}$-tremendous, hyper-tremendous, $oldsymbol{α}$-enormous, and hyper-enormous—placing their consistency strength between $ extsf{I_3}$ and $ extsf{I_2}$ (and above $ extsf{I_1}$ for the enormouses), while examining the limits of $ extsf{I_0}$ relative to these notions. The work also provides $ extsf{ZF}$-provable variants of Theorem 6.19 that reduce or remove the need for $ ext{DC}$, clarifying the dependence on the rank of the embedding. Consequently, most of the choiceless large-cardinal hierarchy becomes incompatible with $ extsf{ZF}$, leaving only a narrowly defined fragment (e.g., a potential $V_{λ+2} prec V_{λ+2}$ under $ extsf{ZFC+I_0}$) that aligns with existing equiconsistency results. Overall, the paper sharpens the boundary between choiceless large-cardinals and $ extsf{ZF}$, illustrating how forcing plus reflection arguments yield decisive inconsistencies for Reinhardt-type hypotheses.

Abstract

A proof will be presented that the existence of a non-trivial $Σ_1$-elementary embedding $j: V_{λ+3} \prec V_{λ+3}$ 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 $Σ_1$-elementary embedding $j:V_{λ+3} \prec V_{λ+3}$ 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 $\textsf{ZFC}$.

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

TL;DR

The paper proves that the existence of a non-trivial -elementary embedding is inconsistent with , using Woodin-style forcing and inverse-limit reflection to force a model where holds below a critical sequence and then deriving a contradiction via a -Jónsson argument. It introduces new large-cardinal notions—-tremendous, hyper-tremendous, -enormous, and hyper-enormous—placing their consistency strength between and (and above for the enormouses), while examining the limits of relative to these notions. The work also provides -provable variants of Theorem 6.19 that reduce or remove the need for , clarifying the dependence on the rank of the embedding. Consequently, most of the choiceless large-cardinal hierarchy becomes incompatible with , leaving only a narrowly defined fragment (e.g., a potential under ) that aligns with existing equiconsistency results. Overall, the paper sharpens the boundary between choiceless large-cardinals and , illustrating how forcing plus reflection arguments yield decisive inconsistencies for Reinhardt-type hypotheses.

Abstract

A proof will be presented that the existence of a non-trivial -elementary embedding is inconsistent with . 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 and , and greater than , respectively. The proof of the inconsistency with of the existence of a non-trivial -elementary embedding shall be given in Section 4. The claims of Sections 2 and 4 are provable in ; those of Section 3, with the exception of the last two theorems, in .
Paper Structure (4 sections, 7 theorems)

This paper contains 4 sections, 7 theorems.

Table of Contents

  1. Introduction
  2. Background from existing literature
  3. Some new large-cardinal concepts, and determination of their consistency strength
  4. Proof of inconsistency of Reinhardt cardinals with $\mathsf{ZF}$

Key Result

Theorem 2.1

Suppose $\lambda$ is an ordinal and there is a $\Sigma_1$-elementary embedding $j:V_{\lambda+3} \prec V_{\lambda+3}$ with $\lambda$ equal to the supremum of the critical sequence of $j$. Assume $\mathrm{DC}_{V_{\lambda+1}}$. Then there is a set generic extension $N$ of $V$ such that if $\delta<\lamb

Theorems & Definitions (23)

  • Theorem 2.1: Theorem 6.19 of Goldberg2020
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Theorem 226 of Woodin2010
  • Definition 2.6
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 13 more