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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2601.10231 |
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| _version_ | 1866918333255778304 |
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| author | McCallum, Rupert |
| author_facet | McCallum, Rupert |
| contents | 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 $\textsf{ZFC}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10231 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inconsistency of Reinhardt cardinals with $\mathsf{ZF}$ McCallum, Rupert Logic 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 $\textsf{ZFC}$. |
| title | Inconsistency of Reinhardt cardinals with $\mathsf{ZF}$ |
| topic | Logic |
| url | https://arxiv.org/abs/2601.10231 |