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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2510.02620 |
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| _version_ | 1866909832129282048 |
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| author | Klazar, Martin |
| author_facet | Klazar, Martin |
| contents | A digraph $D=\langle V,E\rangle$ ($E\subset V\times V$) is Cantor if Cantor's theorem - for no set there is a surjection from it to its power set - holds in $D$, in the sense we explain. We construct a ZF formula $φ$ with length $494$ such that $D\modelsφ$ iff $D$ is Cantor. In order to obtain $φ$, which is a word over the alphabet $$ \{x_1,\,x_2,\,\dots\}\cup \{\in,\,=,\,\neg, \,\to,\,\leftrightarrow,\,\wedge,\, \vee,\,\exists,\,\forall,\,(,\,)\}\,, $$ we devise abbreviation schemes of ZF formulas. We introduce extensive and strongly extensive digraphs and show, by the standard argument, that they are Cantor. We construct a countable strongly extensive digraph with arbitrarily large finite in-degrees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cantor digraphs and abbreviations of formulas Klazar, Martin Logic Combinatorics 03B10 A digraph $D=\langle V,E\rangle$ ($E\subset V\times V$) is Cantor if Cantor's theorem - for no set there is a surjection from it to its power set - holds in $D$, in the sense we explain. We construct a ZF formula $φ$ with length $494$ such that $D\modelsφ$ iff $D$ is Cantor. In order to obtain $φ$, which is a word over the alphabet $$ \{x_1,\,x_2,\,\dots\}\cup \{\in,\,=,\,\neg, \,\to,\,\leftrightarrow,\,\wedge,\, \vee,\,\exists,\,\forall,\,(,\,)\}\,, $$ we devise abbreviation schemes of ZF formulas. We introduce extensive and strongly extensive digraphs and show, by the standard argument, that they are Cantor. We construct a countable strongly extensive digraph with arbitrarily large finite in-degrees. |
| title | Cantor digraphs and abbreviations of formulas |
| topic | Logic Combinatorics 03B10 |
| url | https://arxiv.org/abs/2510.02620 |