Plenitudinous Urelements and the Definability of Cardinality
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911305847275520 |
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| author | Yao, Bokai |
| author_facet | Yao, Bokai |
| contents | The Axiom of Plenitude asserts that every ordinal is equinumerous with a set of urelements, while its stronger form, Plenitude$^+$, extends it to all sets. We investigate these two axioms within ZF set theory with urelements. Assuming that cardinality is definable, Plenitude$^+$ together with the Collection Principle implies the Reflection Principle. If either cardinality is representable or Small Violations of Choice (SVC) holds, Plenitude$^+$ implies the Reflection Principle. In contrast, Plenitude is considerably weaker: SVC + Plenitude does not prove the Collection Principle, and SVC + Plenitude + Reflection Principle does not prove Plenitude$^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Plenitudinous Urelements and the Definability of Cardinality Yao, Bokai Logic 03E25, 03E30, 03E35, 03E65 The Axiom of Plenitude asserts that every ordinal is equinumerous with a set of urelements, while its stronger form, Plenitude$^+$, extends it to all sets. We investigate these two axioms within ZF set theory with urelements. Assuming that cardinality is definable, Plenitude$^+$ together with the Collection Principle implies the Reflection Principle. If either cardinality is representable or Small Violations of Choice (SVC) holds, Plenitude$^+$ implies the Reflection Principle. In contrast, Plenitude is considerably weaker: SVC + Plenitude does not prove the Collection Principle, and SVC + Plenitude + Reflection Principle does not prove Plenitude$^+$. |
| title | Plenitudinous Urelements and the Definability of Cardinality |
| topic | Logic 03E25, 03E30, 03E35, 03E65 |
| url | https://arxiv.org/abs/2508.20641 |