Plenitudinous Urelements and the Definability of Cardinality

Bokai Yao

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arXiv · 2508.20641

Plenitudinous Urelements and the Definability of Cardinality

Bokai Yao

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Abstract

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$^+$.

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BibTeX RIS

Bokai Yao. 2025-08-28. Plenitudinous Urelements and the Definability of Cardinality. https://arxiv.org/abs/2508.20641

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