Plenitudinous Urelements and the Definability of Cardinality
Logic
2025-12-09 v2
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.
Keywords
Cite
@article{arxiv.2508.20641,
title = {Plenitudinous Urelements and the Definability of Cardinality},
author = {Bokai Yao},
journal= {arXiv preprint arXiv:2508.20641},
year = {2025}
}
Comments
Lemma 15 of the previous draft is incorrect. See footnote 3 of the current version