Plenitudinous Urelements and the Definability of Cardinality

Abstract: The Axiom of Plenitude asserts that every ordinal is equinumerous with a set of urelements, while its stronger form, Plenitude$<sup>+$, extends it to all sets. We investigate these two axioms within ZF set theory with urelements. Assuming that cardinality is definable, Plenitude$<sup>+$ unifies the Collection Principle and the Reflection Principle, which are otherwise conjectured to be non-equivalent. If either cardinality is representable or the principle of Small Violations of Choice (SVC) holds, Plenitude$<sup>+$ implies the Reflection Principle. In contrast, Plenitude is considerably weaker: there are models of SVC + Plenitude where Collection fails, and models of SVC + Plenitude + Reflection where Plenitude$<sup>+$ fails.