# Plenitudinous Urelements and the Definability of Cardinality

## TL;DR

This paper explores the implications of the Axiom of Plenitude and its stronger form within ZF set theory with urelements, focusing on their relation to the Reflection Principle and conditions like definability of cardinality.

## Contribution

It establishes the logical relationships between Plenitude axioms, the Reflection Principle, and other set-theoretic assumptions such as definability of cardinality and SVC.

## Key findings

- Plenitude$^+$ plus Collection implies Reflection.
- If cardinality is definable or SVC holds, Plenitude$^+$ implies Reflection.
- Plenitude alone is weaker and does not imply Collection or Reflection.

## 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$^+$.