What type of study is this?

This is a Quantitative Study study.

September 24, 2025Open Access

Plenitudinous Urelements and the Definability of Cardinality

Key Points

Plenitude$^+$ indicates that cardinality can unify various principles within ZF set theory.
The relationship between cardinality and Plenitude$^+$ leads to interesting implications for the Reflection Principle.
Models exist where Plenitude fails while SVC remains true, showcasing its limitations.
Understanding these axioms enhances insights into foundational aspects of set theory.

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^+ 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^+ 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^+ fails.

Read Full Paperexternally

Demander à l'IA

Bookmark

View Full Paper

Cite This Study

Bokai Yao (Thu,) studied this question.

synapsesocial.com/papers/68d6e0fc8b2b6861e4c3f535 https://doi.org/https://doi.org/10.48550/arxiv.2508.20641