Reflection Principles in ZFU

TL;DR

This paper analyzes various reflection principles in ZFU set theory, demonstrating their independence and equivalences under certain axioms, thus clarifying their relationships without relying on the Axiom of Choice.

Contribution

It distinguishes and separates the Collection, Reflection, and Partial Reflection principles in ZFU, showing their independence and conditions for equivalence.

Findings

Collection and Partial Reflection are independent in ZFU.

Collection plus Partial Reflection does not imply Reflection.

Reflection and Collection are equivalent under Tail or Small Violations of Choice.

Abstract

We separate the Collection Principle, the Reflection Principle, and the Partial Reflection Principle in ZF with urelements (ZFU), despite their equivalence under the Axiom of Choice. In particular, Collection and the Partial Reflection Principle are independent of one another, and Collection together with Partial Reflection does not imply the Reflection Principle. We show that Reflection and Collection are equivalent assuming either the Tail axiom or Small Violations of Choice.