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Reflection Principles in ZFU

Elliot Glazer, Bokai Yao

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.

Reflection Principles in ZFU

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.
Paper Structure (10 sections, 35 theorems, 18 equations, 1 figure)

This paper contains 10 sections, 35 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ZFU results
  4. Tail $\land$ Collection $\to$ RP
  5. SVC $\land$ Collection $\to$ RP
  6. Independence results
  7. Permutation models with small kernels
  8. Collection $\not\to$ RP$^-$
  9. Collection $\land$ RP$^-$ $\not\to$ RP$^\sim$
  10. RP$^\sim$ $\land$ Tail $\land$ SVC $\not\to$ Collection

Key Result

Theorem 1

The following are equivalent over ZFCU.

Figures (1)

  • Figure :

Theorems & Definitions (82)

  • Theorem
  • Proposition 1
  • proof
  • Definition 2
  • Definition 3
  • Lemma 4
  • proof
  • Definition 5
  • Lemma 6
  • proof
  • ...and 72 more