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Global choice is not conservative over local choice for Zermelo set theory

Elliot Glazer

Abstract

We prove that Global Choice is not conservative over ZC and that ZF $-$ Union does not prove existence of $x \cup y$ for all $x$ and $y$. Each proof is by constructing a pathological model inside a symmetric extension of $L.$

Global choice is not conservative over local choice for Zermelo set theory

Abstract

We prove that Global Choice is not conservative over ZC and that ZF Union does not prove existence of for all and . Each proof is by constructing a pathological model inside a symmetric extension of
Paper Structure (5 sections, 14 theorems, 31 equations)

This paper contains 5 sections, 14 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Sketch of a failure of conservativity of Global Choice
  3. Construction of $\mathcal{M}$
  4. Adding in Foundation
  5. Failure of the Finite Union Principle

Key Result

Theorem 1.1

There is a sentence in the language $\{\in\}$ which is provable in $\mathrm{ZC} + \mathrm{GC}$ and not provable in $\mathrm{ZC}.$

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Definition 3.1
  • Proposition 3.3
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.7
  • ...and 18 more