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Towards a theory of symmetric extensions

Asaf Karagila, Jonathan Schilhan

Abstract

The technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique remained fairly limited compared to the theory of forcing. Whereas forcing developed products and iterations, no serious attempts at developing any general framework for iterating symmetric extensions were presented before [10], where only finite support iterations are treated. In this paper we develop the theory of symmetric extensions including different types of iterations, quotients, equivalents, and the structural results that can be described in this language. In particular, we give a modern exposition to some of the important theorems of Grigorieff [3], study Kinna--Wagner Principles in symmetric extensions, and show that it is provable from $\mathsf{ZF}$ that every set lies in a symmetric extension of $\operatorname{HOD}$.

Towards a theory of symmetric extensions

Abstract

The technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique remained fairly limited compared to the theory of forcing. Whereas forcing developed products and iterations, no serious attempts at developing any general framework for iterating symmetric extensions were presented before [10], where only finite support iterations are treated. In this paper we develop the theory of symmetric extensions including different types of iterations, quotients, equivalents, and the structural results that can be described in this language. In particular, we give a modern exposition to some of the important theorems of Grigorieff [3], study Kinna--Wagner Principles in symmetric extensions, and show that it is provable from that every set lies in a symmetric extension of .
Paper Structure (24 sections, 66 theorems, 107 equations)

This paper contains 24 sections, 66 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Symmetric systems and extensions
  3. Symmetric systems
  4. Symmetric extensions
  5. Closed names
  6. Two step iterations
  7. Ideal support iterations
  8. General iterations
  9. Ideal support iterations
  10. Factorization
  11. Preservation theorems
  12. Equivalence of symmetric systems
  13. Classes of names
  14. Notions of strong equivalence
  15. Products and reduced two-step iterations
  16. ...and 9 more sections

Key Result

Theorem 2.7

Let $G$ be a $\mathbb{P}$-generic filter over $V$. Then $V[G]_\mathcal{S} \models \mathsf{ZF}$.

Theorems & Definitions (152)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: see, e.g., Jech1973
  • Definition 2.8
  • Theorem 2.9: Symmetric Forcing Theorem
  • proof
  • ...and 142 more