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A General Theory of Class Symmetric Systems

Peter Holy, Emma Palmer, Jonathan Schilhan

Abstract

We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present sufficient conditions for the preservation of the axioms of Gödel-Bernays set theory (without the axiom of choice), and for the forcing theorem to hold for class-sized symmetric systems.

A General Theory of Class Symmetric Systems

Abstract

We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present sufficient conditions for the preservation of the axioms of Gödel-Bernays set theory (without the axiom of choice), and for the forcing theorem to hold for class-sized symmetric systems.

Paper Structure

This paper contains 7 sections, 22 theorems, 44 equations.

Table of Contents

  1. Introduction and Basic Definitions
  2. Atomic Forcing Relations
  3. Forcing Relations
  4. The truth lemma
  5. Preserving GB⁻
  6. Preserving the powerset axiom
  7. Final remarks and open questions

Key Result

Lemma 5

Any two forcing relations $\mathop{\mathrm{\mathop{\mathrm{\Vdash}}\nolimits_0}}\nolimits,\mathop{\mathrm{\mathop{\mathrm{\Vdash}}\nolimits_0}}\nolimits'$ for atomic formulas (for $\mathcal{S}$) will agree on all tuples $\langle p,\dot x,R,\dot y\rangle$ for $p\in P$, $R\in\{\in,\subseteq,=\}$, and

Theorems & Definitions (57)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 47 more