A General Theory of Class Symmetric Systems

TL;DR

This paper develops a comprehensive theory for class-sized symmetric systems, extending existing notions to ensure the preservation of G"odel-Bernays set theory axioms and the forcing theorem.

Contribution

It introduces new conditions for class forcing that guarantee axiom preservation and the validity of the forcing theorem in class-sized symmetric systems.

Findings

Established sufficient conditions for axiom preservation in class forcing.

Proved the forcing theorem holds for class-sized symmetric systems under these conditions.

Extended symmetric system theory to class-sized contexts.

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 G\"odel-Bernays set theory (without the axiom of choice), and for the forcing theorem to hold for class-sized symmetric systems.