Satisfaction relations for proper classes: Applications in logic and set theory

TL;DR

This paper develops a framework for partial satisfaction relations applicable to proper classes, enabling new insights in the metatheory of logic and set theory, including conservative extension results and applications to forcing.

Contribution

It introduces a satisfaction predicate for proper classes and demonstrates its utility in proving conservative extensions and analyzing forcing in set theory.

Findings

Existence of finitely axiomatizable conservative extensions of ZF

Development of a satisfaction predicate for proper classes

Application to characterizing ground models in forcing

Abstract

We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension T of ZF (Zermelo-Fraenkel set theory) there is a finitely axiomatizable extension T' of GB (von Neumann-Bernays-G\"odel class theory) that is a conservative extension of T. We also prove a conservative extension result that justifies the use of this predicate to characterize ground models for forcing constructions.