Class-Forcing in Class Theory

TL;DR

This paper extends the concept of class-forcing from ZFC models to Morse-Kelley class theory models, providing rigorous definitions and analyzing preservation of axioms, while also showing limitations of Laver's Theorem in this context.

Contribution

It introduces a formal framework for class-forcing in Morse-Kelley models and proves key lemmas without restrictions, highlighting differences from ZFC-based forcing.

Findings

Definability and Truth Lemmas hold without restrictions

Axioms are preserved under certain conditions

Laver's Theorem does not hold for class-forcing in this setting

Abstract

In this article we adapt the existing account of class-forcing over a ZFC model to a model $(M,\mathcal{C})$ of Morse-Kelley class theory. We give a rigorous definition of class-forcing in such a model and show that the Definability Lemma (and the Truth Lemma) can be proven without restricting the notion of forcing. Furthermore we show under which conditions the axioms are preserved. We conclude by proving that Laver's Theorem does not hold for class-forcings.