Sufficient conditions for the forcing theorem, and turning proper classes into sets

TL;DR

This paper explores conditions under which class forcing satisfies the forcing theorem and demonstrates how certain properties prevent proper classes from becoming sets in generic extensions.

Contribution

It introduces three combinatorial properties that imply the forcing theorem and links these to the preservation of proper classes as classes in extensions.

Findings

All known sufficient conditions imply proper classes remain classes in extensions.

A class forcing can turn a proper class into a set without satisfying the forcing theorem.

The property of not turning proper classes into sets characterizes pretameness in certain models.

Abstract

We present three natural combinatorial properties for class forcing notions, which imply the forcing theorem to hold. We then show that all known sufficent conditions for the forcing theorem (except for the forcing theorem itself), including the three properties presented in this paper, imply yet another regularity property for class forcing notions, namely that proper classes of the ground model cannot become sets in a generic extension, that is they do not have set-sized names in the ground model. We then show that over certain models of G\"odel-Bernays set theory without the power set axiom, there is a notion of class forcing which turns a proper class into a set, however does not satisfy the forcing theorem. Moreover, we show that the property of not turning proper classes into sets can be used to characterize pretameness over such models of G\"odel-Bernays set theory.