Class forcing, the forcing theorem and Boolean completions

TL;DR

This paper investigates the limitations of the forcing theorem in class forcing, showing it can fail and exploring its connection to Boolean completions and a new combinatorial property, with implications for the uniqueness of Boolean completions.

Contribution

It demonstrates the failure of the forcing theorem in class forcing, establishes its equivalence to Boolean completions, and introduces approachability by projections as a sufficient condition.

Findings

Forcing theorem can fail for class forcing.

Forcing theorem is equivalent to the existence of a Boolean completion.

Boolean completions for class forcing need not be unique.

Abstract

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing. In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.