The exact strength of the class forcing theorem

TL;DR

This paper investigates the precise logical strength of the class forcing theorem, establishing its equivalence to several foundational principles in set theory and class logic, and situating it within the hierarchy of set-theoretic theories.

Contribution

It precisely characterizes the strength of the class forcing theorem, showing its equivalence to principles like ETR_Ord and truth predicates, and compares it to other set-theoretic axioms.

Findings

Class forcing theorem is equivalent to ETR_Ord over GBC.

Existence of truth predicates for infinitary languages is equivalent to the class forcing theorem.

The theorem's strength is situated between GBC and Kelley-Morse set theory.

Abstract

The class forcing theorem, which asserts that every class forcing notion $\mathbb{P}$ admits a forcing relation $\Vdash_{\mathbb{P}}$, that is, a relation satisfying the forcing relation recursion -- it follows that statements true in the corresponding forcing extensions are forced and forced statements are true -- is equivalent over G\"odel-Bernays set theory GBC to the principle of elementary transfinite recursion $\text{ETR}_{\text{Ord}}$ for class recursions of length $\text{Ord}$. It is also equivalent to the existence of truth predicates for the infinitary languages $\mathcal{L}_{\text{Ord},\omega}(\in,A)$, allowing any class parameter $A$; to the existence of truth predicates for the language $\mathcal{L}_{\text{Ord},\text{Ord}}(\in,A)$; to the existence of $\text{Ord}$-iterated truth predicates for first-order set theory $\mathcal{L}_{\omega,\omega}(\in,A)$; to the assertion that every separative class partial order $\mathbb{P}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text{Ord}+1$. Unlike set forcing, if every class forcing notion $\mathbb{P}$ has a forcing relation merely for atomic formulas, then every such $\mathbb{P}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between GBC and Kelley-Morse set theory KM.