The Exact Strength of the Class Forcing Theorem

The class forcing theorem, which asserts that every class forcing notion P admits a forcing relation 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 Gödel-Bernays set theory GBC to the principle of elementary transfinite recursion ETROrd for class recursions of length Ord. It is also equivalent to the existence of truth predicates for the infinitary languages LOrd,ω(∈, A), allowing any class parameter A; to the existence of truth predicates for the language LOrd,Ord(∈, A); to the existence of Ord-iterated truth predicates for first-order set theory Lω,ω(∈, A); to the assertion that every separative class partial order 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 Ord + 1. Unlike set forcing, if every class forcing notion P has a forcing relation merely for atomic formulas, then every such 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.