THE EXACT STRENGTH of the CLASS FORCING THEOREM

The class forcing theorem, which asserts that every class forcing notion admits a forcing relation, 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 to the principle of elementary transfinite recursion for class recursions of length. It is also equivalent to the existence of truth predicates for the infinitary languages, allowing any class parameter A; to the existence of truth predicates for the language; to the existence of-iterated truth predicates for first-order set theory; to the assertion that every separative class partial order 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. Unlike set forcing, if every class forcing notion has a forcing relation merely for atomic formulas, then every such has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between and Kelley-Morse set theory.