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

The exact strength of the class forcing theorem