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It is well known that pretameness implies the forcing theorem, and thatpretameness is characterized by the preservation of the axioms of$\mathsf{ZF}^-$, that is $\mathsf{ZF}$ without the power set axiom, orequivalently, by the preservation of the axiom scheme of replacement, for classforcing over models of $\mathsf{ZF}$. We show that pretameness in fact hasvarious other characterizations, for instance in terms of the forcing theorem,the preservation of the axiom scheme of separation, the forcing equivalence ofpartial orders and their dense suborders, and the existence of nice names forsets of ordinals. These results show that pretameness is a strong dividing linebetween well and badly behaved notions of class forcing, and that it is exactlythe right notion to consider in applications of class forcing. Furthermore, formost properties under consideration, we also present a correspondingcharacterization of the $\mathrm{Ord}$-chain condition.

Characterizations of pretameness and the Ord-cc