The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing.In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.2010 Mathematics Subject Classification. 03E40, 03E70, 03E99. Key words and phrases. Class forcing, Forcing theorem, Boolean completions. The first, third and fifth author were partially supported by DFG-grant LU2020/1-1. The authors would like to thank Joel David Hamkins for the careful reading of the paper and numerous useful comments.1 Note that in the absence of the power set axiom, Collection does not follow from Replacement and many important set-theoretical results can consistently fail in the weaker theory. For further details, consult [GHJ11]. 2 A partial order (or, more generally, a preorder) P is separative if for all p, q ∈ P , if p ≤ q then there exists r ≤ p such that r ⊥ q.

It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of ZF − , that is ZF without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of ZF. We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for sets of ordinals. These results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications of class forcing. Furthermore, for most properties under consideration, we also present a corresponding characterization of the Ord-chain condition. We would like to thank Maurice Stanley and Sy Friedman for providing us with information regarding the definability of the forcing relation in class forcing, and we would like to thank Victoria Gitman for helpful comments and discussions on some of the topics of this paper. We would also like to thank the referee for thoroughly reading through our paper, and for making many useful comments, that enabled us to very much improve this paper. The first and third author were partially supported by DFG-grant LU2020/1-1.1 Arguing in the ambient universe V, we will sometimes refer to classes of such a model M as sets, without meaning to indicate that they are sets of M. In particular this will be the case when we talk about subsets of M . 2 For more details, see [HKL + 16]. For a detailed axiomatization of KM, see [Ant15].

We present three natural combinatorial properties for class forcing notions, which imply the forcing theorem to hold. We then show that all known sufficent conditions for the forcing theorem (except for the forcing theorem itself), including the three properties presented in this paper, imply yet another regularity property for class forcing notions, namely that proper classes of the ground model cannot become sets in a generic extension, that is they do not have set-sized names in the ground model. We then show that over certain models of Gödel-Bernays set theory without the power set axiom, there is a notion of class forcing which turns a proper class into a set, however does not satisfy the forcing theorem. Moreover, we show that the property of not turning proper classes into sets can be used to characterize pretameness over such models of Gödel-Bernays set theory. Basic Definitions and NotationWe will work with transitive second-order models of set theory, that is models of the form M = M, C , where M is transitive and denotes the collection of sets of M, and C denotes the collection of classes of M. 1 We require that M ⊆ C, and that elements of C are subsets of M . We call elements of C \ M proper classes (of M). Classical transitive first-order models of set theory Arguing in the ambient universe V, we will sometimes refer to classes of such a model M as sets, without meaning to indicate that they are sets of M. In particular this will be the case when we talk about subsets of M .

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