Title: Sufficient conditions for the forcing theorem, and turning proper classes into sets,
Abstract: 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.