Classless. (29th June 2019) - Roberts, Sam
Abstract: Classes are a kind of collection. Typically, they are too large to be sets. For example, there are classes containing absolutely all sets even though there is no set of all sets. But what are classes, if not sets? When our theory of classes is relatively weak, this question can be avoided. In particular, it is well known that von Neuman–Bernays–Godel class theory (NBG ) is conservative over the standard axioms of set theory (namely, those of Zermelo–Fraenkel set theory with the axiom of Choice (ZFC )): anything NGB can prove about the sets is already provable in ZFC . In this paper I will prove a new conservativity result for a much broader range of class theories. It tells us that as long as our set theory T contains an independently well-motivated reflection principle, anything provable about the sets in any reasonable class theory extending T is already provable in T.