Class Theories (NBG, MK)

Origin. John von Neumann (1925) axiomatized set theory with functions and a size restriction; Bernays (1937–1954) recast it with classes; Gödel used the result in his 1940 consistency proof for AC and GCH — which is why NBG is the system that constructibility was first done in. Morse and Kelley's stronger variant appeared in the appendix to Kelley's General Topology (1955).

Models. ZFC quantifies over sets and speaks of proper classes only through schemas. Class theories make the classes objects: two sorts, sets and classes, with sets the classes that belong to something. The point is to say "the class of all ordinals" rather than to gesture at it — and the systems differ precisely on how much one may then say about it.

Formalism.

Two sorts: Sets: x, y, ... Classes: X, Y, ... X is a set iff ∃Y (X ∈ Y). Otherwise X is a proper class. Only sets are members.

NBG (von Neumann–Bernays–Gödel): Class comprehension is predicative: ∃X ∀x (x ∈ X ↔ φ(x)), for φ with no quantifiers over classes. Finitely axiomatizable — the comprehension schema reduces to eight class-construction axioms. Conservative over ZFC: NBG ⊢ σ iff ZFC ⊢ σ, for σ in the language of sets.

MK (Morse–Kelley): Class comprehension is impredicative: φ may quantify over classes. Not conservative: MK ⊢ Con(ZFC), and MK ⊢ Con(NBG). Not finitely axiomatizable.

Limitation of size (von Neumann's axiom): X is a proper class iff X is in bijection with V. Implies Replacement, Choice, and global choice at once.

The trade: NBG finitely axiomatizable, conservative, predicative classes MK stronger, impredicative classes, proves NBG consistent ZFC one sort, schemas, no classes as objects Each buys one property with another.

Symbols.

Symbol	Unicode	Name	Meaning
NBG	—	von Neumann–Bernays–Gödel	Predicative class theory
MK	—	Morse–Kelley	Impredicative class theory
V	—	Universe	The class of all sets
Ord	—	Ordinals	The paradigm proper class
∈	U+2208	Membership	Sets only on the right

Metatheory. The conservativity of NBG over ZFC (Novak, Rosser–Wang, Shoenfield) is the result that decides the practice: since NBG proves no new theorems about sets, a set theorist may use classes freely and treat them as a manner of speaking — which is exactly what most do, without adopting the theory. Finite axiomatizability is NBG's other selling point and the reason Gödel chose it: with no schemas, the whole theory is one sentence, and metatheoretic arguments about it are cleaner. MK's extra strength is real but purchased at the cost of both properties, and Con(MK) is strictly above Con(ZFC), which puts it on the large-cardinal scale rather than beside ZFC.

Applies to. Category theory's size problems, where "the category of all sets" wants a class. Gödel's L, first presented in NBG. Global choice and the well-ordering of V. Any argument that needs to quantify over proper classes rather than schematize.

Limitations. NBG's conservativity is also its irrelevance: it says nothing new, so the choice between it and ZFC is one of convenience and not of commitment. MK says more and is correspondingly less believed. Neither solves the size problems it was invented for — a class theory still cannot speak of classes of classes, and category theory's usual answer is Grothendieck universes rather than either of these. The two-sorted language makes ordinary set-theoretic statements longer without making any of them clearer.