NBG or von Neumann–Bernays–Gödel set theory is a material set theory. It is a conservative extension of ZFC and its ontology includes proper classes, like MK. But unlike ZFC and MK, NBG can be finitely axiomatized.


Georg Cantor was well acquainted with the phenomenon that some collections are ‘too big’ to be sets and in his late years made a distinction between consistent and inconsistent multitudes, the former comprising sets, but his ideas were confined to private letters to (Jourdain and) Dedekind that were not published until 1932.

John von Neumann began NBGNBG in the 1920s (von Neumann 1925, von Neumann 1928), but his version was unwieldy. Paul Bernays and Kurt Gödel simplified it later. It also illustrates Gödel's theorem that any first-order theory has a conservative extension? with a finite axiomatization.


NBG is a material set theory, based on a global binary membership predicate \in. The objects of NBG are called classes. If a class xx is a member of another class AA, i.e., xAx \in A, then xx is called set. A class which is not a set is called proper class. NBG can be presented as a two-sorted theory, with lowercase letters denoting sets and uppercase letters denoting classes.

  1. Extensionality: Two classes are equal if and only if the have the same members.

  2. Pairing, Union, Power Sets and Infinity: regard sets and are identical to the ZFC counterparts.

  3. Foundation: For each nonempty class AA there exists a set xAx \in A such that xA=x \cap A = \varnothing.

  4. Class Comprehension schema: For any formula ϕ\phi containing no quantifiers over classes (it may contain quantifiers over sets and it may contain both class and set parameters), there is a class AA whose elements are precisely those sets xx satisfying ϕ(x)\phi(x). In particular, taking ϕ(x)\phi(x) as x=xx = x it follows that there exists the class VV of all sets.

  5. Limitation of Size: A class AA is a set if and only if there is no bijection between AA and the class VV of all sets.

From the axiom of Limitation of Size, it turn out that VV is not a set but a proper class, avoiding Russell's paradox. Moreover, since one can prove that the ordinal numbers form a proper class, there is a bijection beetween VV and Ord\mathrm{Ord}, i.e., the class of all sets can be well-ordered which implies the axiom of global choice.

Finite axiomatization

Every instance of class comprehension can be built out of a few, based on the logical connnectives. This is similar to the finite axiomatization of bounded ZFCZFC (or, in other language, ETCS); indeed, NBGNBG is essentially bounded MK.


  • Gerhard Osius, Kategorielle Mengenlehre: Eine Charakterisierung der Kategorie der Klassen und Abbildungen , Math. Ann. 210 (1974) pp.171-196. (gdz)

Revised on July 28, 2015 04:25:12 by Thomas Holder (