nLab von Neumann–Bernays–Gödel set theory

Idea

Neumann–Bernays–Gödel set theory (NBG) is a material set theory: a conservative extension of ZFC whose ontology includes proper classes, like MK. But unlike ZFC and MK, NBG can be finitely axiomatized.

History

Georg Cantor was well acquainted with the phenomenon that some collections are ‘too big’ to be sets. In his late years he 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 work on NBG in the 1920s (von Neumann 1925, von Neumann 1928), but his version was unwieldy. Paul Bernays and Kurt Gödel simplified it later, also illustrating Gödel’s theorem that any first-order theory has a conservative extension with a finite axiomatization.

Axioms

NBG is a material set theory, based on a global binary membership predicate $\in$. The objects of NBG are called classes. If a class $x$ is a member of another class $A$, i.e., $x \in A$, then $x$ 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 $A$ there exists a set $x \in A$ such that $x \cap A = \varnothing$.

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

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

From the axiom of Limitation of Size, it turn out that $V$ 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 $V$ and $\mathrm{Ord}$, 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 $ZFC$ (or, in other language, ETCS); indeed, $NBG$ is essentially bounded MK.

References

• John von Neumann, “Eine Axiomatisierung der Mengenlehre”, Journal für die Reine und Angewandte Mathematik 154 (1925) 219–240, doi:10.1515/crll.1925.154.219, GDZ

• John von Neumann, “Die Axiomatisierung der Mengenlehre”, Mathematische Zeitschrift 27 (1928) 669–752, doi:10.1007/bf01171122, GDZ

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

An expository account:

• Elliott Mendelson, Introduction to mathematical logic, van Nostrand (1964) CRC Press (1987, 2015) [ark:/13960/t60663n9w, doi:10.1007/978-1-4615-7288-6, ISBN:9781482237726, pdf, ISBN-13:978-1-58488-876-5, ISBN-10:1-58488-876-8]