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In foundations of mathematics, Von Neumann-Bernays-Gödel set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas.
First formulated by John von Neumann in the 1920s, it was, beginning in 1937, modified by Paul Bernays, and further simplified by Kurt Gödel in 1940.
Unlike ZFC, NBG has only finitely many axioms. As Richard Montague showed in 1961, it is not possible to find a finite number of axioms that is logically equivalent to ZFC; the language of NBG is thus one capable of talking about proper classes as well as sets, and a statement about sets is provable in NBG iff it is provable in ZFC (that is, NBG is a conservative extension of ZFC).
The theory
The defining aspect of the theory is the separation of class and set. A class can be very large - indeed, one can speak of "the class of all sets". However, there are structural limitations preventing one from speculating about "the class of all classes" or "the set of all sets".
The membership relation
<math>a \in s<math>
is only defined if <math>a<math> is a set and <math>s<math> is a set or a class.
The development of classes mirrors the development of naive set theory. The principle of abstraction is given, and thus classes can be formed out of any statement of the predicate calculus, with the membership relation. Notions of equality, pairing, subclass, and such, are thus matters of definitions and not of axioms - the definitions denote a particular abstraction of a formula.
The development of sets is carried out very similarly to ZF. There is a predicate "Rp" defined as follows:
<math>\mathrm{Rp}(A,a) := (x \in A \iff x \in a)<math>
That is, a set represents a class if every element of the set is an element of the class. There are classes that do not have representations, such as the class of all sets that do not contain themselves.
The advantage of such a system is that it provides a scaffolding from which to speak about "large objects" without running the risk of paradox. In some developments of category_theory, for instance, one denotes a large category as a category whose collection of objects and collection of morphisms can be represented by proper classes. On the other hand, a small category is a category in which the objects and morphisms "fit" in sets. Thus, we can easily speak of "the category of all small categories" without running into trouble. This would, of course, be a large category.
External links
- von Neumann-Bernays-Gödel set theory (http://planetmath.org/encyclopedia/VonNeumannBernausGodelSetTheory.html)
References
- Template:Book reference
- von Neumann, J.: An Axiomatization of Set Theory, 1925, reprinted in English translation in van Heijenoort (ed.): From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Cambridge, Massachusetts, Harvard University Press 1967
- Montague, R.: Semantic Closure and Non-Finite Axiomatizability I, in Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, (Warsaw, 2-9 September 1959). Oxford, England: Pergamon, pp. 45-69, 1961.