von Neumann-Bernays-Gödel Set Theory

von Neumann-Bernays-Gödel set theory (abbreviated "NBG") is a version of set theory which was designed to give the same results as Zermelo-Fraenkel set theory, but in a more logically elegant fashion. It can be viewed as a conservative extension of Zermelo-Fraenkel set theory in the sense that a statement about sets is provable in NBG if and only if it is provable in Zermelo-Fraenkel set theory.

Zermelo-Fraenkel set theory is not finitely axiomatized. For example, the axiom of replacement is not really a single axiom, but an infinite family of axioms, since it is preceded by the stipulation that it is true "for any set-theoretic formula A(u,v)." Montague (1961) proved that Zermelo-Fraenkel set theory is not finitely axiomatizable, i.e., there is no finite set of axioms which is logically equivalent to the infinite set of Zermelo-Fraenkel axioms. In contrast, von Neumann-Bernays-Gödel set theory has only finitely many axioms, and this was the main motivation in its construction. This was accomplished by extending the language of Zermelo-Fraenkel set theory to be capable of talking about set classes.

See also

Set Class, Set Theory, Zermelo-Fraenkel Axioms, Zermelo-Fraenkel Set Theory

This entry contributed by Matthew Szudzik

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Itô, K. (Ed.). "Bernays-Gödel Set Theory." §33C in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, p. 148, 1986.Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.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.

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von Neumann-Bernays-Gödel Set Theory

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Szudzik, Matthew. "von Neumann-Bernays-Gödel Set Theory." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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