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

Zermelo-Fraenkel set theory is not finitely axiomatized. For example, theaxiom 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." Montague (1961) proved thatZermelo-Fraenkel set theory is not finitely axiomatizable, i.e., there is no finite set of axioms which is logically equivalent to the infinite set ofZermelo-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 ofZermelo-Fraenkel set theory to be capable of talking aboutset classes.

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This entry contributed byMatthewSzudzik

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Szudzik, Matthew. "von Neumann-Bernays-Gödel Set Theory." FromMathWorld--A Wolfram Resource, created byEric W. Weisstein.https://mathworld.wolfram.com/vonNeumann-Bernays-GoedelSetTheory.html

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