High Quality Content by WIKIPEDIA articles! In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes of naive set theory like Russell's paradox. Specifically, ZFC does not allow unrestricted comprehension. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. ZFC has a single primitive ontological notion, that of a hereditary well-founded set, and a single ontological assumption, namely that all individuals in the universe of discourse are such sets. Thus, ZFC is a set theory without urelements (elements of sets which are not themselves sets). ZFC does not formalize the notion of classes (collections of mathematical objects defined by a property shared by their members) and specifically does not include proper classes (objects that have members but cannot be members themselves).

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