ZF set theory

First few Von Neumann ordinals
0		= Ø
1	= { 0 }	= {Ø}
2	= { 0, 1 }	= { Ø, {Ø} }
3	= { 0, 1, 2 }	= { Ø, {Ø} , {Ø, {Ø}} }
4	= { 0, 1, 2, 3 }	= { Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}} }

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Zermelo–Fraenkel set theory with the historically controversial axiom of choice included is commonly abbreviated ZFC, where C stands for choice. Many authors use ZF to refer to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.

ZFC is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of ZFC refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be sets) can only be treated indirectly. Specifically, ZFC does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of ZFC that does allow explicit treatment of proper classes.

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