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INTRODUCTION ZFC consists of a single primitive Ontological notion, that of Set , and a single ontological assumption, namely that all Individual s in the Universe Of Discourse (i.e., all mathematical objects) are sets. There is a single primitive Binary Relation , set membership; that set ''a'' is a member of set ''b'' is written ''a'' ∈ ''b''. ZFC is a first order theory; hence ZFC includes Axiom s whose background logic is First Order Logic . These axioms govern how sets behave and interact. ZFC is the standard form of Axiomatic Set Theory . For an ongoing derivation of a great deal of Ordinary Mathematics using ZFC, see the Metamath online project. In 1908, Ernst Zermelo proposed the first Axiomatic Set Theory , Zermelo Set Theory . This axiomatic theory did not allow the construction of the Ordinal Number s; while most of "ordinary mathematics" can be developed without ever using ordinals, ordinals are an essential tool in most set-theoretic investigations. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not unambiguous. In 1922, Abraham Fraenkel and Thoralf Skolem independently proposed defining a "definite" property as any property that could be formulated in First Order Logic . From their work emerged the Axiom Of Replacement . Appending this axiom, as well as the Axiom Of Regularity , to Zermelo set theory yields the theory denoted by ''ZF''. Adding the Axiom Of Choice (AC) to ZF yields '''ZFC'''. When a mathematical result requires the axiom of choice, this is sometimes stated explicitly. The reason for singling out AC in this manner is that AC is inherently nonconstructive; it posits the existence of a set (the choice set), without specifying just how that set is to be constructed. Hence results proved using AC may involve sets that, although they can be proved to exist (at least if one is not committed to a Constructivist Ontology ), can never be constructed explicitly. ZFC has an Infinite number of axioms because Replacement is in truth an Axiom Schema . In 1957, Richard Montague proved that '''ZF''' (and hence ''a fortiori'' ZFC) cannot be stated without invoking at least one axiom schema; ZFC cannot be finitely axiomatized. On the other hand, the rival ''' NBG ''' set theory can be finitely axiomatized. The ontology of '''NBG''' includes Classes as well as sets; classes are entities that have members but that cannot be members of anything. '''NBG''' and ZFC are equivalent set theories, in the sense that any Theorem about sets (i.e., not mentioning classes in any way) which can be proved in one theory can be proved in the other. Because the axioms of , the Burali-Forti Paradox , and Cantor's Paradox . Drawbacks of ZFC that have been discussed in the literature include:
THE AXIOMS The ZFC axioms are:
eq arnothing \implies \exists B: B \in A \land \lnot \exist C: C \in A \land C \in B
eq B ightarrow \lnot \exist E: E \in B \and E \in D)))
The above symbolic statements of Choice and Replacement are more concise than is usual in the literature, because they employ a device that, while definable in . Replacement also assumes rather than establishes the functionality of ''P''(''X,Y''). SEE ALSO
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