infinity, axiom of
Axiom needed in Russell's development of set theory to ensure that there are enough sets for the purposes of mathematics. In the theory of types, if there were only finitely many objects of the lowest type, then there would only be finite sets at any level. To avoid this Russell postulated that there are infinitely many individuals or objects of the lowest type, but the non-logical nature of this assumption was a grave obstacle to his logicist programme. In Zermelo–Fraenkel set theory the axiom asserts that there is a set of which the null set is a member, and such that if any set is a member, the union of it and its unit set is also a member: (∃S)(ø∈ S & (∀R)(R ∈ S) → R ? R ∈ S).

Philosophy dictionary. . 2011.

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