| Axiom Of Union |
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In the Formal Language of the Zermelo-Fraenkel axioms, the axiom reads: : or in words: : Given Any Set ''A'', There Is a set ''B'' such that, given any set ''C'', ''C'' is a member of ''B'' If And Only If there is a set ''D'' such that ''C'' is a member of ''D'' And ''D'' is a member of ''A''. Thus, what the axiom is really saying is that, given a set ''A'', we can find a set ''B'' whose members are precisely the members of the members of ''A''. By the Axiom Of Extensionality this set ''B'' is unique and it is called the '' Union '' of ''A'', and denoted ∪''A''. Thus the essence of the axiom is: :The union of a set is a set. The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative Axiomatization of set theory. Note that there is no corresponding axiom of , then trying to form the intersection of ''A'' as {''C'' such that for all ''D'' in ''A'', ''C'' is in ''D''} is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the "universe", but the notion of universe is antithetical to Zermelo-Fraenkel set theory.) |
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