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#ZFC

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Maths › 03. Logic and Foundations ›

8.1 Axiom of Choice

The axiom of choice is an additional principle of set theory which says that, given any family of nonempty sets, it is possible to choose one element from each set in the family. The statement sounds simple, but it has deep consequences because it applies even when the family is infinite and no explicit rule for making the choices is given. In ordinary finite mathematics, choosing one element from each...

Maths › 03. Logic and Foundations ›

7.5 ZF and ZFC

Set theory becomes a foundation for mathematics only after its basic principles are stated as axioms. Informal set language is useful, but it can lead to contradictions if every definable collection is allowed to be a set. The Zermelo Fraenkel axioms provide a controlled formal framework in which sets may be built, compared, and used without allowing unrestricted comprehension. Motivation A naive view says that every collection described by a...