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#axiom of choice

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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 ›

8.2 Equivalent Formulations

The axiom of choice has many equivalent forms. Some of these forms speak about selecting elements from sets, some speak about ordering arbitrary sets, and some speak about finding maximal objects in partially ordered sets. Although these statements look different, they express the same underlying principle over the usual axioms of ZF set theory. The purpose of this section is to explain the most important equivalent formulations in detail. The...