English

Noun

axiom of choice ‎(uncountable)

1. (set theory) One of the axioms in axiomatic set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty.
   • 1993, Gary L. Wise, Eric B. Hall, Counterexamples in Probability and Real Analysis, page vii,

     Throughout this work we adopt the Zermelo–Fraenkel (ZF) axioms of set theory with the Axiom of Choice, commonly abbreviated as ZFC. It follows from the work of Gödel and Cohen that if the ZF axioms are consistent, the Axiom of Choice can be neither proved nor disproved from the ZF axioms.

   • 2000, Moses Klein (translator), Bruno Poizat, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, page 169,

     To clarify these ideas for the reader, let us show, without the axiom of choice, that a product of finitely many nonempty sets is nonempty: This is done by induction on the number n of sets. […] The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable.

   • 2004, Michael Potter, Set Theory and its Philosophy: A Critical Introduction, page 259,

     Perhaps what this debate about whether to accept the axiom of choice indicates is that the disjunction between regularity and randomness is as fundamental to our conception of the world as that between discreteness and continuity.

Translations

See also

• ZFC