axiom of choice

plural axioms of choice

1. mathematicsOne of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version of said axiom, for example specifying the cardinality of the number of sets from which choices are made.

   The axiom of choice is logically equivalent to the assertion that every vector space has a basis.

   4 more examples

   If V = L then the axioms of choice and the continuum hypothesis are both true, and the assertion that a measurable cardinal exists is false.

   1993, Thomas Tymoczko, editor, Penelope Maddy: Does V Equal L?: New Directions in the Philosophy of Mathematics: An Anthology, page 357

   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.

   1993, Gary L. Wise, Eric B. Hall, Counterexamples in Probability and Real Analysis, page vii

   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.

   2000, Bruno Poizat, translated by Moses Klein, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, page 169

   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.

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

   Synonyms: choice, AC