Noun
axiom of choice (countable and uncountable, plural axioms of choice)
(set theory) One 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.
The only thing that might have annoyed some mathematicians was the presumption of assuming that maybe the axiom of choice could fail, and that we should look into contrary assumptions. Alonzo Church
After that step, the axiom of choice is not used again. Source: Internet
A function f has a right inverse if and only if it is surjective (though constructing such an inverse in general requires the axiom of choice ). Source: Internet
Also Wacław Sierpiński proved that ZF + GCH (the generalized continuum hypothesis ) imply the axiom of choice and hence a well order of the reals. Source: Internet
Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive. citation. Source: Internet
Because of independence, the decision whether to use the axiom of choice (or its negation) in a proof cannot be made by appeal to other axioms of set theory. Source: Internet