In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to construct a set by arbitrarily choosing one object from each bin, even if the collection is infinite.

<mathematics> (AC, or "Choice") An axiom of set theory: If X is a set of sets, and S is the union of all the elements of X, then there exists a function f:X -> S such that for all non-empty x in X, f(x) is an element of x. In other words, we can always choose an element from each set in a set of sets, simultaneously. Function f is a "choice function" for X - for each x in X, it chooses an element of x. Most people's reaction to AC is: "But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or whatever". Indeed, for any finite set of sets, we can simply consider each set in turn and pick an arbitrary element in some such way. We can also construct a choice function for most simple {infinite sets} of sets if they are generated in some regular way. However, there are some infinite sets for which the construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases. For example, if we express the real number line R as the union of many "copies" of the rational numbers, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are irrational numbers no two of which differ by a rational, and Q+a == q+a : q in Q we cannot pick an element of each of these "copies" without AC. An example of the use of AC is the theorem which states that the countable union of countable sets is countable. I.e. if X is countable and every element of X is countable (including the possibility that they're finite), then the sumset of X is countable. AC is required for this to be true in general. Even if one accepts the axiom, it doesn't tell you how to construct a choice function, only that one exists. Most mathematicians are quite happy to use AC if they need it, but those who are careful will, at least, draw attention to the fact that they have used it. There is something a little odd about Choice, and it has some alarming consequences, so results which actually "need" it are somehow a bit suspicious, e.g. the Banach-Tarski paradox. On the other side, consider Russell's Attic. AC is not a theorem of Zermelo Frankel set theory (ZF). Godel and Paul Cohen proved that AC is independent of ZF, i.e. if ZF is consistent, then so are ZFC (ZF with AC) and ZF(Axiom of ChoiceC) (ZF with the negation of AC). This means that we cannot use ZF to prove or disprove AC. (2003-07-11)

Equivalents of the Axiom of Choice is a book in mathematics, collecting statements in mathematics that are true if and only if the axiom of choice holds. It was written by Herman Rubin and Jean E.

The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.

Axiom of Choice is a southern California (United States) based world music group of Iranian émigrés who perform a modernized fusion style rooted in Persian classical music with inspiration from other classical Middle Eastern and Eastern paradigms.

In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every non-empty set.

In mathematics, the axiom of dependent choice, denoted by \mathsf{DC} , is a weak form of the axiom of choice ( \mathsf{AC} ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements are the same set.

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads:

<mathematics> An axiom schema of set theory which states: if P(x) is a property then x : P is a set. I.e. all the things with some property form a set. Acceptance of this axiom leads to Russell's Paradox which is why Zermelo set theory replaces it with a restricted form. (1995-03-31)