The axiom of countable choice or axiom of denumerable choice, denoted AC<sub>ÃÂ</sub>, 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 with domain (where denotes the set of natural numbers) such that is a non-empty set for every , there exists a function with domain such that for every .
AC<sub>ÃÂ</sub> is particularly useful for the development of mathematical analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. For instance, in order to prove that every accumulation point of a set is the limit of some sequence of elements of , one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to AC<sub>ÃÂ</sub>.
The ability to perform analysis using countable choice has led to the inclusion of AC<sub>ÃÂ</sub> as an axiom in some forms of constructive mathematics, despite its assertion that a choice function exists without constructing it.
As an example of an application of AC<sub>ÃÂ</sub>, here is a proof (from ZF + AC<sub>ÃÂ</sub>) that every infinite set is Dedekind-infinite:
Let be infinite. For each natural number , let be the set of all -tuples of distinct elements of . Since is infinite, each is non-empty. Application of AC<sub>ÃÂ</sub> yields a sequence where each is an -tuple. One can then concatenate these tuples into a single sequence of elements of , possibly with repeating elements. Suppressing repetitions produces a sequence of distinct elements, where
This exists, because when selecting it is not possible for all elements of to be among the elements selected previously. So contains a countable set. The function that maps each to (and leaves all other elements of fixed) is a one-to-one map from into which is not onto, proving that is Dedekind-infinite.
The axiom of countable choice (AC<sub>ÃÂ</sub>) is strictly weaker than the axiom of dependent choice (DC), which in turn is weaker than the axiom of choice (AC). DC, and therefore also AC<sub>ÃÂ</sub>, hold in the Solovay model, constructed in 1970 by Robert M. Solovay as a model of set theory without the full axiom of choice, in which all sets of real numbers are measurable.
Urysohn's lemma (UL) and the Tietze extension theorem (TET) are independent of ZF+AC<sub>ÃÂ</sub>: there exist models of ZF+AC<sub>ÃÂ</sub> in which UL and TET are true, and models in which they are false. Both UL and TET are implied by DC.
Paul Cohen showed that AC<sub>ÃÂ</sub> is not provable in ZermeloâÂÂFraenkel set theory (ZF) without the axiom of choice. However, some countably infinite sets of non-empty sets can be proven to have a choice function in ZF without any form of the axiom of choice. For example, has a choice function, where is the set of hereditarily finite sets, i.e. the first set of non-finite rank in the Von Neumann universe. The choice function is: { ⟨W<sub>n</sub>,W<sub>k</sub>⟩ : k < n < ω ∧ W<sub>k</sub> ∈ W<sub>n</sub> ∧ ∀ j < k ( W<sub>j</sub> ∉ W<sub>n</sub> ) } where W<sub>n</sub> = {W<sub>k</sub> : k<n ∧ (n mod 2<sup>k+1</sup>) âÂÂ¥ 2<sup>k</sup>} for n<ω. W lists every hereditarily finite set exactly once and is based on the binary numeral for n which has a 1 in each place corresponding to a k with W<sub>k</sub> ∈ W<sub>n</sub>. Another example is the set of proper and bounded open intervals of real numbers with rational endpoints.
ZF+AC<sub>ÃÂ</sub> suffices to prove that the union of countably many countable sets is countable. These statements are not equivalent: Cohen's First Model supplies an example where countable unions of countable sets are countable, but where AC<sub>ÃÂ</sub> does not hold.
There are many equivalent forms to the axiom of countable choice, in the sense that any one of them can be proven in ZF assuming any other of them. They include the following: