Each set in the countablesequence of sets (Si) = S1, S2, S3, ... contains a non-zero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ...
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-emptysets 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 .
Applications
ACω 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 ACω.
The ability to perform analysis using countable choice has led to the inclusion of ACω as an axiom in some forms of constructive mathematics, despite its assertion that a choice function exists without constructing it.[1]
Example: infinite implies Dedekind-infinite
As an example of an application of ACω, here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite:[2]
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 ACω 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
, with .
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.[2]
Relation to other axioms
Stronger and independent systems
The axiom of countable choice (ACω) is strictly weaker than the axiom of dependent choice (DC),[3] which in turn is weaker than the axiom of choice (AC). DC, and therefore also ACω, 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.[4]
Urysohn's lemma (UL) and the Tietze extension theorem (TET) are independent of ZF+ACω: there exist models of ZF+ACω in which UL and TET are true, and models in which they are false. Both UL and TET are implied by DC.[5]
Weaker systems
Paul Cohen showed that ACω is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice.[6] 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 in the Von Neumann universe of non-finite rank. The choice function is (trivially) the least element in the well-ordering. Another example is the set of proper and bounded open intervals of real numbers with rational endpoints.
ZF+ACω 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 ACω does not hold.[7]
Equivalent forms
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:[8][9]
Every countable collection of non-empty sets has a choice function.[8]
Every infinite collection of non-empty sets has an infinite sub-collection with a choice function.[8]
Every second-countable space (it has a countable base of open sets) is a separable space (it has a countable dense subset).[8] A metric space is separable if and only if it is σ-compact.[9]
^Tachtsis, Eleftherios (2019), "The Urysohn lemma is independent of ZF + countable choice", Proceedings of the American Mathematical Society, 147 (9): 4029–4038, doi:10.1090/proc/14590, MR3993794
^ abcdefgHoward, Paul; Rubin, Jean E. (1998). Consequences of the axiom of choice. Providence, Rhode Island: American Mathematical Society. ISBN978-0-8218-0977-8. See in particular Form 8, p. 17–18.