Countable set

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> is countable if either it is <a href="/facts/Finite_set/za1newlP">finite</a> or it can be made in <a href="/facts/One_to_one_correspondence/j4gTuTmW">one to one correspondence</a> with the set of <a href="/facts/Natural_number/u65og8JE">natural numbers</a>. Equivalently, a set is countable if there exists an <a href="/facts/Injective_function/5ES6tqf5">injective function</a> from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
In more technical terms, assuming the <a href="/facts/Axiom_of_countable_choice/goSPxEdj">axiom of countable choice</a>, a set is countable if its <a href="/facts/Cardinality/m6VPojwT">cardinality</a> (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite.
The concept is attributed to <a href="/facts/Georg_Cantor/k5v1DvW4">Georg Cantor</a>, who proved the existence of <a href="/facts/Uncountable_set/zl2WqyJQ">uncountable sets</a>, that is, sets that are not countable; for example the set of the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>.

Countable set open-in-new

Countable set