Editing Aleph number (section)

==Aleph-zero<span class="anchor" id="Aleph-null"></span>==
<math>\aleph_0</math> ('''aleph-nought''', '''aleph-zero''', or '''aleph-null''') is the cardinality of the set of all natural numbers, and is an [[transfinite number|infinite cardinal]]. The set of all finite [[ordinal number|ordinals]], called <math>\omega</math> or <math>\omega_0</math> (where <math>\omega</math> is the lowercase Greek letter [[omega]]), also has cardinality <math>\aleph_0</math>. A set has cardinality <math>\aleph_0</math> if and only if it is [[countably infinite]], that is, there is a [[bijection]] (one-to-one correspondence) between it and the natural numbers. Examples of such sets are

* the set of [[natural numbers]], irrespective of including or excluding zero,
* the set of all [[integer]]s, 
* any infinite subset of the integers, such as the set of all [[square numbers]] or the set of all [[prime numbers]],
* the set of all [[rational number]]s, 
* the set of all [[constructible number]]s (in the geometric sense),
* the set of all [[algebraic number]]s, 
* the set of all [[computable number]]s, 
* the set of all [[computable function]]s, 
* the set of all binary [[string (computer science)|string]]s of finite length, and 
* the set of all finite [[subset]]s of any given countably infinite set.

Among the countably infinite sets are certain infinite ordinals,{{efn|This is using the convention that an ordinal is identified with the set of all ordinals less than itself (the so-called [[von Neumann ordinals]]).}} including for example <math>\omega</math>, <math>\omega+1</math>, <math>\omega \cdot 2</math>, <math>\omega^2</math>, <math>\omega^\omega</math>, and [[Epsilon numbers (mathematics)|<math>\varepsilon_0</math>]].<ref>{{cite book | last1=Jech | first1=Thomas | title=Set Theory | publisher= [[Springer-Verlag]]| location=Berlin, New York | series=Springer Monographs in Mathematics | year=2003}}</ref> For example, the sequence (with [[order type]] <math>\omega \cdot 2</math>) of all positive odd integers followed by all positive even integers <math>\{1, 3, 5, 7, 9, \cdots; 2, 4, 6, 8, 10, \cdots\}</math> is an ordering of the set (with cardinality <math>\aleph_0</math>) of positive integers.

If the [[axiom of countable choice]] (a weaker version of the [[axiom of choice]]) holds, then <math>\aleph_0</math> is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.