Editing Regular cardinal (section)

== Examples ==

The ordinals less than <math>\omega</math> are finite. A finite sequence of finite ordinals always has a finite maximum, so <math>\omega</math> cannot be the limit of any sequence of type less than <math>\omega</math> whose elements are ordinals less than <math>\omega</math>, and is therefore a regular ordinal. <math>\aleph_0</math> ([[aleph-null]]) is a regular cardinal because its initial ordinal, <math>\omega</math>, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite.

<math>\omega+1</math> is the [[successor ordinal|next ordinal number]] greater than <math>\omega</math>. It is singular, since it is not a limit ordinal. <math>\omega+\omega</math> is the next limit ordinal after <math>\omega</math>. It can be written as the limit of the sequence <math>\omega</math>, <math>\omega+1</math>, <math>\omega+2</math>, <math>\omega+3</math>, and so on. This sequence has order type <math>\omega</math>, so <math>\omega+\omega</math> is the limit of a sequence of type less than <math>\omega+\omega</math> whose elements are ordinals less than <math>\omega+\omega</math>; therefore it is singular.

<math>\aleph_1</math> is the [[successor cardinal|next cardinal number]] greater than <math>\aleph_0</math>, so the cardinals less than <math>\aleph_1</math> are [[countable set|countable]] (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So <math>\aleph_1</math> cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.

<math>\aleph_\omega</math> is the next cardinal number after the sequence <math>\aleph_0</math>, <math>\aleph_1</math>, <math>\aleph_2</math>, <math>\aleph_3</math>, and so on. Its initial ordinal <math>\omega_\omega</math> is the limit of the sequence <math>\omega</math>, <math>\omega_1</math>, <math>\omega_2</math>, <math>\omega_3</math>, and so on, which has order type <math>\omega</math>, so <math>\omega_\omega</math> is singular, and so is <math>\aleph_\omega</math>. Assuming the axiom of choice, <math>\aleph_\omega</math> is the first infinite cardinal that is singular (the first infinite ''ordinal'' that is singular is <math>\omega+1</math>, and the first infinite ''limit ordinal'' that is singular is <math>\omega+\omega</math>). Proving the existence of singular cardinals requires the [[axiom schema of replacement|axiom of replacement]], and in fact the inability to prove the existence of <math>\aleph_\omega</math> in [[Zermelo set theory]] is what led [[Adolf Abraham Halevi Fraenkel|Fraenkel]] to postulate this axiom.<ref>{{citation
 | last = Maddy | first = Penelope | authorlink = Penelope Maddy
 | doi = 10.2307/2274520
 | jstor = 2274520
 | issue = 2
 | journal = [[Journal of Symbolic Logic]]
 | mr = 947855
 | pages = 481–511
 | quote =  Early hints of the Axiom of Replacement can be found in Cantor's letter to Dedekind [1899] and in Mirimanoff [1917]
 | title = Believing the axioms. I
 | volume = 53
 | year = 1988}}. Maddy cites two papers by Mirimanoff, "Les antinomies de Russell et de Burali-Forti et le problème fundamental de la théorie des ensembles" and "Remarques sur la théorie des ensembles et les antinomies Cantorienne", both in ''L'Enseignement Mathématique'' (1917).</ref>

Uncountable (weak) [[limit cardinal]]s that are also regular are known as (weakly) [[inaccessible cardinals]].  They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom.  Inaccessible cardinals are necessarily [[fixed point (mathematics)|fixed point]]s of the [[aleph number|aleph function]], though not all fixed points are regular.  For instance, the first fixed point is the limit of the <math>\omega</math>-sequence <math>\aleph_0, \aleph_{\omega}, \aleph_{\omega_{\omega}}, ...</math> and is therefore singular.