Editing Von Neumann cardinal assignment (section)

== Initial ordinal of a cardinal ==
Each ordinal has an associated [[Cardinal number|cardinal]], its cardinality, obtained by simply forgetting the order.  Any well-ordered set having that ordinal as its [[order type]] has the same cardinality.  The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal.  Every finite ordinal ([[natural number]]) is initial, but most infinite ordinals are not initial.  The [[axiom of choice]] is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal.  In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal ''is'' a cardinal.

The <math>\alpha</math>-th infinite initial ordinal is written <math>\omega_\alpha</math>.  Its cardinality is written <math>\aleph_{\alpha}</math> (the <math>\alpha</math>-th [[aleph number]]). For example, the cardinality of <math>\omega_{0}=\omega</math> is <math>\aleph_{0}</math>, which is also the cardinality of <math>\omega^{2}</math>, <math>\omega^{\omega}</math>, and [[Epsilon numbers (mathematics)|<math>\epsilon_{0}</math>]] (all are [[Countable set|countable]] ordinals).  So we identify <math>\omega_{\alpha}</math> with <math>\aleph_{\alpha}</math>, except that the notation <math>\aleph_{\alpha}</math> is used for writing cardinals, and <math>\omega_{\alpha}</math> for writing ordinals. This is important because [[cardinal number#Cardinal arithmetic|arithmetic on cardinals]] is different from [[ordinal arithmetic|arithmetic on ordinals]], for example <math>\aleph_{\alpha}^{2}</math>&nbsp;=&nbsp;<math>\aleph_{\alpha}</math> whereas <math>\omega_{\alpha}^{2}</math>&nbsp;>&nbsp;<math>\omega_{\alpha}</math>. Also, <math>\omega_{1}</math> is the smallest [[Uncountable set|uncountable]] ordinal (to see that it exists, consider the set of [[equivalence class]]es of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and <math>\omega_{1}</math> is the order type of that set), <math>\omega_{2}</math> is the smallest ordinal whose cardinality is greater than <math>\aleph_{1}</math>, and so on, and <math>\omega_{\omega}</math> is the limit of <math>\omega_{n}</math> for natural numbers <math>n</math> (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the <math>\omega_{n}</math>).

Infinite initial ordinals are limit ordinals. Using ordinal arithmetic, <math>\alpha<\omega_{\beta}</math> implies <math>\alpha+\omega_{\beta}=\omega_{\beta}</math>, and 1 ≤ ''α'' < ω<sub>''β''</sub> implies ''α''&thinsp;·&thinsp;ω<sub>''β''</sub> = ω<sub>''β''</sub>, and 2 ≤ ''α'' < ω<sub>''β''</sub> implies ''α''<sup>ω<sub>''β''</sub></sup> = ω<sub>''β''</sub>. Using the [[Veblen function|Veblen hierarchy]], ''β'' ≠ 0 and ''α'' < ω<sub>''β''</sub> imply <math>\varphi_{\alpha}(\omega_{\beta}) = \omega_{\beta} \,</math> and &Gamma;<sub>ω<sub>''β''</sub></sub> = ω<sub>''β''</sub>. Indeed, one can go far beyond this. So as an ordinal, an infinite initial ordinal is an extremely strong kind of limit.