Limit ordinal

A limit ordinal is a non-zero ordinal that has no immediate predecessor.MathWorld, Limit Ordinal Formally, an ordinal \(\alpha>0\) is a limit ordinal if and only if there does not exist a \(\beta\) so that \(\beta+1=\alpha\). The least limit ordinal is \(\omega\), some of the next limit ordinals are \(\omega\times2\) and \(\omega\times3\), and limit ordinals can informally be thought of as "multiples of \(\omega\)" due to how all limit ordinals are of the form \(\omega\times \beta\). Some authors allow \(0\) to be a limit ordinal,Thomas Jech, Set Theory. Elsevier Science, 1978. p.15 and hence a limit ordinal is sometimes called a non-zero limit ordinal. The class of limit ordinals is often denoted by \(\textrm{Lim}\).
An important fact about limit ordinals is that not all limit ordinals are of the form \(\beta+\omega\) where \(\beta\) is another limit ordinal. For example, \(\omega^2\) is a limit ordinal that can't be written as \(\beta+\omega\) where \(\beta\) is a limit ordinal \(<\omega^2\).
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== See also ==
Category:Transfinite ordinals
Category:Set theory
Category:Proper class