Limit ordinal

A limit ordinal is a non-zero ordinal that has no immediate predecessor. Formally, an ordinal \(\alpha>0\) is a limit ordinal if \(\nexists\beta:\beta+1=\alpha\). 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 author allows \(0\) to be a limit ordinal, and hence a limit ordinal is sometimes called a non-zero limit ordinal. The class of limit ordinals is often denoted by \(\textrm{Lim}\).

Comparisons of sizes
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 \(<\omega^2\).