Limit ordinal

A limit ordinal is a non-zero ordinal that doesn't have a predecessor. Formally, an ordinal \(\alpha>0\) is a limit ordinal iff \(\nexists\beta:\beta+1=\alpha\). 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 denoted by \(\textrm{Lim}\).

Additive principal numbers
A related concept to limit ordinals are additive principal numbers, which are non-zero ordinals \(\alpha\) where \(\forall(\beta,\gamma<\alpha)(\beta+\gamma<\alpha)\). All additive principal numbers except 1 are limit ordinals. The class of limit ordinals is denoted by \(\textrm{AP}\).