A limit ordinal is a non-zero ordinal that doesn't have a predecessor[citation needed]. 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[citation needed], 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 $$<\omega^2$$.