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A limit ordinal is a non-zero ordinal that has no immediate predecessor.[1] 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,[2] 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 $$<\omega^2$$.

## Sources

1. MathWorld, Limit Ordinal
2. Thomas Jech, Set Theory. Elsevier Science, 1978. p.15