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 and only if there does not exist a $$\beta$$ so that $$\beta+1=\alpha$$. The least limit ordinal is Omega$$\omega$$, 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, 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$$<\omega^2$$.

List of limit ordinals (incomplete）
As mentioned, 0 is not an limit ordinal. However, some authors called 0 as a limit ordinal because by definition there is no ordinal such that its succesor is 0. This has been a debated topic for a while.


 * $$\omega$$
 * $$2\omega$$
 * $$n\omega, (n > 0, n \in N)$$
 * $$\omega^2$$
 * $$\omega^2 + \omega$$
 * $$\omega^2 + n\omega$$
 * $$2\omega^2$$
 * $$2\omega^2 + \omega$$
 * $$2\omega^2 + n\omega$$
 * $$n_{0}\omega^2 + n_{1}\omega, (n_{0}, n_{1} \in N^{*})$$