Factorization of Limit Ordinals

Theorem
Let $x$ be a limit ordinal.

Then:


 * $x = ( \omega \times y )$ for some $y \in \operatorname{On}$

where $\omega$ is the minimal infinite successor set.

Proof
By the Division Theorem for Ordinals,


 * $x = ( \omega \times y ) + z$ for some unique $y$ and $z \in \omega$

Aiming for a contradiction, suppose $z \ne 0$.

Since $z \in \omega$, $z$ is not a limit ordinal.

Therefore, by the definition of limit ordinal, $z = w^+$ for some $w \in \omega$.

But this means that:

This means that $x$ is the successor of some ordinal and cannot be a limit ordinal.

But this contradicts the fact that $x$ is a limit ordinal, so $z = 0$.

Therefore: