Set Theory: Set of limit ordinals does not exist

Question

I am trying to show that a set of limit ordinals do not exists where X as a limit ordinal cannot be a set. Can anyone explain why you can't get a set of limit ordinals?

Answer 1

It is well kwown that $\operatorname{Ord}$, the class of all ordinals, is not a set.

Write $L$ for the class of limit ordinals. We have a surjective functional $F : L \mapsto \operatorname{Ord}$ given by $\lambda \mapsto \alpha \textrm{ such that }\aleph_\alpha = |\lambda|$. Hence, $L$ can't be a set because if it were then the image of $F$, i.e. $\operatorname{Ord}$, would be a set.

Answer 2

Here's a proof that goes like Euclid's proof of the infinitude of the set of primes. For every ordinal $\alpha$, there is a limit ordinal strictly bigger, $\alpha + \omega$. The union of a set of ordinals is an ordinal. So if the class of limit ordinals were a set $LO$, we could take $\bigcup LO + \omega$, a strictly larger limit ordinal, which contradicts $LO$ containing all the limit ordinals.