TITLE: Is the set of countable limit ordinals countable?
QUESTION [6 upvotes]: It is probably a trivial question, but I have no clue about the answer. I know that the elements of $\omega_1$, the first uncountable ordinal, are the countable ordinals, of which there are uncountably many, but I am not sure if that applies to the number of limit ordinals before $\omega_1$ too.

REPLY [5 votes]: Here is a slightly worse proof than that given by Brian.
Suppose that the $A$ is the set of countable limit ordinals, then $\bigcup A=\delta=\sup A$ is a countable limit ordinal, therefore $\delta\in A$, but this means that $\delta=\max A$. It is easy to see that $\delta+\omega>\delta$ and it is a countable limit ordinal, in contradiction to the fact that $\delta$ is the largest countable limit ordinal.