Does every ordinal have a well-defined "next" limit ordinal?

Question

I would like to know whether every ordinal $\alpha$ has a well-defined "next limit ordinal", i.e. a least limit ordinal $\beta$ such that $\beta > \alpha$.

I understand from this discussion that probably not every limit ordinal is such a "next" one. E.g., I assume that $\omega^2$ is the first limit ordinal which is not a "next limit ordinal". But I am still interested whether there is a way to go "from any ordinal $\alpha$ to the next limit ordinal $\beta > \alpha$".

To summarize, my questions are:

1. Does every ordinal have a well-defined "next limit ordinal"?
2. Are there limit ordinals (except 0) which are not a "next limit ordinal"? Is $\omega^2$ the first limit ordinal of that kind?
3. Is the "next limit ordinal" of $\alpha$ always $\alpha + \omega$?

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Update: Questions 1. and 3. are answered below. Question 2. is still unanswered.

Answer 1

Pretty much by design we have that every non-empty collection of ordinals has a least element. In particular, for every ordinal $\alpha$ the collection of limit ordinals greater than $\alpha$ has a least element, which is the ordinal you are looking for.

And we can also see that indeed $\alpha + \omega$ is that least limit ordinal above $\alpha$. To prove this, we only need to check that $\alpha + n$ is never a limit ordinal for $n > 0$ (we know what these are successors of); and that $\alpha + \omega$ is indeed a limit ordinal.