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

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 $$\alpha + \omega$$?

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

## 1 Answer

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.

• Thanks a lot! Can you positively answer my second question, too? (I edited my question)
– blk
Commented May 14 at 12:29