Hello,

I am reading Rosenstein's "Linear Orderings" and I am not sure if I am missing something, or if there is an error.

He gives the definition of a $\beta$-limit ordinal inductively, as follows (his exact wording is given):

- $L_0 = $ all ordinals
- $\alpha\in L_{\beta+1}$ if and only if $\alpha=\lim \{\alpha_m | m<\omega\}$ where each $\alpha_m\in L_\beta$
- If $\lambda$ is a limit ordinal, $\alpha\in L_\lambda$ if and only if $\alpha\in L_\beta$ for all $\beta<\lambda$

He then remarks that the 1-limit ordinals (for example) are the limit ordinals, the 2-limit ordinals are precisely the limit ordinals which are limits of limit ordinals, and so on.

But, what about $\omega_1$? It has uncountable cofinality, so no strictly increasing $\omega$-sequence of ordinals will limit to it. So $\omega_1\not\in L_1$, unless we all constant sequences, in which case $L_\alpha$ is the class of all ordinals, for all $\alpha$.

One obvious modification is to allow any cofinality of a sequence in the definition (2). However, the chapter is especially concerning the finite condensation, which naturally collapses $\omega$-sequences. So, I'm not sure if this is the right answer, or if he really only means ordinals with countable cofinality, or what.

Unfortunately, I have no idea if this terminology is standard, or just useful for this book, so I don't know if people here will have an answer.

To summarize: either the definition above, or the comments proceeding it, or both, are wrong. I would like to know the correct definition, if it is known.Blockquote