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):


1. $L\_0 = $ all ordinals
2. $\alpha\in L\_{\beta+1}$ if and only if $\alpha=\lim \{\alpha\_m | m<\omega\}$ where each $\alpha\_m\in L\_\beta$
3. 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



Definition of $\beta$-limit ordinals
Not only the remark right after the definition, but also Exercise 5.2 ($\alpha$ is a $\beta$-limit ordinal iff the last exponent in the Cantor normal form of $\alpha$ is at least $\beta$) suggest that Rosenstein meant to allow any cofinality, as you suggested. This definition seems very natural, but I do not know if it is used elsewhere.


It seems to me that Rosenstein does not use the notion of $\beta$-limit points a lot; the last usage seems to be in Proposition 5.7, which among other things states: 


* If $x$ and $y$ are distinct $\beta$-limits of a well-order, then they will not be identified by a $\beta$-fold application of the condensation operator. (The condensation operator maps an order to a quotient, identifying two points if they span a finite interval.)


This theorem is true with "your" definition of $\beta$-limits; it is of course also true (but weaker) with Rosenstein's definition read literally, since in this case ordinals of uncountable cofinality are not $\beta$-limits. 


