TITLE: Is there a hierarchy of countable limit ordinals that cannot be reached by repeated successor operations over limit ordinals of the previous level?
QUESTION [5 upvotes]: I am trying to get an intuition on the structure of the countable ordinals (actually, of how they become uncountable), and these kind of questions, which might be trivial for set theorists, can help a lot to the limited mind of a physicist. Let us limit here to countable ordinals.
If I am not wrong, the first limit ordinal that cannot be reached by repeated successor operations over limit ordinals is $\omega^2$. I would call it a limit ordinal of the second level. There are many limit ordinals of the second level(I am not sure what would be the correct definition, but I guess the next one is $\omega^2$.2, then $\omega^2$.3, and so on. (But I am not sure, perhaps after $\omega^2$ comes $\omega^3$?).
Then, we can define the limit ordinals of the third level as the limit ordinals that cannot be reached by repeated successor operations over limit ordinals of the second level. And so on. Can then limit ordinals of any level n be defined? or of a transfinite level? Is this a trivial issue?

REPLY [4 votes]: There are many hierarchies that one can describe.
One can talk about successors, limits of successors, limits of limits of successors, limits of limits of limits of successors, and so on. This property could really be described as a Cantor-Bendixson rank, because when treated as a topological space the rank of the ordinal in this hierarchy is exactly its Cantor-Bendixson rank.
If $\{\alpha_i\mid i<\omega_1\}$ is a closed and unbounded family of ordinals (where $\alpha_0=0$), one can talk about an ordinal having rank $i$ if it lies in the interval between $[\alpha_i,\alpha_{i+1})$. The one can talk about $\varepsilon$ numbers, about admissible ordinals, ordinals which are possible heights of transitive models of $\sf ZFC$, and so on and so forth.
For example, if we take the admissible ordinals, it is a theorem of Sacks that $\alpha$ is a countable admissible ordinal if and only if it is $\omega_{\beta}^{CK}$, for some $\beta<\omega_1$. Then an ordinal $\gamma$ having rank $\beta$ means that $\gamma$ is an ordinal computable by an oracle strong enough, and it tells us how strong this oracle should be.
But what's oh-so-very important to understand is that often these hierarchies are going to have fixed points. So you get $\alpha$ which is its own rank. This tells you that $\alpha$ is mind numbingly huge if you try to explicate its construction by the means you created the hierarchy.
This really tells you that countable ordinals can get very very very hard, and very very very complicated.
And of course, there's the last issue, pointed out by tomasz, if $M$ is a countable transitive model of $\sf ZFC$, then there is some countable $\alpha$ such that $M$ thinks $\alpha$ is uncountable, the least such $\alpha$ is denoted by $\omega_1^M$. In this case we can add a certain bijection of $\alpha$ and $\omega$ to $M$ and generate a new countable transitive model - $N$, which has the same collection of ordinals, but in $N$ the ordinal $\alpha$ is countable. So the notion of being $\omega_1$ of a countable transitive model of $\sf ZFC$ is not absolute, and different models see different ordinals as their [internal] $\omega_1$.
All the above paragraph is a huge technical mumbo-jumbo telling you just one thing, you can't really prove a lot about $\omega_1$ from "below", or from "inside", in the sense that you don't really have an absolute measure of how many ordinals are below $\omega_1$. (And if one tosses large cardinals into the game, things become even more complicated quite easily.)