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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?

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There are successor ordinals; $3$, $\omega+2$, and $\omega^2+\omega\cdot5+1$ are all examples. It you could draw a picture of one of these ordinals, it would end in $\cdot\cdot$.

There are ordinals that are limits of successor ordinals but not of any kind of limit ordinal; $\omega$, $\omega\cdot2$, and $\omega^{\cdot2}+\omega\cdot5$ are all examples. (I write $\omega^{\cdot2}$ instead of $\omega^2$ to emphasize that I’m talking about ordinal exponentiation, not cardinal exponentiation.) A picture of any one of these would end in a sequence of dots $\cdot\cdot\cdot\ldots$, i.e., in a copy of $\omega$. Call these limit ordinals of type $1$.

There are ordinals that are limits of type $1$ ordinals but not of any higher type; $\omega^{\cdot2}$ is indeed the first of these. It’s the limit of the type $1$ limit ordinals $\omega,\omega\cdot2,\omega\cdot3,\dots$, and so on. The next $\omega$ of them are the ordinals $\omega^{\cdot2}\cdot n$, looking like $n$ copies of $\omega^{\cdot2}$ strung end to end. Others are $\omega^{\cdot\omega}+\omega^{\cdot2}\cdot5$ and $\omega^{\cdot4}\cdot2+\omega^{\cdot3}+\omega^{\cdot2}\cdot3$. Call these type $2$ limit ordinals.

There are ordinals that are limits of type $2$ ordinals but not of any higher type; the first of these is $\omega^{\cdot2}\cdot\omega=\omega^{\cdot3}$; its picture looks like $\omega$ copies of $\omega^{\cdot2}$ laid end to end. The next is $\omega^{\cdot3}\cdot2$, and so on.

But there are countable limit ordinals that are not of type $n$ for any finite $n$. The first is $\omega^{\cdot\omega}$, which is the limit of $\omega$, $\omega^{\cdot2}$, $\omega^{\cdot3}$, and so on; call it a limit ordinal of type $\omega$, because it really is the smallest ordinal not of one of the finite types. And of course the next ordinal of this type is $\omega^{\cdot\omega}\cdot2$, two copies of $\omega^{\cdot\omega}$ end to end.

And so on, and so on, and so on, and $\ldots~$. In this rather ad hoc terminology there are countable limit ordinals of type $\alpha$ for every countable ordinal $\alpha$. (It’s all vaguely incestuous!)

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  • $\begingroup$ no comments!!! thanks!!! $\endgroup$ – Wolphram jonny Mar 23 '13 at 3:50
  • $\begingroup$ @julian: You’re welcome! $\endgroup$ – Brian M. Scott Mar 23 '13 at 3:51
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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.)

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It is quite trivial. You can prove by induction that those "n-th level limit ordinals" are just multiples of $\omega^n$, and this generalizes to infinite $n$ in the natural way.

Perhaps interestingly (or insightfully) for you, there are countable ordinals $\alpha$ such that $\omega^\alpha=\alpha$ (this is due to exponentiation being a normal function). The smallest such ordinal is called $\varepsilon_0$, and this notion can be generalized to produce the so-called epsilon hierarchy (see the Wikipedia article on $\varepsilon_0$).

As far as I can tell, you can't really see how ordinals become uncountable -- they just are that, at some point, and uncountability of an ordinal is not inherent in the sense that we can have two (transitive) models of ZFC with the same ordinals, and yet have ordinals which are very uncountable in one of the models, at the same time being actually countable in the other one (this is one of the most basic forcing constructions), so you can't really hope for a way to see uncountable ordinals without more or less explicit use of cardinality.

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  • $\begingroup$ do you mean that $\omega_1$ is not the first uncountable ordinal in all models of ZFC? $\endgroup$ – Wolphram jonny Mar 23 '13 at 3:19
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    $\begingroup$ @julian: He means that different models may think that different ordinals are $\omega_1$. That is, the notion of "$\alpha\text{ is the first uncountable ordinal}$" is not absolute. $\endgroup$ – Asaf Karagila Mar 23 '13 at 10:14

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