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My understanding is that the class of all ordinals is, by definition a proper class. This in the end is done to avoid a paradox: the collection of all sets would be paradoxical if you allow it to be a set, because then you could consider the power set of this set, and so on. But is this solution satisfactory? Is in it equivalent to consider the number of elements in this proper class a potential infinite that can never be reached? Its cardinality is not even defined. It looks to me a lot like $\omega$ (the class of all N): before cantor it was not a "number", just a potential infinity. Today we not only have $\omega$ well defined, but also all the zoo of transfinite ordinals greater than it. But apparently, we reached a new wall with the class or all ordinals. Is this a temporary wall, of is it the largest possible infinite that humans can imagine (at least today?)
Update: I agree that you can have models $V_\kappa$ that do not contain all ordinals. But in a response to a previous question, I was told that V is perhaps the largest possible universe, and "... the class of ordinals should really be seen as a central anchoring point which doesn't change when we do our favourite constructions". If you do not like V, or you do not consider it large enough, we can define a super universe U consisting of all sets of all models of all possible set theories. It will still have at most all the ordinals (just by definition, or am I wrong?). The analogy I made between $\omega$ and the class of all ordinals is because: 1) given a subset of N you can always find a larger n, with the true infinity not reachable (well, today we can, and we call it $\omega$). Same for the ordinals, you can define any universe you like, you can always find a larger ordinal, but never reach the class of all ordinals. And by definition, there exists the collection of all N, and there exists the collection of all ordinals, ORD. But you can find things much larger than N, but not than ORD? The question is if the reason is we could not make a meaningful definition of a collection larger than ORD yet, or it will be always beyond reach?

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  • $\begingroup$ Not a single comment? is this question so dumb? at least let me know if that is the case. Thanks! $\endgroup$ – Wolphram jonny Apr 13 '13 at 18:06
  • $\begingroup$ I wrote a very long answer, but then I figured it may confuse you instead of helping you... so I didn't post it at all. $\endgroup$ – Asaf Karagila Apr 14 '13 at 21:59
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    $\begingroup$ @Asaf if you still have it saved in some file, please post it, it cannot do any any harm, I live in permanent state of confusion! $\endgroup$ – Wolphram jonny Apr 15 '13 at 2:24
  • $\begingroup$ @Asaf I put up an update in the answer, it was too large for a comment. $\endgroup$ – Wolphram jonny Apr 16 '13 at 16:29
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The notions are very different, to my best understanding.

As far as I could understand it, the pre-Cantorian infinity was primarily a notion of a length which is longer than any other. It was closer to the infinity we meet in real analysis, rather than the infinity we deal with in set theory. Despite the informal similarities that one can talk about "$f(\infty)$" and "$\alpha\in\sf Ord$", as if both these were actual objects of their universes.

Cantor noticed that one can consider a queue of infinite length, and that it makes sense to have an infinite queue, and then some. From there he defined the ordinals and the cardinals, and the rest is history.

On the other hand, from a modern point of view, the class of ordinals need not be absolute. One can consider end-extensions, which really add more ordinals to the universe, a typical example is taking an inaccessible $\kappa$, then $\bf V$ is an end-extension of $V_\kappa$.

In one considers theories like the Tarski-Grothendieck, or equivalently the theory $\sf ZFC+$"There is a proper class of inaccessible cardinals" then one can consider the least inaccessible as the universe of sets, and then one can always find a larger and larger universe with more and more ordinals. If one goes to even stronger cardinals (e.g. Woodin cardinals) then these properties of "extending higher and higher" can get stronger and stronger.

From another point of view, one can consider the multiverse approach which was proposed by Joel D. Hamkins (see J. D. Hamkins, The set-theoretic multiverse, Review of Symbolic Logic 5 (2012), 416–449.), which says among other things that every universe of set theory is really a countable model from the point of view of another universe (see also this blog post by Francois Dorais).

This is a much stronger assumption that the proper class of inaccessible cardinals. There we had a sequence of universes, each larger than the last, but none was countable in any of the extensions. In fact they all agreed on their common cardinalities and sets. In this case the universes become smaller and smaller as we go along.

So whereas the potential infinity of the early 19th century was somewhat of "the energy of an unstoppable object"; the class of ordinals is something vastly more frightening in size. But at the same time, the calculus notion of infinity is very coarse and hardly at all manageable. On the other hand, the class of ordinals is a concrete class (for a given universe, of course), which can be managed and manipulated internally. This is added by the above facts that the ordinals can be made a set of a larger universe; or even a countable set of a much larger universe.

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  • $\begingroup$ Thanks! I didn't know about the Hamkins conception, I am gonna read the article. Regarding the first part, I have a few comments, but I am going to post them tonight. $\endgroup$ – Wolphram jonny Apr 15 '13 at 13:53
  • $\begingroup$ julian, I have to admit that I did not yet read Joel's paper in full. I only looked around here and there. I should probably read it sometime soon, but then again my reading list is so damn long... :-) $\endgroup$ – Asaf Karagila Apr 15 '13 at 13:55
  • $\begingroup$ I put up an update in the answer, it was too large for a comment $\endgroup$ – Wolphram jonny Apr 16 '13 at 16:29
  • $\begingroup$ @julian: This was the reason I felt my answer might confuse you. When you want to construct the real numbers, or give a foundation to a particular Hilbert space (or whatever), then you fix one universe of $\sf ZFC$ - which doesn't even have to be a set - and you work inside that one. And that is enough for you to do whatever you want. But on the other hand, equating an "absolute" and "potential" infinity from various philosophical point of view one runs into philosophical foundations on which set theory universes are built upon. $\endgroup$ – Asaf Karagila Apr 16 '13 at 17:31
  • $\begingroup$ Don't worry Asaf, and thanks again. I realized I am trying to understand a concept that is beyond my reach, like trying to understand quantum mechanics without understanding linear algebra, or calculus, or something like that. So, I'll have to find the time to study the topic rigorously (and stop making nonsensical questions). $\endgroup$ – Wolphram jonny Apr 17 '13 at 2:08
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Just a short historical footnote to Asaf's posting. Michael Hallett's rather wonderful book Cantorian set theory and limitation of size (OUP 1984) is a terrific source if you want to understand (1) something more about pre-Cantorian conceptions of the infinite (2) Cantor's distinction between the transfinite and the Absolute, and (3) how Cantor regarded e.g. the collection of all ordinals.

As Hallett intriguingly shows, there was a powerful theological drive behind Cantor's ideas, underpinning his set-theoretic realism. (It is a nice question whether modern realists who drop the underpinnings are thereby better off ...)

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