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The wikipedia page on ordinal numbers says:

An ordinal number is used to describe the order type of a well ordered set (though this does not work for a well ordered proper class).

What is meant with

though this does not work for a well ordered proper class

?

Isn't it possible to extend the notion of ordinal number so that we have ordinal numbers of proper class size?

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To a certain extent, yes, we can treat well-orderings of proper class length; but the resulting objects are no longer sets, and treating them is somewhat difficult. And I see no way of getting proper class ordinals.

Let's recall a bit how ordinals are constructed: we began with the notion of a well ordered set, and then found canonical representatives of each well-ordertype (these are the ordinals).

So let's do the same thing here. A class well-ordering is a class $A$ together with a class $R$ of ordered pairs of elements of $A$, such that every nonempty subset of $A$ has a least element according to $R$. (It might see like I should say "every nonempty subclass, but it turns out that using subsets is enough.)

Now this raises two immediate problems:

  • Talking about specific class well-orderings is easy enough; but how should I write something like "Every class well-ordering is ..."? That would involve quantifying over classes, which can't be done in ZFC.

  • Moreover, there's no obvious way to develop canonical representatives! So even though class well-orderings make sense, there's no obvious notion of "class ordinal".

That said, we can treat class well-orderings to a certain extent; see e.g. this mathoverflow question or this other mathoverflow question. But the point is that things - even really basic things we take for granted - which are true about well-ordered sets and ordinals, can be false for class well-orders.

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  • $\begingroup$ What's the problem with this approach: informally, a class is a collection of sets and a meta-class is a collection of classes. We can write down formal axioms handling classes (Morse-Kelley). Similarly, we can write down formal axioms handling meta classes and meta meta classes and so on. Now we could define an ordinal to be the collection of all isomorphic well orderings. When I say "well ordering", then I allow proper class domains. $\endgroup$
    – Please Help
    Feb 22, 2017 at 21:01
  • $\begingroup$ Is your point that in Morse-Kelley, the approach with represantatives doesn't work? Because I'm assuming you are not working in ZFC (which only handles sets), because there we can't even speak about proper classes (nor about ordinals having proper class size). But the approach with represantatives would work if we had some stages (meta classes, meta meta classes, .....), right? $\endgroup$
    – Please Help
    Feb 22, 2017 at 21:04
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    $\begingroup$ @PleaseHelp I was indeed working in ZFC, since you didn't specify otherwise. And ZFC can handle classes to a certain degree - a class in a model of ZFC is a collection which is definable (with parameters) in that model, and we can reason about individual classes internally and general classes externally. If you invoke "meta$^n$-classes" as you suggest, then yes, we can make sense of ordinals at arbitrary levels; but then the "meta$^\omega$ class" ordinals are still untreated, even though they can be reasonably definable (e.g. the meta$^\omega$-class of all ordinals in all senses). (cont'd) $\endgroup$
    – Noah Schweber
    Feb 22, 2017 at 21:28
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    $\begingroup$ By Burali-Forti, you're never going to escape something like this happening - in any theory you choose, there will be "reasonably definable" well-orderings which are too long to be treated nicely in the theory. $\endgroup$
    – Noah Schweber
    Feb 22, 2017 at 21:28
  • $\begingroup$ "But the point is that things - even really basic things we take for granted - which are true about well-ordered sets and ordinals, can be false for class well-orders." Could you give an example? $\endgroup$
    – Please Help
    Feb 22, 2017 at 22:54

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