TITLE: Is the proper class of all ordinals equivalent to the potential infinity of pre-Cantor times?
QUESTION [10 upvotes]: 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?

REPLY [6 votes]: 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 ...)