It's just a wording question: 


> How does one tell - by a simple
> adjective - that a collection is "of
> the size of a **proper** class"?

Their might be several sizes of proper classes, but on the other side, it's not a problem that there are several sizes of uncountable classes to call all of them "uncountable".

The context is: How do I have to proceed with "there are *finitely* many, *countably* many, *uncountably* many, *weakly inaccessibly* many, *strongly inaccessibly* many, ..." (see [Inaccessible Cardinals][1])?


  [1]: http://en.wikipedia.org/wiki/Inaccessible_cardinal