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 infinite/uncountable classes to call all of them "infinite/uncountable".


The context is: How do I have to proceed with "there are *finitely* many, *infinitely* many, *countably* many, *uncountably* many, *weakly inaccessibly* many, *strongly inaccessibly* many, ..." (see [Inaccessible Cardinals](http://en.wikipedia.org/wiki/Inaccessible_cardinal))?



Simple adjective for "of the size of a proper class"?
**Proper-class-many**.



> 
> “We show that if there exist proper class many Woodin cardinals, then the set of reals x for which there is exists an ordinal α with {a ∈ Pω1 (α) | x ∈ L[a]} stationary is countable.”    —Paul Larson, *Reals constructible from many countable sets of
>  ordinals*.
> 
> 
> 


It’s grammatically ugly, but mathematically transparent and unambiguous, and rolls off the tongue reasonably well. I’ve heard it used by and among set theorists, category theorists, and homotopy theorists, without confusion — I’m pretty sure it’s as widely understood as anything will be for this distinction.


