
Re: Ultrafinitism
well the objects of Cat are categories, so many of the objects themselves are proper classes. it's not even "locally small" because the morphisms are functors. to se how bad this is, realize that for every set X, we have a free group on X, FX, and F is a functor from Set→Grp, so just this one collection of morphisms for these 2 objects that is Hom(Set,Grp) is a proper class itself. and it gets worse: if one considers Cat as an object of Cat, one runs into the same problem as with the Russel set, so Cat cannot be a proper class. Cat, even if you restrict it to having objects that are proper classes or sets, is mindnumbingly HUGE, because of the wide variety of making new categories from old categories.
so a lot of authors only consider "small categories" (the objects form a set), or "locally small categories" (Hom(A,B) is a set for A,B in the category C).
you see, when we invoke a general set A, ZFC requires we have some set B that contains A. well, eventually that leads us to the collection of all sets, which cannot be a set (since no set contains it). so an arbitrary set is an element of something that is not a set, a class. and classes are BIG (the class of all sets has every set we can think of, right?). and there is more than one class, there's LOTS of those, too (any kind of structure which has an associated free structure gives rise to a class).
and what do you call the collection of all proper classes? then rinse, repeat.