How many objects are in $\mathbf{Set}$?

Question

... or does this question even make sense, considering the object "collection" of $\mathbf{Set}$ is a proper class rather than a set?

Answer 1

There are a few ways to make questions like this precise.

The most basic is - "What are the (necessarily proper) classes which are in bijection with the class of all sets?" For instance: is there a bijection from the class of ordinals to the class of sets? This question is independent of ZFC (or NBG), and we can potentially have a rich structure of classes between Ord and Set. (We can prove that there is a surjection from any proper class onto Ord, so in that sense the class of ordinals is the smallest proper class.) Note that we probably want to work in a theory stronger than ZFC here: ZFC alone doesn't even prove trichotomy for proper classes!

Of course, if you're interested in Set, and not just the class of sets, then you're probably interested in a more category-theoretic perspective. So you'd want to define a notion of "same size" for arbitrary toposes. For instance, for toposes $\mathcal{D}, \mathcal{E}$, we could say $\mathcal{D}\le\mathcal{E}$ if there is an appropriately nice functor (perhaps related to geometric embeddings, see http://ncatlab.org/nlab/show/geometric+embedding) from $\mathcal{D}$ to $\mathcal{E}$. (Of course, there's a problem here - we'd want some form of Cantor-Bernstein to hold, so that $\mathcal{D}\le\mathcal{E}$ and $\mathcal{E}\le\mathcal{D}$ implies the existence of some sort of "equivalence" (not necessarily equivalence in the usual sense) between $\mathcal{D}$ and $\mathcal{E}$, and that's not at all obvious - but we could start with this and see where it goes.) I'm not well-versed in topos theory, but maybe someone can chime in here. In general, my suspicion is that most naturally-occurring large toposes would be significantly bigger than Set in this sense.