Largest Useful Sets

Question

I'm just asking this out of curiosity, what are the largest sets that are actually meaningful (infinite sets)? I know that there is no highest cardinal number, but there must come a point where we cannot actually find any interesting set of that size. The largest I can think of is the set of functions from $\mathbb R\rightarrow \mathbb R$, but this is just the power set of $\mathbb R$ and so not particularly big.

Answer 1

First of all, define useful.

The notion of large cardinals, which are sets so large that you cannot even begin to understand how large they are, and then there are even larger large cardinals, is useful. The current known proof of Fermat's Last Theorem uses these cardinals, or rather an equivalent concept called "universes".

The question whether or not all "nicely definable" sets of real numbers have certain properties are also determined by the existence of fairly huge sets in the universe.

And that is before we talk about uses of large cardinals in set theory, or model theory, or the weird appearances of very large cardinals in homotopy theory.

That been said, of course that if your definition of interesting is "what they do in classical measure theory/functional analysis/classical analysis/classical number theory and not a tiny bit of generality more!" then it is true that it's hard to see the point of using sets larger than the continuum, or at most its power set.

Finally, let me just point out that we can't even decide how large is $2^{\aleph_0}$ from the axioms of $\sf ZFC$, and certainly not $2^{2^{\aleph_0}}$ either. So both can be fairly large in terms of $\aleph$ numbers, but of course neither is even remotely as large as the least inaccessible cardinal (which is the smallest notion of a large cardinal).