For a finite set, one can describe a uniform distribution. There isn’t a natural way to do so for a countable set. But for a hyperfinite set, one can describe a uniform distribution through a probability density. So in some ways the countable is “bigger”.
Why would this property mean that it is “bigger”? You can construct a uniform distribution of a uncountable set through a probability density as well. However, using the same measure on a countably infinite subset of the uncountable set would show that the countable set has measure 0.
Or a counterexample from the other direction would be that you can’t describe a uniform distribution of the empty set either (I think). And that would feel even weirder to call “bigger”.
How can the uniform distribution on the natural numbers be used?
The point is that such a distribution (uniform on countable infinite set like naturals), is not internal, and therefore external. it’ll depend on the specific ultrafilter used under the hood.
for how to use it, see either alain roberts or sylvia wenmackers
Could you give a sneak peak on how Sylvia /Alain use these uniform distributions?