Zermelo and Cantor did. They concluded there were countably many, which turned out to be equivalent to the Axiom of Choice.
This isn't right - aleph numbers are indexed by all ordinals, not just natural numbers. What's equivalent to AC is that the aleph numbers cover all infinite cardinals.
This is our monthly thread for collecting arbitrarily contrived scenarios in which somebody gets tortured for 3^^^^^3 years, or an infinite number of people experience an infinite amount of sorrow, or a baby gets eaten by a shark, etc. and which might be handy to link to in one of our discussions. As everyone knows, this is the most rational and non-obnoxious way to think about incentives and disincentives.