[alexey-salmin]

Natural numbers are "natural" enough but N as the "set of all natural numbers" not so much. It only takes N to build the Hilbert's hotel. Uncountable set of all subsets of N is probably even worse.

All that, of course, doesn't make N bad or useless. It just shows that mathematical objects don't have to follow the laws or intuition of the real world to be useful in the real world.