This may have been asked before, since it seems like an obvious question, but I couldn't find it.
I just read some interesting stuff about set theory and how the size of the natural numbers is not the same as the size of the real numbers.
How I understand it is that there is no function $f$ whose domain is all natural numbers and whose range is all real numbers such that $f^{-1}$ also exists.
Obviously, if we allow each natural number to pair with two real numbers instead of just one, there still aren't enough natural numbers. This is pretty intuitive.
But what if we allowed each natural number to pair with countably infinitely many real numbers. Then does a mapping exist?
Mathematically, does there exist a function $f$ (domain: all real numbers, range all natural numbers) such that for every natural number $n$, the set of real numbers $r$ satisfying $f(r) = n$ is countably infinite?