A set is "countably infinite" if you can construct a one to one mapping from the set to the natural (counting) numbers. The natural numbers are of course count

A set is "countably infinite" if you can construct a one to one mapping from the set to the natural (counting) numbers. The natural numbers are of course countably infinite, but so are even numbers, prime numbers, fractions, algebraic numbers and lots of other kinds of numbers that people want to do arithmetic with. "Uncountably infinite" sets can't be put into a one to one correspondence with the naturals, no matter how you produce to produce such a mapping, it's always possible to produce a number in the set which wasn't part of it. Real numbers being the prototypical example of that, but the power set (the set of all the subsets) of the naturals is, also. People do arithmetic with all kinds of things that "aren't numbers". Math isn't restricted to working with numbers.