I think "countably infinite" makes no sense to you because you have a different idea of countable than a mathematician (disclaimer: not a mathematician).
A mathematician compares the size of two sets of stuff by pairing off items from each set, but this is not a mechanical process taking a finite or even unbounded amount of time: they just need to show such a mapping exists or that nonexistence would lead to a contradiction; they don't need to actually carry out the process mechanically. By definition (according to mathematicians), something is countable if it is the same size as the set of natural numbers {0, 1, 2, ...} or smaller, and "countably infinite" just means it is the same size as the naturals (and not smaller, which would make it finite).
A small minority of mathematicians hold the position that proof-by-contradiction is not good enough, and that you really do need to positively prove something. They are called intuitionists.
Presumably, an even smaller minority of mathematicians hold the position that this proof must (theoretically) be able to be carried out in a mechanical manner. They're some flavor of constructivists, but maybe they're better called programmers. <- This is where you are.
A mathematician compares the size of two sets of stuff by pairing off items from each set, but this is not a mechanical process taking a finite or even unbounded amount of time: they just need to show such a mapping exists or that nonexistence would lead to a contradiction; they don't need to actually carry out the process mechanically. By definition (according to mathematicians), something is countable if it is the same size as the set of natural numbers {0, 1, 2, ...} or smaller, and "countably infinite" just means it is the same size as the naturals (and not smaller, which would make it finite).
A small minority of mathematicians hold the position that proof-by-contradiction is not good enough, and that you really do need to positively prove something. They are called intuitionists.
Presumably, an even smaller minority of mathematicians hold the position that this proof must (theoretically) be able to be carried out in a mechanical manner. They're some flavor of constructivists, but maybe they're better called programmers. <- This is where you are.