rationalskepticism.org

LjSpike wrote:(If you have a solid explanation using terms which have a concrete value, to prove the infinities as equal, I'll give in).

First, you should realise that these correspondences have nothing to do with ratios, or proportions, and that none of it generalises the sort of stuff you do in algebra with real numbers. This is discrete mathematics, a completely different branch of the subject, and set theory in particular. The theory was laid down towards the backend of the 19th century, and is now part of the basic vocabulary that mathematicians use to talk about an axiomatic notion called sets.

I suspect you haven't properly done functions yet at school, and knowing exactly what counts as a function is pretty important for getting into the subtleties of this stuff. But the basic idea is that a function is a rule which tells you how to go from one set of objects to another. It's like a function in programming, except all you're allowed to do is take an input and compute an output.

Some functions send two different inputs to the same output. Some functions send any two inputs to two different outputs. The latter are called "injective." Some functions can produce any output in the output set for a suitable input. Such functions are called "surjective." Functions which are both are called "bijective", or "one-one correspondences."

On these definitions, we say that there is a one-one correspondence from the naturals to the rationals.

Why one-one correspondences are of interest is something you learn by studying set theory.