I recently watched a YouTube video on Banach-Tarski theorem (or, paradox). In it, the presenter builds the proof of the theorem on the basis of a non-intuitve assertion that there as as many even numbers as there are whole numbers, which he 'proves' by showing a 1:1 mapping between the two sets.
But would that constitute a valid proof?
To me, the number-density (per unit length of the number-line) for whole numbers is clearly more than that for even numbers. And this, I'm sure, can also be trivially proved by mathematical induction.
Later, in the same video, it is shown how the interval [0,1] contains as many real numbers as there are in the real number line in its entirety. Once again, using the common-sense and intuitive concept of 'number-density', there would be clearly (infinitely) more real numbers in the entire number line than a puny little section of it.
It seems, the underlying mindset in all of this is: Just "because we cannot enumerate the reals in either set, we'll claim both sets to be equal in cardinality." In the earlier case of even and whole numbers, just "because both are infinite sets, we'll claim both sets to be equal in cardinality." And all this when modern mathematics accepts the concept of hierarchy among even infinites! (First proposed by Georg Cantor?)
Is there a good, semi-technical book on this subject that I can use to wrap my head around this theme, generally? I have only a pre-college level of knowledge of mathematics, with the rest all forgotten.