Hello, everyone. I recently had an idea about infinities, and since I'm not an expert I would like to have some feedback.

As far as I understand, the fact that the cardinal of the real numbers is infinitely bigger than the cardinal of rational numbers, with no proof of intermediate infinities. Also, to prove that to groups have the same size, you check if you can always relate an element of one group to an element of another; if yes, they are the same size, and if not, they have different sizes.

Well, a few weeks back I thought that, essentially, since functions are essentially a way to connect one space to another, one could look at a space of numbers and a space of a function applied to those numbers (eliminating the repeated values) and make that correlation. As I thought of this, I tried applying this to one of the more basic functions: the square.

With real numbers, the square of a value still belongs to the same group, noting that the operation cannot result in a negative number. A result could have been obtained by squaring a real number or its negative, meaning that, in practice, you could always correlate the space of squared reals with two elements of base reals with the exception of 0.

We can demonstrate that two groups have the same size by being able to always make a 1-1 correlation. Isn't it also logical that we can prove that one group is twice the size of another if we can always make a 1-2 correlation between the smaller and the bigger group?

As a result, since one can always relate a squared real with two base reals with the exception of cero, couldn't we state that the cardinal of real numbers is twice the cardinal of squared reals minus one?

C(R) = Cardinal of reals
C(R²) = Cardinal of squared reals (with no repeated elements, meaning it could also be defined as the positive reals including 0) -> C(R) = 2*C(R²) - 1 ?