I got it! It was easier than I thought. The following expression is a valid bijection between the set of subsets of N and N. It is only valid if we consider the premise that you have stated before: that any natural number has to be finite.

Given A, a subset of N, its position in the sequence of all subsets of N is calculated as follows:

With your premise, if any natural is finite, then the highest element in A is a finite natural, and if it is, the maximum number of naturals that your A can have is also finite (and equal to its highest natural plus 1), so the sum would be finite, and a finite sum of finite naturals is a finite natural.

So now you choose. Either all naturals are finite, and then I have found a bijection between N and the set of subsets of N, or in fact infinite naturals are allowed into the naturals set, in which case I have found a bijection between N and the Reals.

**WHAT DO YOU CHOOSE?**