>>14
the question is whether or not a set can be put into one to one correspondence with its power set, thats the set of all of its subsets, not whether it can be put into one to one correspondence with each of its subsets. it obviously cant be done for finite sets, so then the question is whether it can be done for infinite sets, and the answer is no, because of similarity to irrationality.
you can pair a natural number with any finite set of natural numbers, and with any patterned set of natural numbers, the problem is subsets that are infinite but follow no discernable pattern, i.e. {4,86,923,924,925,175312,58971768943,6677777777777...}
if you dont have a finite way to represent it, you cant represent it as a finite natural number. you would need a natural number that approaches infinity as you write it or something.