I'm not a set-theorist, but a set-theorist friend of mine once taught me that you can turn a countably-infinite set (such as the integers) into an uncountably-i

I'm not a set-theorist, but a set-theorist friend of mine once taught me that you can turn a countably-infinite set (such as the integers) into an uncountably-infinite one (like the reals) by applying the 'power set' operation (the set of all subsets). Not heaps sure what this really means with respect to whether 1 the integer is really completely related (as in, equal, or the-exact same-thing) to 1.0 the real though. Kinda seems like it might still need a bit more information to fully identify a real, even when it happens to be infinitely-close to an integer?