[scapp]

Problem 1: 1/(2^aleph_0) isn't a real number. The real numbers don't contain infinitesimals. It's possible to formalize a number that behaves like 1/(2^aleph_0) "ought to" (surreals would be one possible approach), but the result won't be a real number.

Problem 2: There's no natural number that maps to (say) 1. Even if you do allow 1/(2^aleph_0), there's no finite number n that would make n/(2^aleph_0) = 1. With any reasonable definitions of the operations involved here, n/(2^aleph_0) would always be infinitesimal, so it would never equal a non-infinitesimal.

Problem 3: You're still skipping over infinitely many numbers. If 1/(2^aleph_0) is a number (and again, this requires going beyond the real numbers) and 1.5 is a number, then 1.5 * 1/(2^aleph_0) = 1.5/(2^aleph_0) is also a number, but no natural number gets mapped to that.