> Let's talk about real numbers between. 0 and 1 (because we can map all of them to this range anyway). I can (count) enumerate all of them. Simply flip the dig

> Let's talk about real numbers between. 0 and 1 (because we can map all of them to this range anyway). I can (count) enumerate all of them. Simply flip the digits over the decimal point, so 0.14159 become integer 95141. Every integer has a finite number of numerals. Proof: by definition, each natural number is either 0 or a successor to 0. 0 is finite. And if n is a finite length, then n + 1 is a finite length. So by mathematical induction, all natural numbers have finite length. A similar argument covers the negative numbers, so all integers have finite lengths. Since the decimal expansion of certain rational numbers are infinite (e.g., 1/3 = 0.333...), the strategy of mapping each number between 0 to 1 to an integer by flipping digits over the decimal point does not work for all rationals, let alone all reals.