The trouble is that with a little cleverness, you can account for all of the rationals by using some of the natural numbers (once each) and still have infinitely many natural numbers left over. (Left as an exercise to the reader.) That’s why your intuitive notion isn’t going to be self-consistent.
The trouble is that with a little cleverness, you can account for all of the rationals by using some of the natural numbers (once each) and still have infinitely many natural numbers left over. (Left as an exercise to the reader.) That’s why your intuitive notion isn’t going to be self-consistent.