0.0...1 is a very interesting number. Normally, one indexes digits by integers (there's a first digit, a second digit,...) But here you have digits seemingly

0.0...1 is a very interesting number. Normally, one indexes digits by integers (there's a first digit, a second digit,...) But here you have digits seemingly indexed by ordinals. The omega-th digit is 1. I don't think you can make a consistent number system this way. Here's the problem: what is 10*0.00...1? It can't have omega-th digit "10" because 10 isn't a digit. It can't have omega-th digit 1 and omega+1-th digit 0, because that should be the same as 0.00...1, under any logic I can think of. So that number system doesn't work. The real numbers, by contrast, have the advantage of working (so far as we know.)