This question is me trying to understand (again) why there can be no one-to-one correspondence between the sets of natural and real numbers.
The source of confusion is this: if we abstract completely from the properties and operations on numbers and just think of them as strings of digits, it appears to be possible to list all the real numbers between $0$ and $1$ and all positive integers having the same digits in the same (or reverse) order, for example:
$$0.1 \to 1$$
$$0.01 \to 10$$
$$0.0101 \to 1010$$
$$0.110101 \to 101011$$
I use binary expansion for simplicity.
Since irrational numbers have infinite expansions, we would have to consider some kind of 'infinite natural numbers', like:
I think the main (and only) reason why such numbers don't exist is this requirement:
- We need the sets of real and natural numbers to be ordered.
- We need to be able to do arithmetics on real and natural numbers.
Obviously, if we consider any of two types of 'infinite natural numbers' and the usual absolute value, we find that it's impossible to order them, or do arithmetics with them.
The only way is to change the norm (to $p-$adic norm for example).
While with the real numbers it's still possible to order them, and do arithmetics with them when they have infinite amount of digits in their expansion (or at least in principle).
Is my reasoning correct? Are the two requirements (ordering and arithmetics) enough to explain why natural numbers with infinite amount of digits do not exist, while real numbers do?
This confusion is not restricted to me alone, see this paper for example