> So then it’s only proving that if you choose the ordering ahead of time, you won’t be able to do it, because real numbers don’t have a predefined order. You can only order them as you produce them.
What's the problem? You just produce the diagonalized number at the same time as you produce your order. Every order you produce them in has its own corresponding diagonalized number that will never be produced.
> If you re-sort after each iteration (or insert them in order), you can count as many real numbers as you want.
You can count as many real numbers as you want, but none of them will ever be equal to the diagonalized number. (That is, in finite time, you will always be able to find a digit that differs between the two numbers.)
> How many bits can be encoded in the universe? That will give the limit of what could ever be counted. And it’s not infinite.
How do you know the universe isn't infinite?
Regardless, I don't see why we shouldn't consider formal systems in which infinite amounts of information can be manipulated. Even in our finite light cone, mathematical statements about infinite sets can help us separate the possible from the impossible. For instance, mathematics tells us that adding 1 to any integer will never produce the same integer. There are infinitely many integers, so we can't experimentally verify this in the real world. But if we accept that abstract statement, then we know what to expect when we do try to physically add one object to a group of objects.
What's the problem? You just produce the diagonalized number at the same time as you produce your order. Every order you produce them in has its own corresponding diagonalized number that will never be produced.
> If you re-sort after each iteration (or insert them in order), you can count as many real numbers as you want.
You can count as many real numbers as you want, but none of them will ever be equal to the diagonalized number. (That is, in finite time, you will always be able to find a digit that differs between the two numbers.)
> How many bits can be encoded in the universe? That will give the limit of what could ever be counted. And it’s not infinite.
How do you know the universe isn't infinite?
Regardless, I don't see why we shouldn't consider formal systems in which infinite amounts of information can be manipulated. Even in our finite light cone, mathematical statements about infinite sets can help us separate the possible from the impossible. For instance, mathematics tells us that adding 1 to any integer will never produce the same integer. There are infinitely many integers, so we can't experimentally verify this in the real world. But if we accept that abstract statement, then we know what to expect when we do try to physically add one object to a group of objects.