I'd like to understand - Can you explain this? It seems like it would be easy to have a Turing machines that uses the other two Turing machines, adding one digit at a time until it finds a difference.
> I mean, I've thought of noncomputable reals like Chaitin constants.
Heh, but how many digits can you actually provide? Not too many. So have you really thought of the number in any meaningful sense when you barely know any of its digits?
Also interesting that computer languages themselves are countable, so while it's hard to specify the digits algorithmically for the Chaitin constant of any computer language, you already know that the set of ALL Chaitin constants are countable.
I assume it runs into problems when you try to check if 2 > 2.
I’ve never thought through very many digits of pi either. Or even 1/3 for that matter!
But perhaps still not as sane as one may hope. It would be very sane to be able to compute, for any two numbers, which one is larger (or whether they're equal), but sadly this is not computable for the computable numbers.
> Every number you ever thought of is computable, but as a mathematician, working with the set of computable numbers is much more tedious than working with real numbers.
I mean, I've thought of noncomputable reals like Chaitin constants.