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> Much more sane is the set of computable numbers, that is the set of numbers for which you can find an algorithm that computes the number to arbitrary precisio

> Much more sane is the set of computable numbers, that is the set of numbers for which you can find an algorithm that computes the number to arbitrary precision. 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.

xscott 2y ago
> 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. 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.
- hifromwork 2y ago
  >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 assume it runs into problems when you try to check if 2 > 2.
  - xscott 2y ago
    It's very funny how that's a really good example and made it click instantly. Thank you.
- tshaddox 2y ago
  > 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? I’ve never thought through very many digits of pi either. Or even 1/3 for that matter!

xscott 2y ago

> 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. 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.

hifromwork 2y ago
>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 assume it runs into problems when you try to check if 2 > 2.
- xscott 2y ago
  It's very funny how that's a really good example and made it click instantly. Thank you.
tshaddox 2y ago
> 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? I’ve never thought through very many digits of pi either. Or even 1/3 for that matter!