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It's a real PITA to work with computable numbers because even their equivalence is not computable. Also, the field is really young. It was only in 2000 that We
by chx 8y ago
It's a real PITA to work with computable numbers because even their equivalence is not computable.
Also, the field is really young. It was only in 2000 that Weihrauch has proven that every computable real function is continuous.
- myWindoonn 8y agoTheir equivalence can be given by a Turing machine, though, so the non-computability comes directly from the Halting Problem. This is worth mentioning because some folks get the incorrect impression that it's not possible to write a program that, say, outputs whether two computable reals are equal to each other. It is possible! But the program might not halt. I shouldn't trivialize the difficulty. It is a PITA. I remember my first foray into this; I wrote a program which computed sqrt(8), which worked fine, and then another program which computed sqrt(8)^2, which didn't halt. The edges are tricky.
- voxl 8y agoAs opposed to classical reals which have a computable equivalence? (Hint: they don't)
- chx 8y agoBut when working with reals we normally don't bother with computability. Whereas with computable numbers you don't even have anything else really.