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by Diogenesian
18 hours ago
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I don't think it makes sense to say anything except "computable real" - the computable / uncomputable distinction seems totally immaterial for the purposes of most real analysis, or even "pathological" topology and set theory involving R (except for puzzles directly involving computability). And the "interesting" transcendental computable reals are a bit of a grabbag. The mean value theorem isn't true for the computable reals, differentiation of computable function isn't always computable, sequences tend to behave poorly, etc. There's still a lot you can say: https://en.wikipedia.org/wiki/Computable_analysis but in general calculus doesn't care about computability, that's a human problem. |
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As in, not even computable in theory. It is isn't about normal "computable" concerns. It is "proven to never be characterizable".
Which is a class of numbers whose "existence", if that can term can even be applied coherently for undefinable things, is contested, in theory. In practice they certainly do not exist.
1/3 has infinite decimal digits, but is definable with a finite number of symbols, so it is a computable real. Even if we had no algorithm yet to compute those digits.
Try and define a specific number, that requires infinite symbols to define. As far as I am aware of, no part of calculus involves specific values that have no finite definition, except when the need for uncomputable/undefineable reals are asserted on a circular basis (i.e. they are needed to resolve problems with assumptions that already assume them.)
(Note that a number defined by interpreting the infinite digits of pi as mathematical relations, would still be considered a definable number, assuming some form of convergence could be proven. Because pi is finitely defined.)