| In this case, the term "uncomputable" also means "undefinable with less than infinite symbols". 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.) |