Theory of Computation wasn't around when all this "exciting" stuff was developed in Mathematics. Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation.
Chaitin has a great paper on this and shows how Cantor's constructions were reflected a half-century later by Turing. https://arxiv.org/abs/math/0411418
Except of-course, while "hyper-Turing" machines that can do magic "post-Turing" "post-Halting" computation are seen as absurd fictions, real-numbers are seen as "normal" and "obvious" and "common-sensical"! It was amusing sometime back to see people pooh-pooh the likes of Hava Siegelmann for being funded for their "super-Turing" machines with "real-number" computation, without realizing that the core issue is the "real"-number itself!
I've always found this quite strange, but I've realized that this is almost blasphemy (people in STEM, and esp. their "allies", aren't as enlightened etc. as they pretend to be tbh).
Some historicans of mathematics claim (C. K. Raju for eg.) that this comes from the insertion of Greek-Christian theological bent in the development of modern mathematics.
Anyone who has taken measure theory etc. and then gone on to do "practical" numerical stuff, and then realizes the pointlessness of much of this hard/abstract construction dealing with "scary" monsters that can't even be computed, would perhaps wholeheartedly agree.
edit: The post has a great link to a note on Cantor's theology,
> Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation.
It is funny you say that when Turing defined Turing machines to compute real numbers (like π for example). In its original definition, a number was computable if its Turing machine did not stop. Which makes sense since π does not have a finite decimal expansion.
Today, we usually define Turing machines to decide problems and a problem is decidable if for every input its Turing machine stops with a ``yes'' or ``no'' answer. I guess this is what makes people think what you said in the quote above. Maybe this definition is more intuitive but this conclusion from it could not be more wrong.
Think about it for a second, if the computable numbers were countable there would be no uncomputable problem (Turing actually used the classic cantor diagonal argument to prove that there were uncomputable numbers)
The set of computable numbers is actually countable (see ref. linked above). It has to be by definition because the set of finite computer-programs is itself countable.
This is the whole point of the un-reality of "real" numbers: "all" of it (= measure 1) is uncomputable except a "tiny" measure-0 set.
This confuses the halting problem with a still running computation.