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,