Diagonalization proceeds from a list of real numbers to another real number (D) that's not on that list (because D's nth digit differs from that of the nth number on the list).
But this argument only works if D is a real number and this does not seem obvious to me!
Especially given that in Cantor's time there was no developed theory of computability, I could imagine someone of that era saying "ah, Cantor, you are pulling a trick here! D seems like a real number at first glance but when you look closely it's infinitely difficult to calculate or really talk about with precision in any way. In this regard it's unlike every real number I've ever encountered and you must make an additional argument to make me consider it a genuine real."
My question is this: was Cantor's diagonalization argument—at the time he made it—rigorous? Either in the sense that it proceeds from axioms or in the sense that it contains a convincing answer the above skeptic? Does D's real number-ness require a justification and did Cantor offer a sufficient one?
Of course I am not questioning whether the reals are actually uncountable. I'm interested in extent to which Cantor actually proved this versus giving an intuitive argument in favor of believing it.