>>16655268
>There was no specific axiom system that Cantor was working with.
True, because his work predates that level of mathematical rigor that we are now accustomed to. But instead his proof and construction are relative to the informal understanding of sets as it existed at the time, which has the same role of a background against which to interpret proofs and constructions, though not one up to modern standards.

Of course, the modern set theory that was created to replace the messy informal understanding that existed in Cantor's time is consistent with Cantor's theorems, so the Diagonal argument and friends can be legitimately reinterpreted on that much firmer foundation. But you are correct that most of what I said in >>16655259 is relative to the modern interpretation of Cantor's arguments, not the 19th century originals.

>In fact, Set theory was more or less invented to validate Cantor's "proof" that there are uncountably many real numbers
Not really; set theory was mostly invented to get rid of unambiguous *in*consistencies that showed up in the unformal conception of sets that existed at the time, such as Russell's paradox. This properly axiomized set theory does not rule out Cantor's constructions, true, because they were deemed legitimate; but that was definitely not the impetus to create this axiom system in the first place.

>The most you can extract from Cantor's attempted proof is that if you have some computation which produces a list of numbers, then you can also find a computation which produces a number not in that list. But that's compatible with the claim that the real numbers are not uncountable.
You can extract a good bit more than that even if you reject set theory. If you apply the diagonal argument to computable sequences, you conclude that there is no computable sequence of all computable digit sequences, which is a close analogue of the uncountability of the reals in the computational setting.

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>P1) All numbers are in the list>P2) We can create a number different from all numbers in the list>C) There is a contradiction, so P1…

>P1) All numbers are in the list
>P2) We can create a number different from all numbers in the list
>C) There is a contradiction, so P1…