Comment by shannifin
5 years ago
I'm not an expert on Wittgenstein, but I think the critique of Cantor's diagonal argument is more to do with its implication that an infinite set can be "bigger" than another. To say so is more of a semantics argument than a mathematical one as it takes for granted the meaning of "infinite". That is, if you define an infinite set as being inherently without size, it makes no sense to then assign it comparitive sizes via cardinality.
I think it has to do with Wittgenstein's constructivism. Since humans can never actually construct an infinite set, one cannot say that such a thing exists as a constructivist. As such, infinity would be a fuzzy without-size concept that Witty was suspicious of. But if one does think mathematical objects are real, then it's not a problem.
I'm guessing constructivists have found better ways to approach infinity than Witty, without conceding that infinity is real.
Wittgenstein was not a constructivist, though. At least in his later years he explicitly argued against any -ism as a position and tried to avoid philosophical theses as a whole. (Whether he succeeded with this undogmatic approach is another question of course, but to say that he was a constructivist would commit him to a position that he did not advocate for, even if many of his remarks can certainly be read in such a way if they are read in isolation.)
I don't get the sense that Wittenstein was suspicious of infinity as an idea / concept in itself, only of its treatment in mathematical logic. Otherwise much of his writings on it could've been far more terse.