Michael, thanks for the Wittgenstein. I especially like his architrave image. I wonder if the question is actually one of philosophy in that Cantor was a mathematical Platonist and Wittgenstein was apparently not (but about this I am not so sure; it sometimes seems like he is). It didn't matter whether it was real to Cantor since to Cantor none of it was real, or at least whether it was real or not was not the concerning question. Wittgenstein, however, was always tormented by meaning, and I guess his rejection of Cantor comes from his belief that mathematical theorems express truth and that something that is meaningless cannot express truth and is therefore not mathematical... or something. Wittgenstein was right about Russell's failure to build a logical foundation for math; ironically, it was that other mathematical Platonist, Godel, who proved Russell wrong.
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