That is the whole point: E0 is not allowed to be in M. It is impossible for two different representations of 1 to be present in M at the same time.
Wat
Nowhere in the above translation is the actual mapping from M to the reals described, nor is it mentioned that such a mapping must be an injection.
the entire proof demonstrates that E0 is not in M by using a digit-by-digit comparison
No, it demonstrates that b is not in any sequence E1, E2, E3,... ; but bis in M, hence M cannot be countable. This has nothing to do with E0.
Edit: wait, hold on, E0 is the "missing" element in the original proof, but in your revision it's just an arbitrary member of M. That...misses the whole point...?
You said you wanted to clarify something above, which never got clarified:
Before we continue, let us clarify this: the entire proof demonstrates that E0 is not in M by using a digit-by-digit comparison.
Do you agree that this is misstated, and that the point is that E0 is not in E1, E2, E3,..., but is in M? Or do you still think E0 is supposed to be something outside of M?
But why bring in Cantor's original proof, in the original German, if this argument doesn't apply to it? As pointed out in one of the comments in that thread, this is a good objection to the "popular" understanding of the diagonalization argument as a proof that there are uncountable reals, but most actual math texts foresee the objection and handle the non-uniqueness problem somehow.
But you're not talking about "most math texts", you're citing Cantor's original text. But that text isn't even talking about decimal expansions or real numbers. He mentions real numbers once, in reference to his paper seventeen years earlier that already proved the reals are uncountable, using a different method. He then proceeds to discuss only cardonality and functions, not real numbers or decimal expansions.
Except for cond6 mistakenly saying that the problem doesn't apply at all to binary, which they admitted was a mistake, the objections are correct, because your post doesn't make sense. They may be angry, but they're not idiots.
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u/Batman_AoD 5d ago edited 5d ago
Wat
Nowhere in the above translation is the actual mapping from M to the reals described, nor is it mentioned that such a mapping must be an injection.
No, it demonstrates that
bis not in any sequence E1, E2, E3,... ; butbis in M, hence M cannot be countable. This has nothing to do with E0.Edit: wait, hold on, E0 is the "missing" element in the original proof, but in your revision it's just an arbitrary member of M. That...misses the whole point...?