On 16 Dez., 21:34, Rupert <rupertmccal...@yahoo.com> wrote:
> > It is mathematics. Mathematics includes discourse about uncountable > sets.
But it does not include discourse using uncountably many characters, because for that sake infinite strings of bits would be required. By finite strings of bits only countably many characters could be used. And if we restrict our conversation to those usable characters and omit the others, then we discuss in a countable language.
> The fact that it would not be possible for a human to use such a > language is irrelevant
It would not only be impossible for a human but impossible per se, because there might be sentences that do never end.
Mathematics may contain many foolish ideas, They can be discussed. But the language applied to discuss them must be free of foolish items and must be usable by humans and other intellects. That's the way mathematics works: It is mainly a discussion with others or with oneself. Every item (including uncountable sets and inaccessible cardinals) must have finite definitions. Therefore there is no uncountable alphabet and there are not uncountably many languages.
And all mathematics that has been done by Cantor can be discussed in any natural (i.e. countable) language. Further Cantor's diagonal argument works solely with digits (i.e., nodes). A proof that nobody can distinguish more than countable many paths in the Binary Tree is a contradiction of uncountability.