On Dec 16, 9:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > 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. >
You're missing the distinction between the metatheory and the object theory. The metatheory in which mathematical discourse takes place, in which we discuss various object theories, is in a countable language. But the object theories could be in an uncountable 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. >
I don't know what the distinction is between "impossible for a human to use" and "impossible per se".
In metamathematics, we can study languages other than the languages which are in fact possible for humans to use.
> 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. >
Not in languages that humans actually use, no. But in the universe of discourse of metamathematics, there are such 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. >