On Dec 16, 10:10 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 16 Dez., 22:01, Rupert <rupertmccal...@yahoo.com> wrote: > > > > > > > > > > > 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. > > Yes, I agree. We may talk about uncountable languages. But we cannot > use them for discussing and for doing mathematics. > > > > > > > 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". > > Simply impossible. > > > > > In metamathematics, we can study languages other than the languages > > which are in fact possible for humans to use. > > But we cannot used them in order to define real numbers. > > > > > > 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. > > There are ideas about such languages like about unicorns. Assume there > were an uncountable language. Then every word of the language must be > defined. Undefined words do not belong to a language, not even in set > theory, do they? But such a book of definitions would be a list of > uncountably many words, i.e., a list of unlistable elements, i.e., a > contradiction, no? > > Regards, WM
This is all irrelevant. I am definitely correct in saying that there are uncountable languages, in the sense that we can define them and talk about them in some sufficiently strong metatheory such as ZFC. But that's neither here nor there. The point is that there is no good reason to think that every real number must be definable in some language. Also, you could have a situation where there is a countable language corresponding to every countable ordinal and every real number could be defined in at least one such language. This possibility is demonstrably consistent, and it still remains true that the real numbers are uncountable. But the main point is that there is no good reason to think that every real number is definable in some language.