On Dec 16, 10:03 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 16 Dez., 21:35, Rupert <rupertmccal...@yahoo.com> wrote: > > > However, you haven't justified your claim that there are only > > countable many countable languages, and in fact it can be shown that > > this isn't true. > > Then there is another contradiction. I can prove the contrary. > > All characters belong to a countable set, because it must be possible > to encode them as finite strings of bits: > > 0 > 1 > 00 > 01 > 10 > 11 > 000 > ... > > The list is countable. The meanings of words formed by these > characters and used for discourse must be defined by men. There will > never be more than a finite set of definitions. But even an infinite > set of languages would not surpass a countable set of words, because > aleph_0 * aleph_0 = aleph_0. We will never leave the countable domain > unless there are infinite words. But infinite words cannot be used in > discussions with others or with oneself, i.e., in mathematics. >
This is an attempt to reason about humanly constructible languages. That's different to what I was talking about.
> Regards, WM > > And languages are practized: A language is something to be defined by > men. There will never be more than a finite set of languages existing. > > Further > > > > > > What cannot be > > > defined in any language is not a path. > > > This, too, is something which needs to be justified. A set theorist > > would not accept that assumption. > > A set theorist would prefer to accept completed infinity. > But mathematics is mainly a language. And Cantor's proof can be done > in that language. Cantor's proof is limited to paths that can be > identified by their strings of digits or nodes. > > Cantor's proof does not prove anything about undefinable numbers. It > proves that the set of definable numbers is uncountable.
No, it doesn't. Definable in what language?
> Otherwise one > should be tempted to insert indefinable numbers in his list. The > result would be: nothing. Not a number, because soe digits were > undefined. > > The true result is: There is no diagonal number defined unless the > complete list is defined by a finite definition in everyday language.
What do you mean by "everyday language"?
The point is that you cannot set a limit in advance on which languages you are going to work with.
Suppose you say "I'm only going to work with the real numbers that are definable in language L". Then there must be some metalanguage in which you can talk about the language L. In that metalanguage, you can define the anti-diagonal for the list of all real numbers definable in the language L. You're never going to be able to set precise limits to which real numbers you regard as "definable" without also being able to show that you can go beyond them. That's the point. There does not exist any countable language in which all of the real numbers are definable.
> But there are only countably many finite definition, hence only > countably many diagonals. > > > > > > Therefore I know that there are > > > not more than countably many paths - at least paths that can be > > > applied in mathematics and can be the result of any diagonalization. > > > No. You don't know that. > > I know it. And if it were wrong, if there were in fact uncountably > many words to be used in mathematics, then they must be defined > somewhere (how else would we know what meanings these words would > have?), then there must be a list of them. But that is impossible. > > Or do you dpropose the use of words that are completely undefined? >
It's nothing to do with the question of what words we use in mathematics. The issue is about what mathematical objects exist.