Am Mittwoch, 11. Dezember 2013 20:01:40 UTC+1 schrieb Zeit Geist:
> > Fine for you. My claim says that by digits it is impossible to prove that d differs from all rationals. That means up to every finite digit there are infinitely many rationals identical with d up to that digit. > > > > > > > No, you use "up to every finite digit there are infinitely many rationals identical with d up to that digit." To show "by digits it is impossible to prove that d differs from all rationals"
by digits that have a finite index. Please don't forget that. Matheology usually claims the handwaving "infinite sequence" and tries to make us forget that even in such an infinite sequence every digit has a finite index. Only that is required for my proof. > > > > This Does Not Follow.
Of course it follows. And if not, then your logic is broken because it is clear that my assertion is right. > > > > > And if you acknowledge that d is nothing more than its digits, then there are infinitely many rational identical with d. > > > > > > > But it is more. It is a certain arrangement of digits.
Yes, but this certain arrangement is never visible before the last digit. (For instance, you cannot know whether I mean 1/9, if I up to every digit give a sequence of 1's.) Alas this does not exist. Therefore we need the limit. The limit however cannot be defined by digits. > > > > > And if you mention that d is more than its digits at finite places then you either need a digit at infinite place or a finite word defining some "structure". > > > > > > > The structure is explicitly defined in the Axioms ( Properties) of the Real Numbers.
No. The structure of the digit sequence of sqrt(2) is not defined in the axioms but in a specific formula for sqrt(2).
> Meaningless and Garbled.
I know that you don't like it, but I am sure than many understand it and that most of those who have not wasted a significant part of their time with matheology will accept it. You have simply a psychological block, cultivated by the frequent occupation with set theory. > >
> Because, as others have said here, if you insist on Constructible Numbers, Functions and "Words" ( the Correspondence above is a Function ) only, There Is No 1:1 Function between the Set of "Words" and the Set of Natural Numbers.
But if you insist on all finite words, then there is a function. I showed it in binary, but it can be translated into every language. And according to set theory, a subset of a countable set is a countable set. And that means it can be listed. > >
> > But you would need digits or something else like that in order to "realize" uncountably many words. > > > Which has nothing to do with "distinguishing by digits".
Of course. Words must be distiguished lexically by their letters if they shall define something or help to understand it. Here the letters are digits. > > Also, I don't need Digits to show the Uncountability of the Set of All Real Numbers.
That is another matter, but not under consideration here.