On Thursday, December 12, 2013 2:39:11 AM UTC-7, WM wrote: > 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. >
But, there are infinitely many of those finite indices. The Axiom of Infinity allow us to account for All these indices.
> > 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. >
You use an Induction type argument which fails at the Infinite.
You claim, For All x e N, phi(x) is True; thus phi(N) is true.
This is Not necessarily the case for All Statements Phi.
Phi(x) here is "There exists a Rational Number of length x, which is Equal to d.".
> > > 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. >
But we know: For All n e N, the n-th Decimal Digit of 1/9 is "1".
If that does Not mean 1/9 is "defined by digits", please explain.
> > > 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). >
However, the Axioms allow us to define 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. >
No, you just wrong. It is you Philosophical Block. You reject the Existence of an Infinite Set on Purely Philosiphical Grounds. You claim Contradiction and show no such thing. The only Contradiction is with you Finite Intuitions.
You are welcome to go count you finite collection of stones and cut you rationally divisible sticks.
> > 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. >
If you have a Formal (Contradiction free) Definition of "Constructible", then there is No such 1:1 Correspondence.
> > > 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. >
Lexically implies Vocabulary which implies Meaning. It is this "Meaning" that makes the Correspondence Non-Existence.
> > 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. >
Typical WMBS! You can't use you "definable by digits" argument here, so you dismiss it.