Re: Cantor's diagonal proof wrong?

From: David Kastrup (dak_at_gnu.org)
Date: 11/27/04 

Date: Sat, 27 Nov 2004 23:00:16 +0100

mueckenh@rz.fh-augsburg.de (W. Mueckenheim) writes:

> David Kastrup <dak@gnu.org> wrote in message news:<x5u0re7z8b.fsf@lola.goethe.zz>...
>
>> >> Your question does not make sense since the diagonal is not associated
>> >> with any natural number.
>> >
>> > Oh, is there any digit d_nn of the diagonal which is not associated
>> > with a natural number n?
>>
>> Each digit is, and so are the partial sums of the corresponding
>> series. But the limit of the series is not associated with any n.
>
> What does distinguish the "limit" from the diagonal?

The diagonal is a sequence of digits, the limit is the value indicated
by it (for example, the digit sequences 0.59999999999... and
0.60000000... are different, but denote the same value).

> Do we need different words here?

Only when somebody like you tries to cause confusion by mixing things
up wildly.

> However, how, then, can Cantor change all the digits of this
> "limit"?

He changes the digits of the sequence, resulting in a different value
as the indicated limit.

> And why can't we consider my proof in the limit?

You are again trying to befuddle people by word games. "considering a
proof in the limit" is a completely meaningless and nonsensical
expression.

If we try to turn your word games into something making even remotely
sense, the question might have been "Why does my proof that is valid
for every element of a sequence say nothing about the limit of the
sequence?". And the answer is, of course, that the limit is not a
member of the sequence. Even if we wanted to consider the limit as
some weird element with index omega or whatever, for whatever crazy
reason, your proof is an induction proof, and induction makes a
statement only for all elements of N, and such a hypothetical omega
could not be a member of N.

So whatever one likes to call the limit of the sequence of truncated
decimals of D, your proof does not extend to it.

>> That the value specified by the diagonal (which is basically the
>> limit of the corresponding series) can't be completely specified if
>> you stop at any digit.
>
> So don't stop, but consider the limit of my proof.

Proofs have no "limit". This is nonsensical. They have validity.
Your proof is valid for all numbers corresponding to a truncated
diagonal. The number corresponding to the entire diagonal is not
covered by it.

> Anyhow, n always remains a natural number.

Right. Which is the reason that you only prove something about the
values corresponding to a truncated diagonal. 

-- 
David Kastrup, Kriemhildstr. 15, 44793 Bochum

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... If infinite, the limits of the ... that the reals are not countable. ... He starts with the sequence of rationals: ... in the building of the diagonal *each* digit has to be changed. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... >>> rightmost zero in an unending sequence of ever more rightward ... >> naturals, you might as well call it something, I suppose. ... > In TO's system of "whole numbers", there is a most significant digit and ... >> infinite unending string of bits, even if most are generally ignored. ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... > to reason is notably flawed. ... > A number which differs from the n-th list element in at ... > least one digit, ... And I know a line which has 13 more digits 1 than the sequence of the ...
    (sci.math)
  • Re: .9 repeating
    ... Represent positive infinitesimals with digits. ... let R' be the set consisting of "digit sequences" ... We map x in [0,1) to the sequence ... Now we make an arbitrary choice: we interpret infinite digit sequences ...
    (sci.math)
  • Re: Cantor Confusion
    ... Cantor's diagonal proof is about infinite sequences. ... > You do not require that one digit represents the number 1/3 in Cantor's ... > infinite sequence? ... I do no want not extend anything to nodes representing numbers. ...
    (sci.math)
Quantcast