Date: May 2, 2013 1:51 PM
Author: dan.ms.chaos@gmail.com
Subject: Re: Matheology § 258
On May 2, 11:55 am, WM <mueck...@rz.fh-augsburg.de> wrote:

> On 1 Mai, 23:31, Dan <dan.ms.ch...@gmail.com> wrote:

>

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> > > Yes, that is true. But (and please read this very attentively!):

> > > Cantor's argument requires the existence of the complete sequence

> > > 0.111.... in digits:

>

> > > You can see this easily here:

>

> > > The list

>

> > > 0.0

> > > 0.1

> > > 0.11

> > > 0.111

> > > ...

>

> > > when replacing 0 by 1 has an anti-diagonal, the FIS of which are

> > > always in the next line. So the anti-diagonal is not different from

> > > all lines, unless it has an infinite sequence of 1's. But, as we just

> > > saw, this is impossible.

>

> > I see no significant difference between referring to a mathematical

> > object by a formula and referring to it by 'writing it down' .

>

> But Cantor's argument is invalid, in this special case, unless it can

> produce 0.111... with actually infinitely many 1's, i.e. more than

> every finite number of 1's.

>

> It does not matter whether 1/9 exists as a fraction or whether it

> exísts in the ternary system as 0.01. In order to differ from every

> entry of my list Cantor's argument needs to produce, digit by digit,

> the infinite sequence. And that does not exist.

>

> Regards, WM

You can substitute in any expression "the first digit of 1/9" with

"0.1111....." with "1" and it wouldn't make any difference .

You can substitute all the digit expansions in Cantor's argument with

formulas , and it wouldn't make any difference.

In what way you choose to write down the number (whether as a

fraction, or as a digit expansion , etc. ) is of no relevance .

Because it's still THE SAME NUMBER . It still has THE SAME

DIGITS ,even if you can't write them down .

The way YOU CHOOSE to write a number is as relevant to Cantor's

argument as the color of the ink in your pen .

You've ran out of red ink, and because it's written in a typewriter

(it has a weird font , 1/9 , not 0.1111.... like you would expect),

you say it's not the same number) .

A single valid name is good enough . The fact that we choose ,from

one of the possible names for an object ("2+2" , "1+3" ... ) , one of

them to call "its value" is of no fundamental relevance . I should

know . I work with infinite lists of numbers all the time .

http://www.techrepublic.com/article/infinite-list-tricks-in-haskell/6310740