Re: limitation to induction on finite bounds

From: |-|erc (gotcha_at_beauty.com)
Date: 06/24/04 

Date: Thu, 24 Jun 2004 05:32:16 GMT

"Barb Knox" <see@sig.below> wrote
> "|-|erc" <gotcha@beauty.com> wrote:
> [snip]
>
> >Is this valid?
>
> NO.

so this is false?

<unsnip>
THERFORE
<
<x>
<xx>
...
>
This list is missing a finite sequence of digits.

>
> [snip]
>
> >0
> >0.3
> >0.33
> >0.333
> >..
> >
> >This sequence contains the diagonal, 0.33..
>
> As I posted out before, which you chose not to respond to:
>
> Ah, I think I see the rule you're wanting to introduce: If one allows an
> infinite list of sequences, then one MUST allow the individual sequences in
> the list to be infinite themselves.
>
> Trying to talk mathematics with you can be an intellectually interesting
> exercise, rather akin to a chess player being introduced to fairy chess. So,
> given your above non-standard "fairy maths" rule about infinite lists of
> sequences, here's another go at showing that the infinite sequence .333... is
> legitimately (by your rules) not in the infinite list .3, .33, .333, ...
>
> First, let every member of the infinite list be an *infinite* sequence:
> . 3 0 0 0 0 0 ...
> . 3 3 0 0 0 0 ...
> . 3 3 3 0 0 0 ...
> . 3 3 3 3 0 0 ...
> ...
>
> That is, the i'th sequence in the list consists of i 3s followed by an
> infinite tail of 0s. Clearly, for every i, the i-digit prefix of 333...
> occurs in the list (as the i'th member in fact). However, EVERY sequence in
> the list has an infinite tail of 0s, which 333... conspicuously lacks.
> Therefore 333... can not be in the list.
>
> So, do you have a further fairy maths rule to exclude this construction? Or
> do you now see that 333... is not in the list? Or maybe you'll just change
> the subject.

let me see...
0.333333333333333 infinite number of 3s 3333333330000000 infinite number of 0s 000000

= 0.3..

Just check that on my infinite precision computer, yep that's fine!

Herc

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