Re: induction vs Cantor

From: Norm Dresner (ndrez_at_att.net)
Date: 11/26/04 

Date: Fri, 26 Nov 2004 15:41:17 GMT

"Poker Joker" <Poker@wi.rr.com> wrote in message
news:8vHpd.1507$XQ2.823@twister.rdc-kc.rr.com...
> Let L_1 be a list of reals that implies a mapping F_1
> between the naturals and reals.
>
> Let D_n be a Cantor anti-diagonal number that can be
> formed using the mapping F_n
>
> Let L_n+1 be a list of reals by inserting D_n into L_n
> at row 2n and shifting down all the previous rows at 2n
> and above. This process is clearly an inductive process
> that creates a new mapping for each natural number.
> (L_n+1 could also be formed by prepending D_n to
> L_n.)
>
> All of the D_n can be found in "infinitely many" mappings
> between the naturals and the reals. This process has no
> hope of proving that there are more reals than naturals.
>

It's a pity that you don't understand what a Proof By Contradiction means.

It works like this.
1. You assume something In your case you're assuming that the list L_1 is
a complete (1-1 onto) mapping between N & R
2. You use standard logic to reach a contradiction. In this case, that the
real number D_1 is not in the supposed exhaustive list.
3. You then conclude that your original premise is faulty. In this case,
that the list is a complete mapping.

That's how it works. It makes no sense to assume that a list is complete
and then add something to it and proclaim that it's more complete.

    Norm

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