Re: Cantor's fatal mistake a la Zenkin

From: Tim Mellor (timm_at_amsta.leeds.ac.uk)
Date: 11/25/04 

Date: 25 Nov 2004 01:42:20 -0800

"Poker Joker" <Poker@wi.rr.com> wrote in message news:<sP7pd.91563$T02.76465@twister.rdc-kc.rr.com>...
> Any list of reals forms a countable set. For every such
> set, there exists a countable set of "Cantor" methods or
> "diagonal numbers" or "anti-diagonal numbers."
> (There's no consistency in what these things are called.)
> that are capable of "describing" a countable set of reals
> that are not included in the original list. Together, the
> two sets form a countable set.
>
> The "fatal mistake" of Cantor is that he assumes that a
> list can only be mapped to N in numerical order;

Not at all. The proof that R is uncountable considers injective
functions from N to R. The order on N is not used, and is irrelevant.
Perhaps you have read a poor account.

Try http://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps

> The
> first element maps to 1, the second element maps to 2,
> etc. Nothing forces lists to be mapped that way.
>

True.

> Here's the start of a possible mapping:
>
> 1 <-> 0.012345
> 3 <-> 0.123456
> 5 <-> 0.234567
> .
> .
> .

Yes.

>
> Note that the mapping between the odd naturals and
> the reals is explicitly given. The mapping between the
> even naturals and the remainder of the reals is implicitly
> given via existence:
>
> The list of reals above has an associated countable
> set of reals that can be constructed using Cantor's
> method and similar methods. The number of these
> methods is countable

I'm afraid not.

> and therefore they can not
> produce more than a countable number of reals. The
> even natural numbers form the "missing" part of the
> mapping.

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