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Message from discussion Cantor diagonalization and base 2

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More options Oct 9 2012, 4:44 pm
Newsgroups: sci.math, sci.logic, sci.physics
From: Graham Cooper <grahamcoop...@gmail.com>
Date: Tue, 9 Oct 2012 13:44:38 -0700 (PDT)
Local: Tues, Oct 9 2012 4:44 pm
Subject: Re: Cantor diagonalization and base 2
On Oct 10, 5:37 am, Curlytop <pvstownsend.zyx....@ntlworld.com> wrote:

> Graham Cooper set the following eddies spiralling through the space-time
> continuum:

> > (snip the bull)

> > f(1) = 0. 314 159 265 ...
> > f(2) = 0. 278 182 845 ...
> > f(3) = 0. 333 333 333 ...
> > f(4) = 0. 123445 54321 ...
> > f(5) = 0. 11 11 11 11 ...
> > ...
> > (nip more bull)

> Every supposed demonstration of Cantor diagonalisation invokes a
> *disordered* subset of the supposedly greater-than-infinite set of all
> reals, as here. If somebody would only put them into some kind of order,
> the absurdity of Cantor's proposition will become immediately apparent to
> all.
> --

Dth digit of the Nth real is UTM(N,D)     (mod 10)

Yes but then you get a logical proof, people like their
*constructable* bigger infinities.

IF you used a FINITE ANTIDIAG() FUNCTION

e.g.
FUNCTION ANTIDIAG( DIAGONAL )
{
RETURN  DIAGONAL + 0.111111111....

}

Then it would input a HYPERREAL DIAGONAL
(DIAGONAL doesn't appear on any computable row)

HYPERREAL                   <DIAG>
+   0.1111111111...             <ANTI>
------------------------
=   HYPERREAL                  <MISSING>

i.e. the computable reals list is not missing any computable real.

ALTERNATIVELY

HYPERREAL              <DIAG>
+   HYPERREAL              <oo ANTI>
------------------------
=   COMPUTABLE           <MISSING>

So EITHER WAY
ANTIDIAG() is FINITE --> no missing computable real
ANTIDIAG() is INFINITE --> no computable missing real

Herc