Re: Cantor's "proof"

From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 09/22/04 

Date: Wed, 22 Sep 2004 21:01:48 +0000 (UTC)

In article <opseqxoypk3uk9lu@cs81133.pp.htv.fi>,
Keckman <keckman@welho.com> wrote:
>On 22 Sep 2004 13:19:48 -0700, Daryl McCullough <daryl@atc-nycorp.com>
>wrote:
>
>> But there is a real number d that is not on *any* of the lists you
>> construct.
>
>Same way as there is an _natural_ number that is not on *any* list you
>construct (of natural numbers).
>
>Let suppose you have listed natural numbers:
>
>1: 3
>2: 7
>3: 99
>.
>.
>
>Now, in every step lets construct a number by adding them: 3+7+99+...This
>number is bigger than any number previously in list (do i have to prove it
>to you?).

Addition is a finitary operation. There is no such thing as the sum of
an infinite number of natural numbers. So you have not defined a
number: you have merely said some more nonsense. You do not have a
natural number not on the list. You have merely said that if you
assume that something which is false is true, then kookiness will
follow. Well, duh.

>The situation is same as in Cantor's list - he construct a
>decimal number in every step that has not yet been on list - but it could
>be next or later.

No. There is no "step". The diagonal number is a number which is
produced at once given the list. The number is not anywhere on the
list by construction. You do not construct the number "successively",
you construct it given the full list.

>It will be different than *any* only if we can go the
>list trough, but we can't. Or if we can, then we can go this list too:
>
>1: 0,1000...
>2: 0,2000...
>.
>.
>999: 0,999000....
>
>33333.... : 0,3333333333....
>
>And the last row is the answer to other Cantorist who ask where is the
>place for the 0,3333333....

There is no such line in the original list. "333333333....." is not a
natural number. 

-- 
======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu

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