Re: induction vs Cantor

From: Poker Joker (Poker_at_wi.rr.com)
Date: 11/26/04 

Date: Fri, 26 Nov 2004 13:45:38 -0600

"Chairman of the David Hilbert Appreciation Society"
<mathgeekxxiiii@hotmail.com> wrote in message
news:8JqdnQxnYLcA6zrcRVn-3w@giganews.com...
> Poker Joker wrote:
>> Let L_1 be a list of reals that implies a mapping F_1
>> between the naturals and reals.
>
> Ok
>
>> Let D_n be a Cantor anti-diagonal number that can be
>> formed using the mapping F_n
>
> Ok
>
>> 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.)
>
> Right. You outline a process to create infinitely many L_n,
> none of which contains all of the reals.
>
>> All of the D_n can be found in "infinitely many" mappings
>> between the naturals and the reals.
>
> This part is a bit fuzzy. At this stage you don't have a proof
> that any one L_n contains all of the reals; which is what you
> would need to declare that Cantor made an error. That you are
> hand-waving about something included in "infinitely many" mappings
> is irrelevant since in no obvious way does such a thing relate
> to a list, or a bijection.

This part is not fuzzy. For all j > n, D_n is in L_j.

The union of all the L_n, (a countable set) contains all the D_n.

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