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Topic: On the diagonal argument again
Replies: 199   Last Post: Jun 10, 2012 10:12 AM

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 Tim Little Posts: 1,793 Registered: 2/7/09
Re: On the diagonal argument again
Posted: May 29, 2012 10:28 PM

On 2012-05-29, Aielyn <gjo1984@gmail.com> wrote:
> Which does make me wonder, though... what would happen if you were
> to construct a set by starting with an enumerable set, then find the
> cantor diagonal and insert into position 1, then find the cantor
> diagonal and insert into position 2, then find the cantor diagonal
> and insert into position 3, then... you see where this is going, I
> assume. The limit is, technically, well-defined

Indeed it is.

> but the cantor diagonal is, technically, also in the set, due to
> the limit.

No, it isn't. You have a sequence of entries that converges toward
the antidiagonal, but none of them are equal to it.

> Obviously, the problem is assigning an integer to it (and you could
> just apply a permutation of finite extent and find a new cantor
> diagonal)... but what properties would that set have, relative to
> the initial enumerable set?

Nothing particularly special, as far as I can tell. The "lower
triangle" of the list will be completely determined by the original
list's diagonal digits and your antidiagonal function, and the "upper
triangle" by the digits above/right of that diagonal.

--
Tim

Date Subject Author
5/15/12 LudovicoVan
5/15/12 LudovicoVan
5/15/12 William Hughes
5/15/12 LudovicoVan
5/15/12 Virgil
5/16/12 Graham Cooper
5/15/12 Shmuel (Seymour J.) Metz
5/15/12 LudovicoVan
5/15/12 Virgil
5/16/12 mueckenh@rz.fh-augsburg.de
5/16/12 Daryl McCullough
5/16/12 LudovicoVan
5/16/12 LudovicoVan
5/16/12 Virgil
5/16/12 LudovicoVan
5/16/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 Graham Cooper
5/18/12 Michael Stemper
5/18/12 Daryl McCullough
5/18/12 Shmuel (Seymour J.) Metz
5/18/12 LudovicoVan
5/16/12 Graham Cooper
5/16/12 Shmuel (Seymour J.) Metz
5/16/12 LudovicoVan
5/16/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/18/12 Virgil
5/18/12 Graham Cooper
5/18/12 David Bernier
5/18/12 Graham Cooper
5/18/12 K_h
5/18/12 |-| E R C
5/18/12 K_h
5/18/12 Graham Cooper
5/18/12 Graham Cooper
5/19/12 K_h
5/19/12 Graham Cooper
5/19/12 Graham Cooper
5/19/12 David Bernier
5/19/12 Graham Cooper
5/19/12 Graham Cooper
5/20/12 K_h
5/20/12 Graham Cooper
5/21/12 K_h
5/21/12 Graham Cooper
5/29/12 mueckenh@rz.fh-augsburg.de
5/29/12 Virgil
5/19/12 K_h
5/19/12 Graham Cooper
5/18/12 LudovicoVan
5/18/12 LudovicoVan
5/18/12 Shmuel (Seymour J.) Metz
5/17/12 Graham Cooper
5/17/12 Shmuel (Seymour J.) Metz
5/18/12 LudovicoVan
5/19/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/20/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/19/12 LudovicoVan
5/20/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/19/12 LudovicoVan
5/16/12 Graham Cooper
5/16/12 Virgil
5/18/12 K_h
5/15/12 dilettante
5/16/12 LudovicoVan
5/16/12 dilettante
5/16/12 LudovicoVan
5/16/12 dilettante
5/16/12 Graham Cooper
5/16/12 dilettante
5/16/12 Graham Cooper
5/16/12 dilettante
5/16/12 Graham Cooper
5/16/12 LudovicoVan
5/16/12 dilettante
5/16/12 LudovicoVan
5/16/12 Virgil
5/17/12 William Hughes
5/17/12 LudovicoVan
5/17/12 William Hughes
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/16/12 Graham Cooper
5/16/12 Virgil
5/16/12 |-| E R C
5/16/12 LudovicoVan
5/16/12 Virgil
5/16/12 LudovicoVan
5/16/12 Jim Burns
5/16/12 LudovicoVan
5/16/12 Graham Cooper
5/16/12 Jim Burns
5/16/12 LudovicoVan
5/16/12 Jim Burns
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5/17/12 LudovicoVan
5/17/12 Jim Burns
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/16/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/18/12 Virgil
5/17/12 |-| E R C
5/18/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/16/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 Graham Cooper
5/18/12 Daryl McCullough
5/18/12 Graham Cooper
5/18/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 YBM
5/17/12 LudovicoVan
5/18/12 Virgil
5/18/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/17/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/20/12 Shmuel (Seymour J.) Metz
5/16/12 Virgil
5/29/12 Aatu Koskensilta
5/29/12 Graham Cooper
5/30/12 K_h
5/30/12 Virgil
5/30/12 Curt Welch
5/30/12 Graham Cooper
5/30/12 ross.finlayson@gmail.com
5/30/12 Graham Cooper
5/30/12 K_h
6/1/12 ross.finlayson@gmail.com
6/8/12 Albert van der Horst
5/29/12 Aielyn
5/29/12 Aielyn
5/29/12 Graham Cooper
5/29/12 Aielyn
5/29/12 Mike Terry
5/31/12 Aielyn
5/31/12 Virgil
5/31/12 Aielyn
5/31/12 Virgil
5/31/12 Aielyn
5/31/12 Shmuel (Seymour J.) Metz
5/31/12 Shmuel (Seymour J.) Metz
5/31/12 Mike Terry
6/1/12 Graham Cooper
6/1/12 Guest
6/1/12 LudovicoVan
6/1/12 LudovicoVan
6/1/12 LudovicoVan
6/1/12 LudovicoVan
6/1/12 Graham Cooper
6/1/12 Virgil
6/1/12 Graham Cooper
6/1/12 LudovicoVan
6/1/12 |-| E R C
6/10/12 Aielyn
6/10/12 Shmuel (Seymour J.) Metz
5/29/12 Tim Little
5/31/12 LudovicoVan
6/5/12 Shmuel (Seymour J.) Metz
6/4/12 Aatu Koskensilta
6/4/12 Graham Cooper
6/4/12 ross.finlayson@gmail.com
6/4/12 ross.finlayson@gmail.com
6/4/12 Graham Cooper
6/4/12 ross.finlayson@gmail.com
6/4/12 |-| E R C