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Topic: Diagonal Argument
Replies: 157   Last Post: Jun 22, 2000 7:44 AM

 Messages: [ Previous | Next ]
 Chuck Cadman Posts: 16 Registered: 12/12/04
Re: Diagonal Argument
Posted: Jun 9, 2000 11:44 AM

Flavio Borges Botelho wrote in message <3940E639.64179351@pronet.net>...

>Well, i know that the real numbers produced by this arrangement does not
>seem get to the irrational numbers, but the arrangement does provide
>aproximations as good as necessary.
> For any e we choose, it's possible to find a number n E N so that:
> |Irracional Number - F(n)| > e

Yes, it is true that the reals have a countable dense subset, i.e., a set S
which has the cardinality of the naturals and the property that every real
number is the limit of a sequence of elements of S. But that doesn't make
the reals countable.

Here's something else to consider. Since S is countable, it can be
expressed as a sequence: s_1, s_2, ....
Now let U_n be the open interval (s_n - 2^{-n-1}, s_n + 2^{-n-1}) Then U_n
has length
2^{-n} so the union of all these sets, call it U, has total measure <= 1.
Furthermore, U is an open set containing this dense subset S. But somehow U
does not equal all the real numbers.

Date Subject Author
6/9/00 Flavio Borges Botelho
6/9/00 Dr. Michael Ulm
6/9/00 Flavio Borges Botelho
6/12/00 Dik T. Winter
6/12/00 Flavio Borges Botelho
6/13/00 Dik T. Winter
6/9/00 Simon Kristensen
6/9/00 Flavio Borges Botelho
6/9/00 Clive Tooth
6/9/00 Flavio Borges Botelho
6/12/00 Jonathan Hoyle
6/9/00 Axel Harvey
6/9/00 Flavio Borges Botelho
6/9/00 Axel Harvey
6/9/00 Flavio Borges Botelho
6/9/00 Richard Carr
6/9/00 Flavio Borges Botelho
6/12/00 Jonathan Hoyle
6/13/00 Flavio Borges Botelho
6/13/00 bobg0
6/14/00 Flavio Borges Botelho
6/16/00 bobg0
6/17/00 Flavio Borges Botelho
6/17/00 bobg0
6/18/00 Flavio Borges Botelho
6/18/00 bobg0
6/19/00 Flavio Borges Botelho
6/19/00 Flavio Borges Botelho
6/19/00 denis-feldmann
6/19/00 Flavio Borges Botelho
6/19/00 bobg0
6/19/00 Flavio Borges Botelho
6/20/00 bobg0
6/20/00 Flavio Borges Botelho
6/20/00 Flavio Borges Botelho
6/20/00 Flavio Borges Botelho
6/20/00 denis-feldmann
6/20/00 Flavio Borges Botelho
6/21/00 denis-feldmann
6/21/00 Flavio Borges Botelho
6/21/00 denis-feldmann
6/21/00 david_ullrich@my-deja.com
6/21/00 denis-feldmann
6/22/00 Flavio Borges Botelho
6/22/00 Flavio Borges Botelho
6/20/00 denis-feldmann
6/20/00 Flavio Borges Botelho
6/20/00 Flavio Borges Botelho
6/14/00 Jonathan Hoyle
6/14/00 Flavio Borges Botelho
6/14/00 Jonathan Hoyle
6/15/00 Flavio Borges Botelho
6/15/00 Jonathan Hoyle
6/17/00 Flavio Borges Botelho
6/14/00 Virgil Hancher
6/15/00 Flavio Borges Botelho
6/9/00 Kent
6/9/00 Flavio Borges Botelho
6/10/00 Virgil Hancher
6/10/00 Flavio Borges Botelho
6/10/00 Fred W. Helenius
6/13/00 Flavio Borges Botelho
6/13/00 Fred W. Helenius
6/10/00 Virgil Hancher
6/13/00 Flavio Borges Botelho
6/14/00 Dave Seaman
6/14/00 Flavio Borges Botelho
6/14/00 Dave Seaman
6/15/00 Flavio Borges Botelho
6/15/00 Dave Seaman
6/16/00 Flavio Borges Botelho
6/16/00 Pertti Lounesto
6/16/00 denis-feldmann
6/16/00 Dik T. Winter
6/16/00 Dave Seaman
6/16/00 Mike Mccarty Sr
6/17/00 Flavio Borges Botelho
6/17/00 bobg0
6/18/00 Flavio Borges Botelho
6/19/00 Mike Mccarty Sr
6/20/00 Flavio Borges Botelho
6/14/00 bobg0
6/14/00 Mike Mccarty Sr
6/14/00 Fred W. Helenius
6/15/00 Flavio Borges Botelho
6/15/00 Dik T. Winter
6/15/00 Mike Mccarty Sr
6/15/00 Edward C. Hook
6/15/00 Mike Mccarty Sr
6/16/00 Flavio Borges Botelho
6/16/00 Dik T. Winter
6/17/00 Flavio Borges Botelho
6/17/00 Dik T. Winter
6/19/00 Mike Mccarty Sr
6/16/00 hale@mailhost.tcs.tulane.edu
6/17/00 Flavio Borges Botelho
6/16/00 Mike Mccarty Sr
6/15/00 Daryl McCullough
6/12/00 Jonathan Hoyle
6/13/00 Flavio Borges Botelho
6/14/00 Flavio Borges Botelho
6/10/00 Kent
6/10/00 Richard Carr
6/10/00 Kent
6/10/00 Archimedes Plutonium
6/11/00 Jos Bergervoet
6/12/00 Jonathan Hoyle
6/11/00 Flavio Borges Botelho
6/11/00 David C. Ullrich
6/13/00 Flavio Borges Botelho
6/14/00 Flavio Borges Botelho
6/14/00 David C. Ullrich
6/14/00 Pertti Lounesto
6/14/00 denis-feldmann
6/15/00 Flavio Borges Botelho
6/15/00 David C. Ullrich
6/11/00 Kent
6/14/00 Flavio Borges Botelho
6/11/00 Virgil Hancher
6/13/00 Flavio Borges Botelho
6/14/00 denis-feldmann
6/14/00 Flavio Borges Botelho
6/14/00 Dave Seaman
6/11/00 Virgil Hancher
6/13/00 Flavio Borges Botelho
6/13/00 Nicolas Bray
6/14/00 Flavio Borges Botelho
6/14/00 David C. Ullrich
6/17/00 Flavio Borges Botelho
6/14/00 Dave Seaman
6/17/00 Flavio Borges Botelho
6/13/00 hale@mailhost.tcs.tulane.edu
6/14/00 Flavio Borges Botelho
6/14/00 Virgil Hancher
6/14/00 Flavio Borges Botelho
6/19/00 anonymous
6/19/00 Flavio Borges Botelho
6/19/00 Dave Seaman
6/20/00 Flavio Borges Botelho
6/20/00 anonymous
6/20/00 Flavio Borges Botelho
6/12/00 Jonathan Hoyle
6/10/00 John Savard
6/13/00 Flavio Borges Botelho
6/14/00 John Savard
6/14/00 Flavio Borges Botelho
6/17/00 John Savard
6/18/00 Pertti Lounesto
6/18/00 david_ullrich@my-deja.com
6/18/00 Pertti Lounesto
6/18/00 Virgil Hancher
6/19/00 Mike Mccarty Sr
6/19/00 Mike Mccarty Sr
6/14/00 Jon Haugsand
6/14/00 Pertti Lounesto
6/14/00 denis-feldmann
6/14/00 Jonathan Hoyle