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Topic: Cantor's diagonal argument: Simplified
Replies: 107   Last Post: Nov 12, 2012 9:41 PM

 Messages: [ Previous | Next ]
 David R Tribble Posts: 3,426 Registered: 7/21/05
Re: Cantor's diagonal argument: Simplified
Posted: Nov 9, 2012 6:51 PM

LudovicoVan wrote:
>> But, have you got any proof or argument to support your claim
>> that "[it is a] fact that technically speaking the collection
>> of ALL real number does exhaust the collection of all infinite
>> binary sequences"?

Jack Campin wrote:
>> Given any binary sequence f: N ->> {0,1}, define
>>
>> r(f) = sum (n = 0 ->> inf) 2 ^ (-n)
>>
>> This is always a real number in [0, 1].
>>
>> Done.

LudovicoVan wrote:
>> You have done nothing that proves the claim above.

Uirgil wrote:
> Do you dispute that every r(f) is necessarily a real number?
> That would be the only basis for any objection.

It's true that every r(f) is a real in [0,1].

However the phrase "the collection of real numbers exhausts the
collection of all infinite binary sequences" seems to imply
that the mapping should be the other way around (although I
could simply be reading it wrong); i.e., that every real in
[0,1] maps to a unique infinite binary sequence.

Of course, that should not be too hard to show, either.
(Except for the possible complication that every rational real
has two different binary representations.)

-drt

Date Subject Author
11/4/12 Zaljohar@gmail.com
11/4/12 J. Antonio Perez M.
11/4/12 LudovicoVan
11/4/12 Uirgil
11/5/12 LudovicoVan
11/5/12 Uirgil
11/5/12 Graham Cooper
11/5/12 LudovicoVan
11/5/12 zuhair
11/5/12 LudovicoVan
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11/6/12 LudovicoVan
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11/6/12 LudovicoVan
11/6/12 LudovicoVan
11/6/12 Uirgil
11/6/12 LudovicoVan
11/6/12 Uirgil
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11/6/12 LudovicoVan
11/6/12 Zaljohar@gmail.com
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11/6/12 LudovicoVan
11/6/12 Zaljohar@gmail.com
11/6/12 LudovicoVan
11/6/12 LudovicoVan
11/6/12 LudovicoVan
11/6/12 Uirgil
11/6/12 Uirgil
11/6/12 LudovicoVan
11/6/12 Uirgil
11/6/12 Shmuel (Seymour J.) Metz
11/9/12 David R Tribble
11/6/12 Uirgil
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11/7/12 Zaljohar@gmail.com
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11/7/12 LudovicoVan
11/7/12 Zaljohar@gmail.com
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11/7/12 Zaljohar@gmail.com
11/7/12 LudovicoVan
11/7/12 Zaljohar@gmail.com
11/7/12 LudovicoVan
11/7/12 Zaljohar@gmail.com
11/7/12 Uirgil
11/7/12 LudovicoVan
11/7/12 Jesse F. Hughes
11/7/12 LudovicoVan
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11/7/12 LudovicoVan
11/7/12 Uirgil
11/7/12 Jesse F. Hughes
11/7/12 LudovicoVan
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11/7/12 LudovicoVan
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11/7/12 Uirgil
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11/7/12 Zaljohar@gmail.com
11/5/12 LudovicoVan
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11/7/12 Charlie-Boo
11/7/12 Charlie-Boo
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11/7/12 LudovicoVan
11/7/12 Zaljohar@gmail.com
11/7/12 LudovicoVan
11/7/12 Uirgil
11/5/12 Graham Cooper
11/5/12 Rupert
11/5/12 Zaljohar@gmail.com
11/7/12 Charlie-Boo
11/7/12 Zaljohar@gmail.com
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11/8/12 Zaljohar@gmail.com
11/8/12 Charlie-Boo
11/8/12 William Hale
11/8/12 Charlie-Boo
11/8/12 William Hale
11/8/12 Uirgil
11/9/12 Shmuel (Seymour J.) Metz
11/9/12 Uirgil
11/12/12 Charlie-Boo
11/12/12 billh04
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11/12/12 William Hale
11/12/12 Uirgil
11/12/12 Uirgil
11/12/12 billh04
11/12/12 Charlie-Boo
11/12/12 William Hale
11/12/12 Uirgil
11/9/12 Zaljohar@gmail.com
11/10/12 Charlie-Boo
11/12/12 Charlie-Boo
11/12/12 Charlie-Boo
11/10/12 Charlie-Boo
11/10/12 Zaljohar@gmail.com
11/12/12 Charlie-Boo
11/12/12 Zaljohar@gmail.com