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Topic: Uncountability of the Real Numbers Without Decimals
Replies: 110   Last Post: Dec 10, 2013 4:33 AM

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 wolfgang.mueckenheim@hs-augsburg.de Posts: 3,394 Registered: 10/18/08
Re: Uncountability of the Real Numbers Without Decimals
Posted: Dec 3, 2013 6:19 AM

Am Dienstag, 3. Dezember 2013 11:21:46 UTC+1 schrieb christian.bau:

> Shouldn't really replying at all?

Yes, if you prefer to stay blind, you should only talk to other blinds.

>"Zeit Geist"s fine post proved that the real numbers are uncountable by starting with an assumption that there is a sequence containing them all (which is _by definition_ the opposite of the reals being uncountable) and went on to demonstrate a contradiction, which means the assumption of the existence of such a sequence must be false.

It seems you have not yet understood the main point. To contradict a nonsensical definition does not prove anything: There is no sequence containing "all" rational numbers.

> You seem to object to the existence of such a sequence - not realising that this is _exactly_ the point.

This *would* be the point if such sequence could exist at all. But that is impossible. Please try at least to understand that you are mistaken if you think that you can conclude from

"every rational number can be enumerated"

on

"all rational numbers can be enumerated"

The full text of the first line reads

"every rational number can be enumerated if and only if beyond that rational number infinitely many are following."

Can you understand that it is impossible to enumerate any rational number violating this requirement? Can you understand that those following beyond are also rational numbers?

There is no "countable" set of rationals. Therefore the contradiction of countability is a futile exercise.

But if countability *would* be a sensible notion, then there is a simple proof that all definable reals belong to a countable set. Since all "uncountability proofs" prove the existence of a further definable number, all are trash.

Regards, WM

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12/2/13 Tucsondrew@me.com
12/2/13 William Elliot
12/2/13 Tucsondrew@me.com
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12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 gnasher729
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
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12/3/13 wolfgang.mueckenheim@hs-augsburg.de
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12/5/13 wolfgang.mueckenheim@hs-augsburg.de
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12/5/13 wolfgang.mueckenheim@hs-augsburg.de
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12/5/13 wolfgang.mueckenheim@hs-augsburg.de
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12/7/13 wolfgang.mueckenheim@hs-augsburg.de
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12/6/13 wolfgang.mueckenheim@hs-augsburg.de
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12/7/13 wolfgang.mueckenheim@hs-augsburg.de
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12/7/13 Virgil
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12/8/13 Virgil
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12/2/13 Tucsondrew@me.com
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
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12/2/13 Virgil
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12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 gnasher729
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 gnasher729
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/3/13 Virgil
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12/5/13 gnasher729
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 Tucsondrew@me.com
12/5/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
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12/3/13 albrecht
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