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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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William Elliot

Posts: 1,618
Registered: 1/8/12
Re: Uncountability of the Real Numbers Without Decimals
Posted: Dec 2, 2013 4:50 AM
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On Mon, 2 Dec 2013, Zeit Geist wrote:

> The following is a Proof of the Uncountability of the Set of real Numbers.

> The proof proceeds by Contradiction. We assume the Set of Real Numbers is
> Countable, and thus can be exhausted in a Sequence.


> Take any such Sequence of Real Numbers, X = { x_n | n e N }. We begin by
> choosing Real Numbers, a_1 and b_1, such that a_1 < b_1 and x_1 ~e [ a_1,
> b_1 ]. Next, we choose Real Numbers, a_2 and b_2, such that a_1 < a_2 < b_2
> < b_1 and x_2 ~e [ a_2, b_2 ]. We continue by choosing Real Numbers a_k and
> b_k for every k e N, such that for every k e N, we have a_k-1 < a_k < b_k <
> b_k-1 and, of course, x_k ~e [ a_k, b_k ].


Prove that you can construct the sequences as described.
What property of real numbers would you be using?

I think it's easier to use open intervals instead of closed intervals.

> Doing so for every Natural Number, we define A = { a_n | n e N }. Now, A is
> a Set of Real Numbers that is bounded above, since any b_n is an upper bound
> of A. Hence, a = sup(A) is a Real Number. Since our Sequence, X, exhausts
> all Real Numbers, a e X and there is a Natural Number, m, such that x_m = a.


> Now, we have previously defined Real Numbers, a_m and b_m, such that a = x_m
> ~e [ a_m, b_m ]. However, we know a_m <= a, since a = sup(A) and a e A; and
> a <= b_m, since any b_n is an upper bound of A. These together give us that
> we must have x_m e [ a_m, b_m ]. This results in a Contradiction. Hence,
> we must have that the Set of Real Numbers is Uncountable.
> qed


> I find the proof rather straight forward. Question, comments, suggestions
> and corrections are welcome.


An order is order dense when for all a,b,
. . if a < b, there some t with a < t < b.

Show that any ordered set, even if it's not linearly nor totally order,
that's order dense, has two elemens a,b with a < b,
and for which every bounded above set has a supremum, is uncountable.




Date Subject Author
12/2/13
Read Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
William Elliot
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
G. A. Edgar
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
gnasher729
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/10/13
Read Re: Uncountability of the Real Numbers Without Decimals
Robin Chapman
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Michael F. Stemper
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/6/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/6/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/6/13
Read Re: Uncountability of the Real Numbers Without Decimals
Brian Q. Hutchings
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Brian Q. Hutchings
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Brian Q. Hutchings
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
fom
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
albrecht
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
fom
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
ross.finlayson@gmail.com
12/8/13
Read Re: Uncountability of the Real Numbers Without Decimals
albrecht
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
fom
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
fom
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
fom
12/8/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/8/13
Read Re: Uncountability of the Real Numbers Without Decimals
albrecht
12/6/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/6/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Michael F. Stemper
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
fom
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/2/13
Read Re: Uncountability of the Real Numbers
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers
wolfgang.mueckenheim@hs-augsburg.de
12/3/13
Read Re: Uncountability of the Real Numbers
Virgil
12/5/13
Read Re: Uncountability of the Real Numbers
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Uncountability of the Real Numbers
Virgil
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
gnasher729
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
gnasher729
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
gnasher729
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/8/13
Read Re: Uncountability of the Real Numbers Without Decimals
wolfgang.mueckenheim@hs-augsburg.de
12/8/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Virgil
12/2/13
Read Re: Uncountability of the Real Numbers Without Decimals
ross.finlayson@gmail.com
12/4/13
Read Re: Uncountability of the Real Numbers Without Decimals
ross.finlayson@gmail.com
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
albrecht
12/3/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
albrecht
12/5/13
Read Re: Uncountability of the Real Numbers Without Decimals
Tucsondrew@me.com

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