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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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 Tucsondrew@me.com Posts: 1,161 Registered: 5/24/13
Re: Uncountability of the Real Numbers Without Decimals
Posted: Dec 2, 2013 3:07 PM

On Monday, December 2, 2013 2:50:51 AM UTC-7, William Elliot wrote:
> 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?
>

The Sequence X is given, and hence so are all its members.

Choose a Real Number a_1 such that a_1 ~= x_1.

If x_1 < a_1 then choose b_1 > a_1, else choose b_1 e ( a_1, x_1 ).

Now, if x_2 e ( a_1, b_1 ) choose a_2 e ( x_2, b_1 ), and then choose b_2 e ( a_1, b_1 ).
OTOH, if x_2 ~e ( a_1, b_1 ), simply choose a_2 and b_2 from ( a_1, b_1 ) such that a_2 < b_2.

The method used to find a_2 and b_2 can be used for all other a_k and b_k.

All Sets chosen from are non-empty, so the chosen do exist.

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

What would make that easier?

If we were to use open intervals, we could end up with a = sup(A) being one of x_n's.
At least, I think so. Still working out the logic.
But anyway, if she ain't broke don't fix her.

> > 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.

We should be able to use the same basic proof structure.
Just take your a and b, for which you know a < b, and show that there are uncountably many members, t, such that a < t < b.

The denseness of the order gives the non-empty subsets and its completeness give the supremum.

ZG

Date Subject Author
12/2/13 Tucsondrew@me.com
12/2/13 William Elliot
12/2/13 Tucsondrew@me.com
12/2/13 G. A. Edgar
12/2/13 Tucsondrew@me.com
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 gnasher729
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/2/13 Tucsondrew@me.com
12/2/13 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 Virgil
12/3/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/5/13 Virgil
12/10/13 Robin Chapman
12/2/13 Virgil
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/2/13 Tucsondrew@me.com
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
12/5/13 Michael F. Stemper
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Tucsondrew@me.com
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/6/13 Brian Q. Hutchings
12/7/13 Brian Q. Hutchings
12/7/13 Brian Q. Hutchings
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 albrecht
12/7/13 fom
12/7/13 ross.finlayson@gmail.com
12/8/13 albrecht
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/7/13 fom
12/8/13 Virgil
12/7/13 Virgil
12/7/13 Virgil
12/7/13 Virgil
12/8/13 albrecht
12/6/13 Virgil
12/6/13 Virgil
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/5/13 Virgil
12/3/13 Virgil
12/3/13 Michael F. Stemper
12/3/13 Virgil
12/3/13 fom
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
12/5/13 Virgil
12/2/13 Virgil
12/2/13 Tucsondrew@me.com
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
12/3/13 Virgil
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
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/8/13 wolfgang.mueckenheim@hs-augsburg.de
12/8/13 Virgil
12/5/13 Virgil
12/3/13 Virgil
12/2/13 ross.finlayson@gmail.com
12/4/13 ross.finlayson@gmail.com
12/3/13 albrecht
12/3/13 Tucsondrew@me.com
12/5/13 albrecht
12/5/13 Tucsondrew@me.com