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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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 albrecht Posts: 1,136 Registered: 12/13/04
Re: Uncountability of the Real Numbers Without Decimals
Posted: Dec 3, 2013 6:09 AM

On Monday, December 2, 2013 9:09:39 AM UTC+1, Zeit Geist wrote:
> The following is a Proof of the Uncountability of the Set of real Numbers. Please note that it avoids the use the Representation of the Real Numbers as Infinite Decimals. It uses only the Property of Completeness (that every Set of Real Numbers that is bounded above has a Least Upper Bound which is a Real Number); and some Properties due to the Ordering of the Real Numbers, such as if x is Real Number then there exist Real Numbers a and b such that a < b and x ~e [ a, b ].
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> The proof proceeds by Contradiction. We assume the Set of Real Numbers is Countable, and thus can be exhausted in a Sequence.
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> 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
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> a_k-1 < a_k < b_k < b_k-1 and, of course, x_k ~e [ a_k, b_k ].
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> 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.
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> 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.
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> qed
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> I find the proof rather straight forward. Question, comments, suggestions and corrections are welcome.
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> ZG

You are right with the fact that there is no sequence X = { x_n | n e N } of all reals. If you define this fact as Uncountability, the reals are uncountable. Okay.
But does this mean that there are more reals than naturals? No.

Without Axiom of Infinity and Axiom of Powerset: Anyway: No.

And with AoI and AoP: No. Because ZFC has to have a countable model and the quantifiers "less" and "more" are irrelevant in the domain of infinity.

Infinity/Ceaselessness is a quality , not a quantity.

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