Date: Dec 2, 2013 8:34 AM
Author: G. A. Edgar
Subject: Re: Uncountability of the Real Numbers Without Decimals

In article <38ef6b5b-8df9-430a-9e5d-2867d4fde621@googlegroups.com>,
Zeit Geist <tucsondrew@me.com> 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 ].
>
> 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 ].
>
> 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.
>
> ZG


Similar to Cantor's first proof of uncountability of the real line.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/