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/