Re: Non-countability of R

On Wed, 6 Aug 2008, Virgil wrote:
William Elliot <marsh@xxxxxxxxxxxxxxxxxx> wrote:

prove that "the set R of real numbers is uncountable".

PROOF. It is enough to show that the set I of all real
numbers r such that 0 <= r <= 1 is uncountable: this is
because |I| <= |R|. Assume that I is countable, so that it
can be written in the form {r_1,r_2,r_3,...}. Write each r_i
as a decimal, say

r_i = 0.r_{i1} r_{i2} ...

where 0 <= r_{ij} <= 9. We shall get a contradiction ny
producing a number in the set I which does not equal any r_i.
Define

s_i = 0 if r_{ii} <> 0; 1 if r_{ii} = 0

and let s be the decimal 0.s_1 s_2 ...; then certainly s \in
I. Hence s=r_i for some i, so that s_i=r_{ii}; but this is
impossible by the definition of s_i. QED

We remove all sequences that end in 999... to assure a bijection between
sequences and reals in [0,1).

There are some problems with base 3 as well, but none for bases of 4 or
more.

What goes wrong with base 3 if I first insist that no trimal
in [0,1) end in 2222....?

You eliminate a priori infinitely many sequences of digits.

I don't get it. One can lament the same about
tossing out those ending in 999....

Riddle of the day. How much longer before
the national debt becomes uncountable?

It is already unaccountable.

The truth is classified.
You MAY NOT know this.
.

Relevant Pages

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  • Re: Cascading vs. Specified Systems
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