Re: Non-countability of R

In article <Pine.BSI.4.58.0808062259310.22510@xxxxxxxxxxxxxxxxx>,
William Elliot <marsh@xxxxxxxxxxxxxxxxxx> wrote:

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).

If your construction rule does not ever produce either a 0 or a 9,
which is quite easy to arrange, you may allow all digit sequences in the
list from which you construct a non-member of that list.

Anyway, Cantor's original proof was that there were more that countably
many binary sequences, functions from N to , say, {m,w} where m =/= w.
.

Relevant Pages

  • Re: Non-countability of R
    ... We remove all sequences that end in 999... ... If your construction rule does not ever produce either a 0 or a 9, ... that method requires an integer base> 3 and has the advantage ... The restricitve method can work however for base 3. ...
    (sci.math)
  • Re: Non-countability of R
    ... We remove all sequences that end in 999... ... If your construction rule does not ever produce either a 0 or a 9, ... that method requires an integer base> 3 and has the advantage ... The restricitve method can work however for base 3. ...
    (sci.math)
  • Re: Cascading vs. Specified Systems
    ... somehow impose "magic" on the number 1000 aa? ... the odds of producing a specific ... arrangement of smaller sequences. ... I know you aren't big on statistics, ...
    (talk.origins)
  • Re: Cascading vs. Specified Systems
    ... somehow impose "magic" on the number 1000 aa? ... the odds of producing a specific ... arrangement of smaller sequences. ... I know you aren't big on statistics, ...
    (talk.origins)
  • Re: Cascading vs. Specified Systems
    ... sequence is much much less likely than producing a non-specific ... arrangement of smaller sequences. ...  I know you aren't big on statistics, ... The worst math is no math. ...
    (talk.origins)
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