Re: Non-countability of R

In article <Pine.BSI.4.58.0808060142370.7465@xxxxxxxxxxxxxxxxx>,
William Elliot <marsh@xxxxxxxxxxxxxxxxxx> wrote:

On Tue, 5 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

I think that this method won't work because imho the author
should have spoke about the fact that the decimal representation
of reals is not unique (as other authors do).

You are correct. To correct for this, require that the
reals in [0,1] be expressed without ending in 9999....

Now however to write 1 in the form of 0.d1 d2 d3 ...
requires 1 = 0.9999.... So we divest ourselves of 1
by just listing the reals in [0,1) expressed as decimals
not ending in 999....

It should work because the diagonal decimal is entirely of 0's and 1's.
Now if perhaps, it ends in d 0000..., (0 < d) then it should be in the
list because decimals ending in (d-1) 9999... have been omitted as a
representation of a real in [0,1).

What do you see wrong with the method? Yes, if one used 9's instead
of 1's in the diagonal decimal definition, then it wouldn't work.
Also this method won't work in binary decimals but it will for all
other bases b in N - {1,2}.

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.

But base two and three can both be fixed by taking digits in pairs and
limiting the 'diagonal' to the pairs '01' and '10'

Tsk tsk. You're using base 4 instead of base 2 or
base 9 instead of base 3.

Or at least properties of those bases.

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

It is already unaccountable.
.

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