Re: Proof that real numbers are not countable

On Feb 1, 4:23 pm, "larry.free...@xxxxxxxxx" <larry.free...@xxxxxxxxx>
wrote:
Hi Math Experts,

I am a math amateur who runs a math web site and recently, someone
asked me a question that I found myself unable to answer in a clear
and concise way.  I would greatly appreciate your help in answering
the question.

The question regards Cantor's proof that the real numbers are
uncountable.

A reader of my blog argues that the real numbers in (0,1) are
countable using the following construction:

0.1
...
0.9
0.01
...
0.09
0.11
...
0.19
...
0.91
...
0.99
0.001
...
0.009
0.010
...

I would appreciate it if someone could clearly state what is wrong
with this.  The reader wants to know why the above construction
doesn't disprove Cantor's Diagonal argument.

I had argued that there will always be irrational numbers missing from
the construction and that this was the essence of Cantor's argument.
The questioner responded that it was no different from the whole
numbers.  No matter how many you list out, there is always a number
that is not included.

I would like to answer the question in a definitive way that will
encourage the reader to ask these types of questions but at the same
time give the person a more detailed understanding of Cantor's insight
about real numbers.

Here is the link to the blog entry and the question asked:http://mathrefresher.blogspot.com/2006/09/countability.html

Thanks very much,

-Larry

Hi, All the numbers numbers on your list (terminating decimals) are
rational and in fact of the form m/10^n so even 1/3 =.3333....is not
on your list and of course all irrationals are missing.Regards,smn
.

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