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In article <33588D23.3FDF@ix.netcom.com>,
The Silent Observer  <silent1@ix.netcom.com> wrote:

[....]
>By contrast, between 1 and 2, there are an infinite number of rational 
>numbers -- just in the series 1/2, 2/3, 3/4, 4/5, 5/6, ... there is a 
>countably infinite number of members, and there are an infinite number of 
>such series of rational numbers between every two integeres.  Hence, the 
>rational numbers cannot be mapped onto the natural numbers, and is called 
>"uncountably infinite."

This is wrong.   

Proof:

Any rational number can be written in the form of N/D.  If each rational 
number can be mapped to a integer there is a countable number of them.  
For any rational number an integer can be written using the following 
procedure:

Write down the value in the form N/D.  If one of the numbers has less 
digits than the other add leading zeros to that number to make the number 
of digits equal.

Starting at the left end of the number write down the left most digit of 
N and then the left most digit of D.  Next write beside those digits the 
next digit of N and the next digit of D.  Continue this process of taking 
one digit from each number until all of the digits have been used.

Example:

           12/345  ...>   012/345   ....>  304152

Each rational number can thus be mapped to an integer and hence the 
number of rationals must equal the number of integers.


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