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   From: sylvia@not.at.this.address   
      
   _Schwann_ wrote:   
      
   > That's news to me. I thought infinity meant infinite, which means that   
   > it can be ascribed a symbol, but not a number. I suppose, like most   
   > else, it depends how it was 'meant' and I meant 'infinite, without   
   > number or end...   
   >   
   > thanks for the interesting comment....   
      
   An example of a countable infinity is the set of rational numbers. You   
   can write them all down this way   
      
   1/1 2/1 3/1 4/1 5/1 ...   
      
   1/2 2/2 3/2 4/2 5/2 ...   
      
   1/3 2/3 3/3 4/3 5/3 ...   
      
   1/4 2/4 3/4 4/4 5/4 ...   
      
   1/5 2/5 3/5 4/5 5/5 ...   
      
   etc.   
      
   Now you take ever larger diagonal sweeps through them   
      
   1/1  1/2 2/1  1/3 2/2 3/1  1/4 2/3 3/2 4/1  1/5 2/4 3/3 4/2 5/1 ...   
      
   So there's a systematic way of putting them into a list, and you can   
   count them in the sense that you can assign an integer to each of them   
      
   1  1/1   
   2  1/2   
   3  2/1   
   4  1/3   
   5  2/2   
   6  3/1   
   7  1/4   
   8  2/3   
   9  3/2   
   10 4/1   
   11 1/5   
   12 2/4   
   13 3/3   
   14 4/2   
   15 5/1   
      
   etc.   
      
   OK - big deal.   
      
   But try it with all numbers, including the irrational ones. Suppose we   
   wrote their decimal expansions down in a list, like this. Some end with   
   a (countably) infinite number of zeros, but most don't.   
      
   1  abcdef.....   
   2  ghijkl.....   
   3  zyxwv......   
   4  mnopq......   
      
   where the letters stand for arbitrary digits (not necessarily different).   
      
   Now, write a number whose first digit is not the first digit of the   
   first number (ie, not a), whose second digit is not the second digit of   
   the second number (ie, not h), and so on.   
      
   Since it has at least one digit that differs from the corresponding   
   digit of a number in the list, our constructed number cannot be in the   
   list, despite the list supposedly containing all the numbers.   
      
   The conlusion must be that it is not possible to write down all the   
   numbers in a list, even if the list is infinitely long. This is an   
   uncountable infinity.   
      
   Sylvia.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)