while (a=b()) ... infinite sets digression

[Bryan VanDeVen]

Graham Matthews wrote:
> 
> Gordon McMillan (gmcm at hypernet.com) wrote:
> : Moshe, Chad & I straightened this out in email. Chad is correct
> : (well, if you take him as saying "more than Aleph-0").
> :
> : Instead of the integers, take the numbers between 0 and 1. Use your
> : mapping. Now realize that most of the numbers you've created are
> : transcendentals, and the rationals, by comparison, amount to a hill
> : of beans. So it _is_ a denser form of infinity than the integers.
> 
> I am not sure I understand this. Surely the set of strings on a finite

Me either, the real numbers beween 0 and 1 are all infinite strings -
limits of infinite cauchy sequences (all of them, 0.1 <=>
0.100000000...) which is why Cantor's Diagonalization argument works. 
But not all of the strings in a set of all strings over a finite
alphabet are infinite so Cantor's argument is not applicable.  

Besides, the set of all strings of single digits (0 through 9) is an
infinite set of strings over a finite alphabet, and it can easily  be
mapped to the integers.  In fact, most people do it without even
thinking.  QED. :)

When's the last time they talked set theory in that other newsgroup. ;)
  
-- 
Bryan Van de Ven
Applied Research Labs
University of Texas, Austin