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

[Gordon McMillan]

Blake blunders into the tarpit:

> On Wed 19 May, Chad Netzer <chad at vision.arc.nasa.gov> wrote:
> In comp.lang.python, you wrote:
> >Greg Ewing wrote:
> >> Chad Netzer wrote:
> >In fact, both are probably Aleph-1 sets which means they are larger than
> >the set of integers...  I'm out of my area, here, so I'll let it go at
> >that. ;)
> 
> Hmmm...  I can't quite believe that, since there is an obvious
> mapping from the set of all strings to the set of all integers. 
> (Think base n, where n=the number of possible characters.)  I
> suppose this assumes a less-than-infinite number of characters, but
> I can only think of about 104 different characters, so the total
> must be finite, right?  :)

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.

Well, as long as you leave 42 out of it.

my-density-is-calling-me-ly y'rs

- Gordon