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

[Gordon McMillan]

Chad Netzer wrote:

> Well, I am not a Set Theorist, 

I was working my way towards that conclusion <wink>.

/snip/
> Assuming a character set of size N, the number of possible strings
> that start with a particular character (assuming unbounded string
> length) is:
> 
> 1 + N + N**2 + N**3 + N**4 + ... + N**infinity

Bzzt. Arithmetic on infinities works just as well as arithmetic on 
NaNs. But that's really [i from 0..infinity]sumof(N**i), so N**i 
converges to infinity rather more quickly than i, but it takes a good 
deal more than that to say it _passes_ Aleph-0.

> ...  My
> understanding is that both sets have the same cardinality, and are
> thus both are the same size (in the formal sense, despite being
> infinitely large).  

Whew. You got _something_ right!

> 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. ;)

Ay yi yi. They are countable (Aleph-0). Proof: you can sort 
them. In fact, using an insertion sort, I can sort them a 
_whole_lot_faster_ than you can create them. (See what happens when 
you compare infinities?).

thank-god-it-doesn't-take-set-theory-to-launch-a-shuttle-ly y'rs

- Gordon