You should be able to solve this.

Create a method to choose a random integer between 0 and infinity such that no integer is more likely to be chosen than any other.

Yes it is possible, I don't care what your probability book says (Notice I didn't even use the word "probability").  You may assume the axiom of choice.

>>39
>cheating

OP here: A little bit, maybe. ^^ (notice I avoided the word "probability")  It makes intuitive sense, though, imo.  I think maybe if you threw Vitali sets in as extra generators of your sigma field, and used outer measure instead of lebesgue measure for your probability, you could still get a finitely additive probability space such that the probability of any single integer is zero. I don't know much about how finitely additive probability works (does your sigma field still have to be closed under infinite unions?) so maybe I'm wrong.

I know it's possible to get a uniform distribution on the natural numbers with finitely additive probability (if someone wants, I can try to find the paper I read about it in).  What I was trying to get at though is some kind of algorithm to actually pick one.  This doesn't really work for that, though, since you can't explicitly describe Vitali sets or figure out which one some given real number is in.