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.

>>37
>It does...
You don't need AC to pick an element from one infinite set.  Besides, AC doesn't say anything about the distribution of the element chosen.

>What the fuck? No it's not, it a valid answer to OP's question. "Create a method to..." well there's the fucking method!

The problem was to pick an element uniformly at random from the integers (a countable set) which is the same thing as picking an element at random from the set of sequences of coinflips that are eventually all zero, since the two sets are in bijection.