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.

>>63

No, everyone's been getting in a fucking tizzy about flipping coins, and whether constructing a random number in [0,1] uses AC (It doesn't)

It's been hinted at informally, which is why everyone's been getting confused, I thought it best to state it properly.

Me saying it doesn't use the axiom of choice is referring to that problem specifically, if you're OP didn't you make that silly post about bins in which you claimed to pick an element out of each bin uses the axiom of choice?

Because it doesn't. You don't need AC if the sets are well-ordered, cause you can just specify the choice function, and also the existence of a choice function doesn't guarantee you a set of ones uniformly distributed. Those are specific points people are being ridiculous about.