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.

>>36
You're looking at the (countable) set of all infinite sequences of coinflips which are eventually always zero
Yeah...

and just claiming that AC lets you pick a random element of that set
It does...

which is basically the same problem we started with.
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!