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 here again, with a more serious objection to the alleged "solution" in
>>16
You assert that if a subset of [0,1] is obtained from another by partitioning it into finitely many pieces and translating them, then if we choose a random number in [0,1] it is "as likely" to appear in either one.  If we apply the same reasoning to [0,1]^3 we run into problems with Banach-Tarski.

Namely, if we have three disjoint balls in [0,1]^3 of the same radius, and we choose a random point in the unit cube, it should be "equally likely" for this point to lie in balls 1, 2, or 3.  But with AC, balls 2 and 3 can be obtained from 1 by partitioning 1 into finitely many pieces, and translating them; so by your reasoning it should be "as likely" for this point to be in ball 1 as in the union of balls 2 and 3.

Anyway, I sort of doubt what OP is suggesting is possible without an utterly trivial notion of "likelihood", but this is still a fun question to think about.