The reason you can’t sample uniformly from the integers is more like “because they are not compact” or “because they are not bounded” than “because they are infinite and countable”. You also can’t sample uniformly at random from the reals. (If you could, then composing with floor would give you a uniformly random sample from the integers.)
If you want to build a uniform probability distribution over a countable set of numbers, aim for all the rationals in [0, 1].
How do you sample uniformly from the integers?
The reason you can’t sample uniformly from the integers is more like “because they are not compact” or “because they are not bounded” than “because they are infinite and countable”. You also can’t sample uniformly at random from the reals. (If you could, then composing with floor would give you a uniformly random sample from the integers.)
If you want to build a uniform probability distribution over a countable set of numbers, aim for all the rationals in [0, 1].