Quote by chronon
This is wrong. It is not possible to have a uniform probability distribution over the natural numbers.

If you read my post, I'm not talking about the probability of selecting a given number. I'm talking about the probabilities of selecting numbers wholly divisible by 10, 100, 1000, etc. as examples. If you take finite random samples of size N from the set of natural numbers, the relative proportions of such numbers in the sample will tend to match the frequency in the set as N grows large. In probability theory this holds even if the set being sampled is infinite provided you can define the set by induction. So the probability of selecting a number divisible by 10 is 0.1.