choosing a random integer from $\mathbb{N}$

Question

I heard that if we choose a random positive integer, no matter how big it is, it will be closer to $0$ than $∞$. I have an interesting question: if we assume that we had an infinite computer which had infinite power and memory, then what would happen if I make it choose a random integer from all $\mathbb{N}$? The probability of choosing an integer from say $n$ numbers is $\frac{1}{n}$, and since $\lim _{n \to \infty} \frac{1}{n} =0 $ , that means any number we choose has a probability of $0$ of appearing. So that means any number we choose won’t appear. Then what will appear on the screen of the computer? There is a contradiction here, but I don’t know what it means. Does this mean that we can’t choose a random number from all $\mathbb{N}$?

Answer 1

Does this mean that we can’t choose a random number from all $\mathbb{N}$?

Yes, sort of. More formally, there is no way to choose a random number from $\mathbb{N}$ in such a way that every number has the same probability of being chosen (since that probability can't be $0$ and it also can't be positive). There are probability distributions you can define on $\mathbb{N}$ but they aren't and can't be uniform.

A probability distribution on $\mathbb{N}$ is just a sequence $p_1, p_2, \dots$ of non-negative real numbers satisfying $\sum_{i=1}^{\infty} p_i = 1$. Examples are given by the zeta distributions for $s > 1$.

Answer 2

It means that you cannot apply the uniform distribution to the set of natural numbers, period. A computer that does cannot possibly exist. Therefore you'll a) have to be more specific about what "random" means in this context and b) discard the uniform distribution as a possibility when you do so

Answer 3

Suppose you have written a computer program that is able to generate (in finite time) any natural number with equal probability of every number occuring. Now for the computer to be able to display that information, it will first print the first digit, then the next and so on. However there is no upper bound to the length of a natural number. Thereby any attempt to display a number will take an amount of time which has no upper bound. Therefore the time taken is infinite in essence. But our original assumption was that the algorithm be completed in finite time. Therefore we have reached an impasse.