Context: I saw a comment that if we choose a random positive integer, no matter how big it is, it will be closer to $0$ than $\infty$. Although this statement is far from rigorous this made me wonder: if we assume that we had an 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\limits _{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 $\mathbb{N}$?