In the frequentist interpretation of probability, we define probability as follows
$$P(x) = \lim_{n_t \to \infty}\frac{n_x}{n_t}$$
Where $n_t$ is the number of trials we preform of a certain experiment, and $n_x$ is the number of times we observe the event $x$ occurring during those experiments.
Let's say we have some experiment $E$ that outputs a random natural number, and we are trying to determine the probability distribution on it. So we decide to preform it an infinite number of times, and collect statistics. The first time (whose result we will call $E_0$), we get $0$. The second time ($E_1$), we get $1$. The third time we get $2$. After preforming all the experiments and collecting statistics, we see that $E_n=n$ for all $n \in \mathbb N$.
Now, for any $n$, the probability that result of $E$ is $n$ equals $0$ (i.e. $P(\text{$E$ results in $n$})=0$). This seems to violate the axioms of probability though! The total probability is $0$, by the third axiom, which contradicts the second axiom.
So, what happened here? Why doesn't the frequentest probabilities corresponding to $E$ form an axiomatically correct probability distribution?
(If you do not like events with infinite domains, you can also use a finite domain. For example, let's say you have a coin that you flip infinitely many times. You get $1$ heads, then $2$ tails, then $4$ heads, then $8$ tails, etc...)