A number is chosen by random from the set of natural numbers. what is the probability that number will be even? [closed]

I know that there is no answer to this question, but how to explain that the probability doesnt exist? I spent a lot of time reading answers to this question. Almost all people say that we cant define the probability because we cant choose a number uniformly in natural set. But what if we say that $P(n) = 2^{-n}$ where $n = \{1,2,3...\}$. Then there will be no contradictions with the axioms of the probability theory

The point is not that one can't choose a natural number at random, it's that there are many different ways of doing so. When one is given a finite set $\{1,2,\ldots,n\}$ then there is a natural way to choose a number at random: uniformly. (This is by no means the only way to choose a number at random from a finite set.)

When the set is $\mathbb{N}$ it is not possible to choose a number uniformly in the sense that if $P(n=n_0)=x$ for all $n_0\in\mathbb{N}$ then by additivity we have that $P(n\in\mathbb{N})=\infty>1$ which contradicts the axioms of probability.

So there are many ways to choose a random number from $\mathbb{N}$, but uniformly is not one of them. For each such probability distribution you choose you will get a different chance that the chosen number is even.