$\textbf{The Problem:}$ Is it possible to set up a probability space with sample space $\Omega=\{1,2,\dots\}$ to model a 'uniformly chosen positive integer'?
$\textbf{My Thoughts:}$ Suppose that we can indeed set up such a probability space $(\Omega,\mathcal F,P)$. Observing that outcomes must be equally likely due to the uniformly chosen hypothesis, we may assume that there exists some $0<\varepsilon<1$ such that for all $n\in\Omega$ we have that $P(\{n\})=\varepsilon$. Then since $\bigcup_{n\in\mathbb N}\{n\}=\Omega$ is a union of pairwise disjoint sets, countable additivity implies that $$P(\Omega)=P\left(\bigcup_{n\in\mathbb N}\{n\}\right)=\sum_{n=1}^\infty\varepsilon=\infty>1,$$ and we have a contradiction. Therefore, no such probability space can be set up.
Could anyone please provide some feedback on the proof above?
Any comments are much appreciated, and thank you very much for your time.
Answering for the sake of being able to resolve the question, but making this a community wiki since I'm not contributing anything of substance:
You've got it!