Can there be a lottery of the natural numbers, so that every natural number is chosen equally likely?
The standard answer would be "No" because: If we define a measure $\mathbf{P}$ on $\mathbb{N}$ so that $\mathbf{P}(n) = r \in (0,1] \; \forall \, \mathbb{N}$, then $\mathbf{P}(\mathbb{N}) = \infty$. If we define a measure so that $\mathbf{P}(n) = 0$, then $\mathbf{P}(\mathbb{N}) = 0$.
But why can we conclude from that, that a lottery of the natural numbers (with every natural number equally likely) is impossible?
Note: the question is not if there can be a uniform probability distribution (satisfying all axioms of probability, including countable additivity) over the natural numbers but if there can be a lottery of the natural numbers so that every number is chosen equally likely!
The probability that some number is drawn cannot be positive because the sum of the probabilities would be infinite.
Probability $0$ for an event not impossible is possible, if the number of events is uncountable. But for countably many events, $P(X)=0$ is equivalent to $X$ is the impossible event.