Assume a unit square to be sample space (infinite points inside it being its elements). Let the points are $\{p_1, p_2, ...\}$ then, by probability axioms, $$1 = Pr(p_1 \cup p_2 \cup \cdots ) = Pr(\{p_1\}) + Pr(\{p_2\}) + \cdots + Pr(\{p_n\}) = \\ = Pr(p_1) + Pr(p_2) + \cdots + Pr(p_n) = 0 + 0 + \cdots $$ (as Pr of individual point in space is zero) $= 0$
Where do I lack in understanding the logic of axioms?
There is no sequence $(p_k)_{k\in\Bbb N}$ of points of the unit square $S$ such that $S=\{p_1,p_2,p_3,\ldots\}$. In other words, $S$ is not countable.