The additivity axiom of probabiity One of the axioms of probability is:
If the sample space is finite and $A$ and $B$ are disjoint events then $Pr[A\cup B]=Pr[A]+Pr[B]$, and if the sample space is infinite, then for for any (possibly infinite number of) disjoint events, like $A_1,A_2,\ldots$, then $Pr[\cup_iA_i]=\sum_iPr[A_i]$
I cannot understand why the first one cannot imply the second one. I have seen several probability books and they have mentioned it and skipped its detail. Can anyone give an example that the union of just two disjoint sets does not imply the union of any (possibly infinite) number of disjoint events?

This is a good example to show that induction can prove things "for all positive integers $n$" but that does not necessarily imply the result "for $n=\infty$." 
There are still some things you can say with only the finitely additive axiom: Let $\{A_n\}_{n=1}^{\infty}$ be an infinite sequence of disjoint events.  Fix $m$ as a positive integer. Since $\cup_{n=1}^{\infty} A_n \supseteq \cup_{n=1}^m A_n$ we conclude: 
$$P[\cup_{n=1}^{\infty} A_n] \geq P[\cup_{n=1}^m A_n] = \sum_{n=1}^m P[A_n] $$
where the final equality holds by finite additivity.  Taking a limit as $m\rightarrow \infty$ gives: 
$$ P[\cup_{n=1}^{\infty} A_n]  \geq \sum_{n=1}^{\infty} P[A_n]$$
The reverse inequality requires the countable additivity axiom (as described in the aduh answer).  As a sanity check, notice that the above inequality indeed holds for the "fair integer lottery" example from aduh's answer, using $A_n = \{n\}$ for $n \in \{1, 2, 3, ...\}$ (and redefining that example only over positive integers, rather than all integers, for simplicity), since $1 \geq \sum_{n=1}^{\infty} 0 = 0$. 
