What does it really mean when we say that the probability of something is zero? Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible.
But when we look at the probability from a mathematics perspective, probability is defined as the frequency of occurrence over the how many times the experiment is performed, limit as the number of trials goes to infinity. 
Doesn't this mean a probability of zero is an occurrence that is arbitrarily small but possible? What are some of the ways to make this line of argument more rigorous?

I have an interesting analogy in mind. I think it might have some value for further understanding. I think about the the problem of reaching from point $A$ to point $B$. The strategy is to take the half of the way at each step between the current position and the target position $B$. If we continue taking the half of the way at each step, we we will have the half way left all the time. But when we take the limit of this strategy the distance between the target $B$ and us will go to zero, meaning that we will eventually reach $B$ but in order to reach we need to take infinitely many number of steps.
Regarding the comment by @Cameron Williams we can hit the target on a say line, not the disk, for some subset (say a closed interval) of this line with some probability $p$. Then an equivalent way regarding the example above suggests that to talk about such a probability for a single point on the line we must try infinitely many number of times, while this probability is defined for any finite number of trial for the subset line (any measuralbe subset of the sample space).
