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?

Many events which mathematically have probability zero are possible. The standard source of examples is random variables with continuous distribution. Here the probability of taking on any given real value is 0, but certainly the variable always takes on some real value! In these cases, with repeated sampling, you will, with probability 1, get a sequence of distinct numbers, so the frequency of the first value will look like $1,1/2,1/3,\dots$.
