In case I understand your concern, here's how I address it: We like something like probability theory because in practice it works great for saying that some r

In case I understand your concern, here's how I address it: We like something like probability theory because in practice it works great for saying that some roulette wheel is crooked, for confidence intervals on measurements, ..., for various stochastic processes and their power spectra, e.g., for the 3 degree K background radiation. Roughly we know quite well what we want in our probability theory. From what we know we want in our probability theory, we're essentially pushed into the axioms whether we like it or not. Or, the axioms are basically what the heck we need in order to talk about probability. E.g., we want to talk about events, say, event A, so that we can consider the probability of events, say, P(A), the probability of A. And given event B, we want to be able to consider the new event A or B. So, if in our foundations we have that events A and B are subsets of the set of trials Omega, then the new event A or B is just the set union of sets A and B. So, we want to be able to consider the new event A or B so go ahead and accept the set theory because it lets us get what we want. So, basically we got pushed into the set theory. For more, if event A is 'the next flip of our fair coin comes up heads' and event B there is 'a magnitude 1 quake in SF 10 days from now', then we jump at saying that events A and B have nothing to do with each other and are independent so that we know that P(A and B) = P(A)P(B). So, in practice, how do we justify independence? Sadly, in my view, mostly just intuitively as in the little example of coins and quakes I gave. The axioms ask for a little more than we might, first cut, have believed we need to specify. E.g., so that we will have plenty of ability to take old events and create new ones, we want the events actually to form a sigma algebra. Next, so that we can talk about the event 'the coin never comes up heads', again we want a sigma algebra, and we want our probability P to be countably additive. What about uncountably additive? Quickly we see that that causes us serious problems so don't ask for it. Net, we really do like probability and its applications, say, to statistics, and the axioms I gave (thank you A. Kolmogorov and H. Lebesgue) are basically about the minimum we want. Really, we just get pushed into those axioms; if we like probability, and we do, then we don't have much choice but to accept those axioms. Curious universe we live in.