Traditional probability is based on Kolmogorov's three axioms of probability. Because they are axioms they don't require a proof and although they are intuitive I am wondering if there is a more rigorous way to be convinced of their validity other than intuition.
Think of the properties of relative frequency. The first two axioms then are trivial if we consider the probabilities to be the counterparts of relative frequencies.
The third axiom is also trivial in the mirror of relative frequencies if we talk only about a finite number of pairwise excluding events.
The theory requires the generalization for an infinite number of such events. This cannot be made more intuitive.