The atomless property and only contradictions taking a 0 value could both be consequences of the axioms in question. The Kolmogorov paper (translated from French by Jeffrey) has the details, but from skimming it I don’t immediately understand how it works.
Wait, how does the atomless property ensure that if the probability of an event is 0, then the event can never happen at all, as a matter of logic?
The atomless property and only contradictions taking a 0 value could both be consequences of the axioms in question. The Kolmogorov paper (translated from French by Jeffrey) has the details, but from skimming it I don’t immediately understand how it works.