Suppose F is a finite set of propositions such that, for every proposition A in F and every proposition B in F such that A is distinct from B, P(A) = P(B) and A is inconsistent with B.
By using the axiom of unit measure and the axiom of countable additivity, one can show that, for every proposition A in F, P(A) <= 1/n where n is the cardinality of F. (Please tell me if you don't see why or if you think that's false.)
So, as the cardinality of F approaches infinity, P(A) tends toward 0 for every A in F.
Now, doesn't this suggest that, in the limit where F is infinite, P(A) = 0 for every A in F? (I don't think one can prove this, because it would require a division by infinity and that's strictly speaking undefined, but doesn't it make sense to say that?)
And, if that's right, assuming that P(Z), where Z is the union of the elements of F, is greater than 0, we are left with the conclusion that P(Z) > 0 but P(A) = 0 for every A in F, even though Z = the union of the elements of F. This seems really weird.
Can someone explain to me what's going on here? I'm not a mathematician, so I'm just puzzled by what seems to be a consequence of the axioms of probability theory.