Short answer: The Kolmogorov axioms are just mathematical. They have nothing inherently to do with the real world. P(A|B)=P(A∩B)/P(B) is the definition of P(A|B). There is a compelling argument that the beliefs of a rational agent should obey the Kolmogorov axioms, with P(A|B) corresponding to the degree of belief in A when B is known.
Long answer: I have a sequence of posts coming up.
There is a compelling argument that the beliefs of a rational agent should obey the Kolmogorov axioms, with P(A|B) corresponding to the degree of belief in A when B is known.
Are you thinking of this one, or something else?
From what I understand, the Kolmogorov axioms make no mention of conditional probability. That is simply defined. If I really want to show how probability actually works, I'm not going to argue "by definition". Does anyone know a modified form that uses simpler axioms than P(A|B) = P(A∩B)/P(B)?