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)?
Your question doesn't make any sense to me. I don't know what it means to "prove" a definition. Did you mean to ask for an (informal) argument that the concept is useful?
My confusion is compounded by the fact that I find P(A|B) = P(A∩B)/P(B) pretty self-explanatory. What seems to be the difficulty?
I find this set of answers being top rated quite disturbing to be honest.
There are several people in the same main thread pointing out that
a) There ways to define it that would make it obviously violating basic intuition and hence a disconnection of it from intuition does have limits
b) There are intuitive solutions to the problem that may reach a proof for it and hence the whole argument to be unfounded.