 Countable additivity is not as innocent as it looks: it rules out the possibility that any agent is indifferent over a countably infinite set of mutully exclusive possibilities. De Finetti (1970, 1972) famously argued that we ought to reject countable additivity since it is conceivable that God could pick out a natural number “at random” and with equal (zero) probability. For another example, suppose you assign 50% credence to the proposition \(\neg B\) that not all ravens that will ever be observed are black. Let \(\neg B_i\) be the proposition that the \(i^{th}\) observed raven is the first non-black raven to appear. Then \(\neg B = \cup_{i=1}^\infty \neg B_i.\) Countable additivity entails that for all \(\epsilon&gt;0\) there is a finite \(n\) such that \(p(\cup_{i=1}^n \neg B_i) = 1/2 -\epsilon.\) So you must be nearly certain that if not all ravens are black, the first non-black raven will appear among the first \(n\) ravens. But is it really a requirement of rationality that you must be opinionated about when the first non-black raven must appear? The only way to assign equal probability to all \(\neg B_i\) is to violate countable additivity by setting \(p(\neg B_i)=0\) for all \(i\). This solution has its own drawbacks. On all standard models of Bayesian update it will be impossible to become convinced that the \(i^{th}\) raven is indeed non-black, even if you are looking at a white one. For more on countable additivity, see Chapter 13 in Kelly (1996).