What's objectionable about the Axiom of Countable Additivity?

Casella & Berger (2001) write:

the Axiom of Countable Additivity, is not universally accepted among statisticians. ... [It] is rejected by a school of statisticians led by deFinetti (1972), who chooses to replace this axiom with the Axiom of Finite Additivity.

What might possibly be wrong or objectionable about the Axiom of Countable Additivity?

(Simple examples would be helpful!)

The Axiom of Countable Additivity is the 3rd condition below:

Solution 1:

De Finetti was a subjectivist, which means that he interpreted probabilities as judgments of a certain kind. On his view, countable additivity is too strong to be a reasonable constraint on judgment. Basically, de Finetti thought that countable additivity amounts to a prohibition on judgments that ought to be allowed.

The classic example is the following. A natural number is going to be chosen according to some distribution, you know not which. In your state of ignorance, surely it is permissible for you to judge that the selection will be fair. That is, it doesn't seem like you're making an error in judgment by taking the selection to be fair (though, of course, you're not required to think it's fair; this is just one permissible judgment among perhaps many others). In order for the selection to be fair, each natural number must have the same probability of being chosen. But that means the distribution of the selection is not countably additive. To see this, consider two cases: (1) if each number has a positive probability of being chosen, the sum of the probabilities diverges; (2) if each number has probability zero of being chosen, the sum of the probabilities is likewise zero. In both cases, countable additivity is violated.

On the other hand, there are finitely, but not countably, additive distributions that assign every natural number probability zero while still assigning the set of all natural numbers probability 1. De Finetti objected to requiring countable additivity in this case because it precludes the seemingly permissible judgment that the selection is fair. This judgment is consistent, however, with a probability theory that requires finite but not countable additivity, and it is precisely this kind of probability theory that de Finetti endorsed.