 But some, starting with de Finetti (1937, 1972, 1974) have argued on other grounds that we should give up even countable additivity and only accept finite additivity, with a correspondingly broader class of probability distributions. One of de Finetti’s chief arguments involves an infinite lottery with each natural number appearing on exactly one ticket. We would like to assign each ticket the same probability of being drawn. Under countable additivity, this is not possible. For if we assign probability 0 to each number’s being picked, then the sum of all these probabilities is again 0; yet the union of all of these events has probability 1 (since it is guaranteed that some number will be picked), and \(1 \ne 0\). On the other hand, if we assign some (real-valued) probability \(\varepsilon \gt 0\) to each number being picked, then the sum of these probabilities diverges to \(\infty\), and \(1 \ne \infty\). If we drop countable additivity, however, then we may assign 0 to each event and 1 to their union without contradiction. In the