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According to standard Bayesianism, rational agents should have degrees of belief (i.e. credences) that satisfy the Kolmogorov axioms of probability. In particular, rational credences should be countably additive. One of the strongest arguments for countable additivity is the conglomerability argument. Roughly, the idea is that rational agents should have credences that are countably additive because otherwise they would be disposed to change their credences in a predictable way. One objection against the conglomerability argument is that it begs the question: given the coherent theory of conditional probability, countable additivity is sufficient for conglomerability in countable partitions, but does not guarantee conglomerability in partitions of higher cardinality. But why should rational credences be conglomerable only in countable partitions, and not in uncountable partitions?

This talk explores whether the conglomerability argument succeeds if one adopts Kolmogorov’s theory of conditional probability instead. I argue that the answer is not obvious. In particular, under plausible assumptions, a natural generalization of the conglomerability argument fails because it entails both the continuum hypothesis and its negation.