Some technical remarks on topological aspects of the paper:

The notion of a "coherent probability distribution" and the use of the product topology on the set of all such distributions are a bit ideosyncratic. Here's a way to relate them to more standard notions: Consider a countable first-order language L and the Stone space of complete theories over it. Equip this with the Borel-σ-algebra. (1.) A coherent probability distribution is (equivalent to) a probability distribution on the resulting measure space. (2.) The (AFAICT) most commonly used topology on the space of all such distributions is the one of weak convergence of probability measures. It turns out that this is equivalent to the product topology on the coherent probability distributions.

(1.) The main unusual feature of coherent probability distributions is that they're only defined on the sentences of L, which don't form a σ-algebra. Their equivalence classes under logical equivalence form a Boolean algebra, though, which is in particular a ring of sets (in the measure-theoretic sense) of complete theories (we identify each equivalence class with the set of complete theories containing the sentences in the equivalence class). Furthermore, this ring generates our σ-algebra: by the definition of "Stone space", the sets in the ring form a base of the Stone space, and since this base is countable, every open set is a countable union of base sets (meaning that the smallest σ-algebra containing the open sets is also the smallest σ-algebra containing the base).

A coherent probability distribution, by the alternative characterization from the paper, is a finitely additive probability measure. But a finitely additive measure on a ring is already a premeasure (i.e., σ-additive on the ring) if for every descending sequence of elements of the ring, implies , and a premeasure on a ring extends uniquely to a measure on the generated σ-algebra. Now, by the assumption, we have and therefore for all ; since , this means that the family has the finite intersection property, and so since Stone spaces are compact and each is clopen, the intersection of all is non-empty as desired.

(2.) The Stone space of a Boolean algebra is metrizable if and only if the Boolean algebra is countable, so since we're interested in countable languages, the notion of weak convergence of probability measures on our space is well-defined. One of the equivalent definitions is that if for all open sets . We want to show that this is equivalent to convergence in the product topology on coherent probability distributions, which amounts to pointwise convergence of to for all in the base (i.e., for for some sentence ).

Suppose first that for all base sets and let be an arbitrary open set. can be written as a countable union of base sets; since the base is closed under Boolean operations, it follows that it can be written as a countable disjoint union (let ). For any , there is an such that . By pointwise convergence, for sufficiently large we have for all . Therefore,

Since this holds for all , the desired inequality follows.

Suppose now that for all open sets . We must show that for all base sets , . But base sets are clopen, so we have both and

implying .