We say that P is coherent if there is a probability measure μ over models of L such that
Annoying pedantic foundational issues: The collection of models is a proper class, but we can instead consider probability measures over the set of consistent valuations L→T,F, which is a set. And it suffices to consider the product σ-algebra.
I assumed that this meant “there is some probability space whose set of outcomes is some set of models”, rather than “there is a measure over ‘the canonical’ probability space of all models”, whatever that could mean.
And it suffices to consider the product σ-algebra.
This is the smallest σ-algebra such that for every sentence phi of L, {A : phi in A} is measurable, right? If so, it is pleasingly also the Borel-σ-algebra on the Stone space of consistent valuations (aka complete theories) of L. (Assuming L is countable, but I believe the paper does that anyway.)
I assumed that this meant “there is some probability space whose set of outcomes is some set of models”, rather than “there is a measure over ‘the canonical’ probability space of all models”, whatever that could mean.
Yes, I asked Paul about this and this is the interpretation he gave.
Annoying pedantic foundational issues: The collection of models is a proper class, but we can instead consider probability measures over the set of consistent valuations L→T,F, which is a set. And it suffices to consider the product σ-algebra.
I assumed that this meant “there is some probability space whose set of outcomes is some set of models”, rather than “there is a measure over ‘the canonical’ probability space of all models”, whatever that could mean.
This is the smallest σ-algebra such that for every sentence phi of L, {A : phi in A} is measurable, right? If so, it is pleasingly also the Borel-σ-algebra on the Stone space of consistent valuations (aka complete theories) of L. (Assuming L is countable, but I believe the paper does that anyway.)
Yes, I asked Paul about this and this is the interpretation he gave.