Talk:General frame

Sigma algebra? Cardinality?
In the definition (quoting the article), I see this:
 * A modal general frame is a triple $$\mathbf F=\langle F,R,V\rangle$$, where $$\langle F,R\rangle$$ is a Kripke frame (i.e., $$R$$ is a binary relation on the set $$F$$), and $$V$$ is a set of subsets of $$F$$ that is closed under the following:
 * the Boolean operations of (binary) intersection, union, and complement,

Based on my reading, this means that $$V$$ is a sigma algebra (the elements of $$V$$ are Borel sets). Is there some reason technical reason not to state this? OK, well, I see one: sigma algebras are closed under countable intersections and unions, whereas this article makes no statements about cardinality, one way or the other.

Am I supposed to assume that the statements in this article are valid for sets of arbitrary cardinality? e.g. for $$\aleph_\alpha$$? Or is this intended to work for only $$\aleph_0$$ or $$\aleph_1$$? Defining the set $$V$$ correctly seems to require a walk up the Borel hierarchy and you'll immediately bump into analytic sets.

The reason I ask is because in Bayesian inference, each Bayesian "prior" is an element of $$F$$, each possible inference is in $$R$$, and then $$V$$ is just the normal probability space. So its all hunky-dorey for small-enough sets. Whether any of this works out for higher order logic is not clear. 67.198.37.16 (talk) 21:31, 31 May 2024 (UTC)