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)