Talk:General frame

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In the definition (quoting the article), I see this:

A modal general frame is a triple ${\displaystyle \mathbf {F} =\langle F,R,V\rangle }$, where ${\displaystyle \langle F,R\rangle }$ is a Kripke frame (i.e., ${\displaystyle R}$ is a binary relation on the set ${\displaystyle F}$), and ${\displaystyle V}$ is a set of subsets of ${\displaystyle F}$ that is closed under the following:

• the Boolean operations of (binary) intersection, union, and complement,

Based on my reading, this means that ${\displaystyle V}$ is a sigma algebra (the elements of ${\displaystyle V}$ 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 ${\displaystyle \aleph _{\alpha }}$? Or is this intended to work for only ${\displaystyle \aleph _{0}}$ or ${\displaystyle \aleph _{1}}$? Defining the set ${\displaystyle V}$ 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 ${\displaystyle F}$, each possible inference is in ${\displaystyle R}$, and then ${\displaystyle V}$ 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)Reply