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Sigma algebra? Cardinality?

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

A modal general frame is a triple , where is a Kripke frame (i.e., is a binary relation on the set ), and is a set of subsets of that is closed under the following:

Based on my reading, this means that is a sigma algebra (the elements of 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 ? Or is this intended to work for only or ? Defining the set 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 , each possible inference is in , and then 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]