A question about modal logic

September 5, 2008

Say that a class of Kripke frames, \mathfrak{C}, is compact just in case, for every set of propositional modal formulae, \Gamma, \Gamma has a model in \mathfrak{C} (i.e. a model based on a frame in \mathfrak{C}) iff every finite subset of \Gamma has a model in \mathfrak{C}.

We know some sufficient conditions for \mathfrak{C} to be compact, for example, if \mathfrak{C} is closed under taking ultraproducts. Does anyone know any necessary conditions (or even necessary and sufficient conditions) for compactness in this sense?


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