This will be a talk for the CUNY Logic Workshop on September 7, 2012.
Abstract. The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, with “true in all forcing extensions” and“true in some forcing extension” as the accompanying modal operators. In this modal language one may easily express sweeping general forcing principles, such asthe assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, orthe assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC. Similarly, the dual modal logic of grounds concerns the modalities “true in all ground models” and “true in some ground model”. In this talk, I shall survey the recent progress on the modal logic of forcing and the modal logic of grounds. This is joint work with Benedikt Loewe and George Leibman.
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Can one use modal logic to reinterpret the universal quantifier ‘all’ in the power set axiom for ZFC as ‘all possible’ subsets of a set S where possible means ‘true in some forcing extension’ or would it make more sense to use this type of interpretation over a class of models consisting of a ground model and its forcing extensions iff the ground model and its forcing extensions share the same ordinals? Perhaps more conditions would be needed (eg no collapsing cardinals) for such an interpretation to be viable.