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- W2951554701 abstract "The modal logic of forcing arises when one considers a model of set theory the context of all its forcing extensions, interpreting necessity as in all forcing and possibility as in some forcing extension. In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC (Stavi-Vaananen, Hamkins). Every definable forcing class similarly gives rise to the corresponding forcing modalities, for which one considers extensions only by forcing notions that class. In previous work, we proved that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2. In this article, we prove that the provably valid principles of collapse forcing, Cohen forcing and other classes are each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CH-preserving forcing and others are each exactly S4.2; and the provably valid principles of $omega_1$-preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise, including the connection between various control statements, such as buttons, switches and ratchets, and their corresponding forcing validities. These structural connections therefore support a forcing-only analysis of other diverse forcing classes." @default.
- W2951554701 created "2019-06-27" @default.
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- W2951554701 date "2012-07-24" @default.
- W2951554701 modified "2023-09-26" @default.
- W2951554701 title "Structural connections between a forcing class and its modal logic" @default.
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