PP-2012-19: Structural connections between a forcing class and its modal logic

PP-2012-19: Hamkins, Joel David and Leibman, George and Löwe, Benedikt (2012) Structural connections between a forcing class and its modal logic. [Report]

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Abstract

Every denable forcing class Gamma gives rise to a corresponding forcing modality, for which Box_Gamma phi means that phi is true in all Gamma extensions, and the valid principles of Gamma forcing are the modal assertions that are valid for this forcing interpretation. For example, [9] shows 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 similarly that the provably valid principles of collapse forcing, Cohen forcing and other classes are in 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 identied between a forcing class and the modal logic of forcing to which it gives rise.

Item Type: Report
Report Nr: PP-2012-19
Series Name: Prepublication (PP) Series
Year: 2012
Uncontrolled Keywords: forcing; modal logic; forcing class
Subjects: Logic
Depositing User: Benedikt
Date Deposited: 12 Oct 2016 14:37
Last Modified: 12 Oct 2016 14:37
URI: https://eprints.illc.uva.nl/id/eprint/461

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