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