# The Minimality Principle for the Modal Logic of Forcing

**Date**: Monday 19 November 2018, 14:00 – 15:00**Location**: Roger Stevens LT 01 (7.01)**Type**: Postgraduate Logic**Cost**: 0.00

#### Francesco Gallinaro, University of Leeds The Minimality Principle for the Modal Logic of Forcing Part of the Postgraduate Logic seminar series

The fruitful investigation of the modal logic of the generic multiverse carried out over the last two decades by several authors (primarily Hamkins and Löwe) was kicked off by a paper by Hamkins, published in 2003 and called "A Simple Maximality Principle". Here Hamkins discussed the consistency with ZFC of a forcing principle which informally stated "any sentence that is true in some forcing

extension and all its forcing extensions is true". If we try to dualize this, by looking at submodels rather than extensions, we can then state two minimality principles: a strong one, informally stating "any sentence that is false in some inner model and all its inner models is false", and a weak one, informally stating "any sentence that is false in some ground model

and all its ground models is false". After introducing modal logic and briefly sketching the main result about its application to forcing, we will see how these versions of the minimality principle can be interpreted in formal terms, and study their relevance in the multiverse: we'll see that the strong version is actually equivalent to the axiom of constructibility, while the weak version has deep connections to set-theoretic geology.