37 research outputs found
The modal logic of forcing
What are the most general principles in set theory relating forceability and
truth? As with Solovay's celebrated analysis of provability, both this question
and its answer are naturally formulated with modal logic. We aim to do for
forceability what Solovay did for provability. A set theoretical assertion psi
is forceable or possible, if psi holds in some forcing extension, and
necessary, if psi holds in all forcing extensions. In this forcing
interpretation of modal logic, we establish that if ZFC is consistent, then the
ZFC-provable principles of forcing are exactly those in the modal theory known
as S4.2.Comment: 31 page
On Subcomplete Forcing
I survey an array of topics in set theory in the context of a novel class of
forcing notions: subcomplete forcing. Subcompleteness was originally defined by
Ronald Jensen. I have attempted to make the subject somewhat more approachable
to set theorists, while showing various properties of subcomplete forcing which
one might desire of a forcing class, drawing comparisons between subcomplete
forcing and countably closed forcing. In particular, I look at the interaction
between subcomplete forcing and -trees, preservation properties of
subcomplete forcing, the subcomplete maximality principle, the subcomplete
resurrection axiom, and show that generalized diagonal Prikry forcing is
subcomplete.Comment: This is my PhD dissertatio
Moving up and down in the generic multiverse
We give a brief account of the modal logic of the generic multiverse, which
is a bimodal logic with operators corresponding to the relations "is a forcing
extension of" and "is a ground model of". The fragment of the first relation is
called the modal logic of forcing and was studied by us in earlier work. The
fragment of the second relation is called the modal logic of grounds and will
be studied here for the first time. In addition, we discuss which combinations
of modal logics are possible for the two fragments.Comment: 10 pages. Extended abstract. Questions and commentary concerning this
article can be made at
http://jdh.hamkins.org/up-and-down-in-the-generic-multiverse