28 research outputs found
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
Indestructibility of Vopenka's Principle
We show that Vopenka's Principle and Vopenka cardinals are indestructible
under reverse Easton forcing iterations of increasingly directed-closed partial
orders, without the need for any preparatory forcing. As a consequence, we are
able to prove the relative consistency of these large cardinal axioms with a
variety of statements known to be independent of ZFC, such as the generalised
continuum hypothesis, the existence of a definable well-order of the universe,
and the existence of morasses at many cardinals.Comment: 15 pages, submitted to Israel Journal of Mathematic