text: 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.
  @cite_4: This article surveys two recent developments in set theory sharing an essential second-order nature, namely, the modal logic of forcing, oriented upward from the universe of set theory to its forcing extensions; and set-theoretic geology, oriented downward from the universe to the inner models over which it arises by forcing. The research is a mixture of ideas from several parts of logic, including, of course, set theory and forcing, but also modal logic, finite combinatorics and the philosophy of mathematics, for it invites a mathematical engagement with various philosophical views on the nature of mathematical existence.
  @cite_7: The Necessary Maximality Principle for c.c.c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. The Necessary Maximality Principle for c.c.c. forcing, denoted 2mpccc(R), asserts that any statement about a real in a c.c.c. extension that could be- come true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal. The principle is one of a family of principles considered in (Ham03) (build- ing on ideas of (Cha00) and overlapping with independent work in (SV01)). MSC: 03E55, 03E40. Keywords: forcing axiom, ccc forcing, weakly compact cardinal. The first author is affiliated with the College of Staten Islandof CUNY and The CUNY Graduate Center, and his research has been supported by grants from the Research Foun- dation of CUNY and the National Science Foundation DMS-9970993. The research of the second author has been partially supported by National Science Foundation grant DMS-9970255.
  @cite_8: 
  @cite_9: The motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for κ -closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞ ), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.
  @cite_1: The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, interpreting necessity as "in all forcing extensions" and possibility as "in some forcing extension". In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC (Stavi-V "a "an "anen, Hamkins). Every definable forcing class similarly gives rise to the corresponding forcing modalities, for which one considers extensions only by forcing notions in that class. In previous work, we proved 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 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 @math -preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise, including the connection between various control statements, such as buttons, switches and ratchets, and their corresponding forcing validities. These structural connections therefore support a forcing-only analysis of other diverse forcing classes.
  @cite_0: 
  @cite_2: I investigate versions of the Maximality Principles for the classes of forcings which are < -closed, < -directed-closed, or of the form Col(;< ). These principles come in many variants, depending on the parameters which are allowed. I shall write MP (A) for the maximality principle for forcings in , with parameters from A. The main results of this paper are: The principles have many consequences, such as < -closed-generic 1 2(H ) absoluteness, and imply, e.g., that holds. I give an application to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing. The principles can be separated into a hierarchy which is strict, for many . Some of the principles can be combined, in the sense that they can hold at many dierent simultaneously. The possibilities of combining the principles are limited, though: While it is consistent that MP< closed(H +) holds at all regular below any xed , the " maximality principle, stating that MP< closed(H [ f g) holds at every regular , is inconsistent. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also
  @cite_15: 
  @cite_10: 
  @cite_12: Hamkins and Lowe proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.
  @cite_11: 
summary: In our previous paper @cite_0 , we introduced the modal logic of forcing and proved that the -provably valid principles of forcing were exactly those in the modal theory known as S4.2 . The modal logic of forcing corresponds to the monomodal fragment of the modal logic discussed in , that only uses the relation @math ; or to the part of the generic multiverse that is generated only by the operation of taking forcing extensions and not ground models. In @cite_0 , we not only consider the @math -provable modal logic of forcing, but also the modal logic of forcing of particular universes @math . We show that this modal logic always lies between @math and @math and that the two extreme values are realized (for more details, cf. ,). Various other aspects of the modal logic of forcing are considered in @cite_15 @cite_7 @cite_2 @cite_9 @cite_10 @cite_11 @cite_8 @cite_12 @cite_1 . The paper @cite_4 presented at ICLA 2009 gives an overview of the status of research and creates a connection between the modal logic of forcing and set-theoretic geology'', i.e., going down from a universe to its ground models. This connection is further developed in this paper.