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

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by $Loc({\rm ZFC})$, says that every set belongs to a transitive model of ZFC. LZFC consists of $Loc({\rm ZFC})$ plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All $\Pi_2$ consequences of ZFC are provable in ${\rm LZFC}$. LZFC strongly extends Kripke-Platek (KP) set theory minus $\Delta_0$-Collection and minus $\in$-induction scheme. ZFC+``there is an inaccessible cardinal'' proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define $\alpha$-Mahlo models and $\Pi_1^1$-indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form $Loc({\rm ZFC}+\phi)$ are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.Comment: 38 page

The modal logic of set-theoretic potentialism and the potentialist maximality principles

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and L\"owe, including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_\beta$); Grothendieck-Zermelo potentialism (true in all larger $V_\kappa$ for inaccessible cardinals $\kappa$); transitive-set potentialism (true in all larger transitive sets); forcing potentialism (true in all forcing extensions); countable-transitive-model potentialism (true in all larger countable transitive models of ZFC); countable-model potentialism (true in all larger countable models of ZFC); and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.Comment: 36 pages. Commentary can be made about this article at http://jdh.hamkins.org/set-theoretic-potentialism. Minor revisions in v2; further minor revisions in v

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

First-order modal logic in the necessary framework of objects

I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument

De Jongh's Theorem for Intuitionistic Zermelo-Fraenkel Set Theory

We prove that the propositional logic of intuitionistic set theory IZF is intuitionistic propositional logic IPC. More generally, we show that IZF has the de Jongh property with respect to every intermediate logic that is complete with respect to a class of finite trees. The same results follow for CZF.Comment: 12 page

Fragments of Frege's Grundgesetze and G\"odel's Constructible Universe

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting.Comment: Forthcoming in The Journal of Symbolic Logi

The modal logic of arithmetic potentialism and the universal algorithm

I investigate the modal commitments of various conceptions of the philosophy of arithmetic potentialism. Specifically, I consider the natural potentialist systems arising from the models of arithmetic under their natural extension concepts, such as end-extensions, arbitrary extensions, conservative extensions and more. In these potentialist systems, I show, the propositional modal assertions that are valid with respect to all arithmetic assertions with parameters are exactly the assertions of S4. With respect to sentences, however, the validities of a model lie between S4 and S5, and these bounds are sharp in that there are models realizing both endpoints. For a model of arithmetic to validate S5 is precisely to fulfill the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently characterized as those satisfying a maximal $\Sigma_1$ theory. The main S4 analysis makes fundamental use of the universal algorithm, of which this article provides a simplified, self-contained account. The paper concludes with a discussion of how the philosophical differences of several fundamentally different potentialist attitudes---linear inevitability, convergent potentialism and radical branching possibility---are expressed by their corresponding potentialist modal validities.Comment: 38 pages. Inquiries and commentary can be made at http://jdh.hamkins.org/arithmetic-potentialism-and-the-universal-algorithm. Version v3 has further minor revisions, including additional reference