Tag Archives: modal logic

The modal logic of forcing

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[bibtex key=HamkinsLoewe2008:TheModalLogicOfForcing]

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.

Follow-up article:  Structural connections between a forcing class and its modal logic

A simple maximality principle

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[bibtex key=Hamkins2003:MaximalityPrinciple]

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence$\varphi$ holding in some forcing extension $V^{\mathbb{P}}$ and all subsequent extensions $V^{\mathbb{P}*\mathbb{Q}}$ holds already in $V$. It follows, in fact, that such sentences must also hold in all forcing extensions of $V$. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\Diamond\Box\varphi)\to\Box\varphi$, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in $\varphi$, is equiconsistent with the scheme asserting that $V_\delta$ is an elementary substructure of $V$ for an inaccessible cardinal $\delta$, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is the Necessary Maximality Principle, which asserts that the boldface MP holds in V and all forcing extensions. From this, it follows that $0^\sharp$ exists, that $x^\sharp$ exists for every set $x$, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.

The set-theoretic multiverse: a model-theoretic philosophy of set theory, Paris, 2010

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A talk at the Philosophy and Model Theory conference held June 2-5, 2010 at the Université Paris Ouest Nanterre.

Set theorists commonly regard set theory as an ontological foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets, enjoying a real mathematical existence as sets accumulate to form the universe of all sets. The Universe view—perhaps it is the orthodox view among set theorists—takes this universe of sets to be unique, and holds that a principal task of set theory is to discover its fundamental truths. For example, on this view, interesting set-theoretical questions, such as the Continuum Hypothesis, will have definitive final answers in this universe. Proponents of this view point to the increasingly stable body of regularity features flowing from the large cardinal hierarchy as indicating in broad strokes that we are on the right track towards these final answers.

A paradox for the orthodox view, however, is the fact that the most powerful tools in set theory are most naturally understood as methods for constructing alternative set-theoretic universes. With forcing and other methods, we seem to glimpse into alternative mathematical worlds, and are led to consider a model-theoretic, multiverse philosophical position. In this talk, I shall describe and defend the Multiverse view, which takes these other worlds at face value, holding that there are many set-theoretical universes. This is a realist position, granting these universes a full mathematical existence and exploring their interactions. The multiverse view remains Platonist, but it is second-order Platonism, that is, Platonism about universes. I shall argue that set theory is now mature enough to fruitfully adopt and analyze this view. I shall propose a number of multiverse axioms, provide a multiverse consistency proof, and describe some recent results in set theory that illustrate the multiverse perspective, while engaging pleasantly with various philosophical views on the nature of mathematical existence.

Slides  | Article | see related Singapore talk