This talk, taking place October 19, 2011, is part of the year-long Exploring the Frontiers of Incompleteness (EFI) series at Harvard University, a workshop focused on the question of determinateness in set theory, a central question in the philosophy of set theory. Streaming video will be available on-line, and each talk will be associated with an on-line discussion forum, to which links will be made here later.
In this talk, I will discuss the multiverse perspective on determinateness in set theory. The multiverse view in set theory is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.
Workshop materials | Article | Slides | EFI discussion forum | Video Stream
A talk at the Asian Initive for Infinity: Workshop on Infinity and Truth, July 25-29, 2011, National University of Singapore.
I shall outline and defend the Multiverse view in set theory, the view that there are many set-theoretic universes, each instantiating its own concept of set, and contrast it with the Universe view, the view that there is an absolute background set-theoretic universe. In addition, I will discuss some recent set-theoretic developments that have been motivated and informed by a multiverse perspective, including the modal logic of forcing and the emergence of set-theoretic geology.
Slides | Article
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