THE REVIEW OF SYMBOLIC LOGIC Volume 5, Number 3, September 2012
THE SET-THEORETIC MULTIVERSE JOEL DAVID HAMKINS Department of Philosophy, New York University, Mathematics Program, The Graduate Center of The City University of New York, and Department of Mathematics, The College of Staten Island of CUNY
Abstract. The multiverse view in set theory, introduced and argued for in this article, 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 range of set- theoretic possibilities, a phenomenon that challenges the universe view. In particular, I 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.
§1. Introduction. Set theorists commonly take their subject as constituting an ontological foundation for the rest of mathematics, in the sense that abstract mathematical objects can be construed fundamentally as sets, and because of this, they regard set theory as the domain of all mathematics. Mathematical objects are sets for them, and being precise in mathematics amounts to specifying an object in set theory. These sets accumulate transfinitely to form the cumulative universe of all sets, and the task of set theory is to discover its fundamental truths. The universe view is the commonly held philosophical position that there is a unique absolute background concept of set, instantiated in the corresponding absolute set-theoretic universe, the cumulative universe of all sets, in which every set-theoretic assertion has a definite truth-value. On this view, interesting set-theoretic questions, such as the continuum hypothesis and others, have definitive final answers. Adherents of the universe view often point to the increasingly stable consequences of the large cardinal hierarchy, particularly in the realm of projective sets of reals with its attractive determinacy and regularity features, as well as the forcing absoluteness properties for L(R), as evidence that we are on the right track towards the final answers to these set theoretical questions. Adherents of the universe view therefore also commonly affirm these large cardinal and regularity features in the absolute set-theoretic universe. The pervasive independence phenomenon in set theory is described on this view as a distraction, a side discussion about provability rather than truth—about the weakness of our theories in finding the truth, rather than about the truth itself—for the independence of a set-theoretic assertion from ZFC tells us little about whether it holds or not in the universe. In this article, I shall argue for a contrary position, the multiverse view, which holds that there are diverse distinct concepts of set, each instantiated in a corresponding set- theoretic universe, which exhibit diverse set-theoretic truths. Each such universe exists
Received: September 17, 2009.