The Mathematics of Skolem’s Paradox

In 1922, Thoralf Skolem published a paper entitled “Some Remarks on Axiomatized Set Theory.” The paper presents a new proof of a model-theoretic result originally due to Leopold Löwenheim and then discusses some philosophical implications of this result. In the course of this latter discussion, the paper introduces a model-theoretic puzzle that has come to be known as “Skolem’s Paradox.” Over the years, Skolem’s Paradox has generated a fairly steady stream of philosophical discussion; nonetheless, the overwhelming consensus among philosophers and logicians is that the paradox doesn’t constitute a mathematical problem (i.e., it doesn’t constitute a real contradiction). Further, there’s general agreement as to why the paradox doesn’t constitute a mathematical problem. By looking at the way firstorder structures interpret quantifiers—and, in particular, by looking at how this interpretation changes as we move from structure to structure—we can give a technically adequate “solution” to Skolem’s Paradox. So, whatever the philosophical upshot of Skolem’s Paradox may be, the mathematical side of Skolem’s Paradox seems to be relatively straightforward. In this paper, I challenge this common wisdom concerning Skolem’s Paradox. While I don’t argue that Skolem’s Paradox constitutes a genuine mathematical problem (it doesn’t), I do argue that standard “solutions” to the paradox are technically inadequate. Even on the mathematical side, Skolem’s Paradox is more complicated—and quite a bit more interesting—than it’s usually taken to be. Further, because philosophical discussions of Skolem’s Paradox typically start with an analysis of the paradox’s mathematics—and only then examine how the interpretation of this mathematics reveals the paradox’s philosophical significance—it is important to get the mathematics itself right before we start in on our philosophy. From a structural standpoint, this paper breaks into six sections. In section 1, I formulate a simple version of Skolem’s Paradox and try to disentangle the roles that set theory, model theory and philosophy play in making it look plausible. In section 2, I sketch a generic solution to Skolem’s Paradox—a solution which explains, in rough outline, why no version of the paradox generates a genuine contradiction. Sections 3–5 examine different ways of “filling out” this generic solution. Section 3 focuses on the role quantification sometimes plays in Skolem’s Paradox and includes a discussion of the so-called “transitive submodel” version of the paradox. Sections 4 and 5 look at some cases where quantification doesn’t help to explain Skolem’s Paradox. Finally, section 6 presents some concluding philosophical reflections.