The Mathematics of Skolem's Paradox

Abstract

This chapter presents a discussion on Skolem's paradox. In its simplest form, Skolem's Paradox involves a (seeming) conflict between two theorems of modern logic: Cantor's theorem from set theory and the Lowenheim–Skolem theorem from model theory. Cantor's theorem says that there are uncountable sets—sets that are too big to be put into one-to-one correspondence with the natural numbers. The Lowenheim–Skolem theorem says that if a countable collection of first-order sentences has a model, then it has a model whose domain is only countable. Skolem's Paradox arises when it is noted that the standard axioms of set theory are themselves a countable collection of first-order sentences. This chapter formulates a simple version of Skolem's paradox and attempts to disentangle the roles that set theory, model theory, and philosophy play in making it look plausible. It sketches a generic solution to Skolem's paradox—a solution that explains, in rough outline, why no version of the paradox generates a genuine contradiction. This chapter examines different ways of “filling out” this generic solution. This chapter 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. Some cases are discussed where quantification does not help to explain Skolem's Paradox.