Reflections on Skolem’s Paradox

The Löwenheim-Skolem theorems say that if a first-order theory has infinite models, then it has models which are only countably infinite. Cantor's theorem says that some sets are uncountable. Together, these two theorems induce a puzzle known as Skolem's Paradox: the very axioms of (first-order) set theory which prove the existence of uncountable sets can themselves be satisfied by a merely countable model. Ever since this paradox was formulated in the early 1920's, philosophers have argued that issues of deep philosophical importance hang on the paradox's resolution. Skolem himself used the paradox to argue that set theory provides an inadequate foundation for mathematics. Later authors have used the paradox to argue that " every set is countable from some perspective " (Wang) or that Quine's theory of ontological reduction is hopelessly flawed (Grandy and Chihara). Most recently, Hilary Putnam has claimed that the paradox has, in his words, " profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language. " The present dissertation examines Skolem's Paradox from three perspectives. After a brief introduction, chapters two and three examine a number of different formulations of the paradox in order to disentangle the roles which set theory, model theory, and philosophy play in these formulations. In these two chapters, I accomplish three things. First, I clear up some of the mathematical ambiguities which have all too often infected discussions of Skolem's Paradox. Second, I isolate a key philosophical assumption upon which Skolem's Paradox rests, and I show why this assumption has to be false. Finally, I argue that there is no single explanation as to how a countable model can satisfy the axioms of set theory (in particular, no explanation in terms of quantifier-ranges can ever be fully adequate). In chapter four, I turn to a second puzzle. Why, even though philosophers have known since the 1920's that Skolem's Paradox has a relatively simple technical solution, have they continued to find this paradox so troubling? I argue that philosophers' attitudes towards Skolem's Paradox have been shaped by the acceptance of some fairly specific claims in the philosophy of language. I then tackle these philosophical claims head on. In some cases, I argue that the claims are ill-motivated, as they depend on an incoherent account of mathematical language. In other cases, I argue that the claims are so powerful that they render …