Skolem’s Paradox
Skolem's Paradox involves a seeming conflict between two theorems from classical logic. The Löwenheim-Skolem theorem says that if a first-order theory has infinite models, then it has models whose domains are only countable. Cantor's theorem says that some sets are uncountable. Skolem's Paradox arises when we notice that the basic principles of Cantorian set theory—i.e., the very principles used to prove Cantor's theorem on the existence of uncountable sets—can themselves be formulated as a collection of first-order sentences. How can the very principles which prove the existence of uncountable sets be satisfied by a model which is itself only countable? How can a countable model satisfy the first-order sentence which says that there are uncountably many mathematical objects—e.g., uncountably many real numbers?
Philosophical discussion of this paradox has tended to focus on three main questions. First, there's a purely mathematical question: why doesn't Skolem's Paradox introduce an outright contradiction into set theory? Second, there's a historical question. Skolem himself gave a pretty good explanation as to why Skolem's Paradox doesn't constitute a straightforward mathematical contradiction; why, then, did Skolem and his contemporaries continue to find the paradox so philosophically troubling? Finally there's a purely philosophical question: what, if anything, does Skolem's Paradox tell us about our understanding of set theory and/or about the semantics of set-theoretic language?
