More on Skolem's Paradox

Abstract

The Ldiwenheim-Skolem Theorem to effect that every consistent formal has a countable model is often interpreted as demonstrating that prima facie uncountable sets-such as set of all real numbers-are actually countable. In an earlier paper I questioned arguments for this Skolemite position. The gist of my earlier contention was this. One of Skolemite arguments is based upon an equivocation between terms the sets of system and what can be taken as sets of system. The more rigorous arguments skirt issue by dealing only with predicative subsets of sets whose countability is in question or else by simply postulating countability of latter. In any case, such arguments are faced with seemingly overwhelming difficulty of establishing that sets that have been shown to be countable are identical with ones that most mathematicians hold to be uncountable.' Arthur Fine's recent paper in which Skolemite position is defended anew has prompted me to reexamine my earlier arguments.2 I now see that some of them were predicated upon existence of intended (prima facie) uncountable models for