ABSTRACT
Formally the Lowenheim-Skolem theorem is unproblematical: any firstorder theory which has an infinite model has a denumerable model. But recent controversy over the interpretation of this result indicates that its philosophical significance remains unclear.1 The Skolemite sees it as ontologically significant, as showing that there is no such thing as “absolute” uncountability, but only uncountability relative to a formal system, and hence that there are only finite or denumerable sets in the universe. For to claim uncountability for a set, the Skolemite argues, is just to assert that there is no enumerating function for the set, but since this non-existence claim is proved within a particular formalization, all it shows is that the formal theory itself is not powerful enough to generate the enumerating function. It does not exclude the possibility of such a function outside the formal theory, and that the required function does exist, according to the Skolemite, is shown by the Lowenheim-Skolem theorem, which provides a denumerable model for any first-order theory. For in the Skolem model, the term which designates the supposedly uncountable set is provided with a referent which is at most denumerable. Thus the set, though uncountable within the formal theory, is countable outside the theory, in the meta language, and hence no term of a formal theory can be taken as designating a set which is anything more than relatively uncountable. The Platonist, on the other hand, discounts the ontological significance
of the theorem, claiming that the Skolem models, though denumerable, do not provide either an enumeration of sets (but only map statements about sets onto statements about numbers)2 or an acceptable rendering of the concept of membership (but only a particular unintuitive fragment of number theory), and that furthermore, the Skolemite has overlooked the other side of the coin — the “upward” Lowenheim-Skolem theorem, which tells us that any theory with an infinite model has models of any infinite cardinality, including vastly nondenumerable models. And there is absol utely no reason, according to the Platonist, to allow the countable models greater significance than their uncountable cousins, except for a stubborn
preference for countability which has nothing to do with the proof itself. Thus the theorem rules out nothing, and ontological questions must be decided on other grounds.3 Much of the controversy centers around the question of what constitutes
an intuitively acceptable model for set theory. The Platonist rejects the Skolem model as the intended interpretation of set theory on the grounds that it provides only an unintuitive number-theoretic relation which cannot be taken as a true representation of the concept of membership. The Skolemite replies (not entirely consistently, perhaps) that it is not surprising that the models are unintuitive, since the whole notion of an intended model for set theory is unclear,4 and that anyway, it is possible to obtain intuit ively acceptable denumerable models by a proof of the Skolem theorem which generates an elementary submodel of whatever model the Platonist finds acceptable to begin with.5 I shall argue in the following pages that neither the Platonist nor the
Skolemite can make a convincing case: that the Skolemite cannot really establish that there are only denumerable sets, but that on the other hand, neither can the Platonist establish that there are nondenumerable sets. My own non-conclusion is that the question is undecided, and perhaps undecidable except by fiat, but this does not mean that the discussion is philo sophically unrewarding. There may be significance in the lack of ontological significance of the Lowenheim-Skolem theorem: evidence for some sort of Formalist view of mathematics.6
