VIRGINIA KLENK

INTENDED MODELS AND THE

LOWENHEIM-SKOLEM THEOREM”

Formally the Lowenheim-Skolem theorem Is unproblematical: any first- order theory which has an inlinite model has a denumerable model. But

recent controversy over the interpr-ctation of this result indicates that its philosophical significance remains unclear. ’ The Skolemite sees it as onto- logically 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 nonexistence claim is proved within a particular formalization, all it shows is that the formal theory itself is not powerful enough to gencratc 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 Skolemitc. 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 forma1 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 Skolemitc 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

Journal of Philosophical Logic 5 (1976) 475-489. All Rights Reserved Copyright @ 1976 by D. Reidel Publishing Company, Dordrecht-Holland

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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.s

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 undecid- able 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

I shall first examine certain aspects of the Skolemite’s claim that the Skolem theorem shows that there are only denumerable sets.7 One common objection to the Skolemite position is that the denumerable models do not give us an intuitive interpretation of the concept of membership at all, but merely of a complex number-theoretic predicate which would never have seen the light of day except for its isomorphism with the membership predi- cate as it appears in formalized set theory. As Resnik puts it, “How does it make sense to speak of sets here? The membership predicate has been given a numerical reinterpretation, and set-theoretic statements have been absorbed into number theory. Thus we are no longer dealing with sets . . .“? In other words, under the Skolem interpretation the theorems of formal- ized set theory must be taken as theorems about the natural numbers, and there is a serious question whether the notion of set has not entirely dis- appeared.

THELijWENHEIM-SKOLEYTHEOREM 477

The Skolemite might reply that even if this were a legitimate objection to the original Skolem model, it is hardly a definitive refutation of the Skolemite position, since this particular model is an accident of this particu- lar proof, and there are several different methods of proving the Skolem theorem. Indeed, in what is now perhaps a more familiar proof the domain consists of a subset of the set of terms of the formalized theory, and the predicates are modeled nicely by just those sets of n-tuples of terms between which the predicate is said to hold. It is hard to see how this Mcnkin model could be considered less than intuitive, since it simply mirrors the sentences of the formal theory itself.

But the Platonist would no doubt rejoin at this point that such a mirror- ing is not what he has in mind by “intuitively acceptable”, and that the IIenkin model is no more acceptable than the number-theoretic interpret- ation. If our statements are now taken to be about terms rather than numbers, little has been gained from the Platonist’s point of view. It is plausible to assume that the only sort of model the Platonist will find acceptable is one in which the domain is a set of sets and the membership predicate is given its “normal” interpretation.

It is possible, however, to find a denumerable model which fulfills these conditions. Through a proof of the Skolem theorem which utilizes the Axiom of Choice, we may obtain a denumerable submodel of the Platonist’s original interpretation,” and since this submodel is just a restriction of the original model to a denumerable domain, even the Platonist is willing to admit that it is intuitively adequate,” and it seems the Skolemite is home free.

The Skolemite, however, wants to show that there are ody denumerable sets, and the mere existence of a denumerable submodel does little to shore up this claim. For by hypothesis, the denumerable domain of the Skolemite’s model is a subset of a larger domain, the existence of which must be assumed in the first place. One reply which has been given to this argument is that we no more need accept the existence of the original model than we riced accept the truth of a statement which we assume for a Rcductio proof. WC simply “assume all of U for the space of the argument. We show thereby that if all of U were needed then not all of U would be needed; . . .“.*’ There arc really two questions being confused here, how- ever: whether the domain exists, and whether it is needed for the model. No contradiction is generated by supposing that the original set exists

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(indeed, if this could be done there would be no need for these endless discussions about the significance of the Skolem theorem); thus the con- clusion cannot be that the set does not exist. Rather, all we show is that a model can be constructed without the entire domain, i.e., the original set is not all needed for the model. Since the existence of the original set has not been refuted the only question is the cardinality of this set, and the Skolemite has not claimed that this set is denumerable; presumably he grants its nondenumerability at this juncture. The Skolemite can get his intuitive denumerable model, then, but in the process he seems to be cutting the ground from under his own feet.

But even if this dispute could somehow be resolved in favor of the Skolemite, there is a more serious problem with his argument. The Skolemite, and the Platonist as well, have been simply assuming that the submodel, which is just a restriction of the original interpretation to a denumerable domain, will retain the intuitive properties of sets and member- ship, but unfortunately for the Skolemite, there is some reason to doubt that this is the case. A simple example from number theory may illustrate the problem. If we consider the relation of properly less than, and restrict it to the first ten natural numbers, the resulting sets of pairs {(I, 2), (2,3), . . (4,6). . . (9, lo>> is certainly still intuitively the less than relation. But if we restrict the original relation to a different domain, say {2,4,8, 16 . . .> the resulting set of pairs will look like ((2,4), (2,8>, (4,8), (2, 16). . .} and in this case we would likely say that the predicate represented by the relation was “divides evenly by 2x”, rather than “properly less than”. There are similar situations in set theory. By an appropriate restriction we might end up with a domain consisting of just those sets which are both members of and subsets of each other, e.g., the natural numbers in the Zermclo construction. And in this case we would have no reason to take ‘E’ as mem- bership rather than subset. Of course fomalZy the relation can still be said to be that of membership, but that is not the point. The point was whether such a restriction would yield a relation which was intuitively the same as the original, and it is clear that this is not always the case. The existence of such submodels, then, should be of no solace to the Skolemite; if denumcrable models are unintuitivc without the Axiom of Choice, there is little reason to think the situation is much improved by applying it in this way.

The Skolemite has one last resort. He can simply deny, as Thomas does,

THE LOWENHEIM-SKOLEM THEOREM 479

that there is any need for an intuitive interpretation of ‘E’, on the grounds that the notion of an intended model for set theory is none too clear in the first place. Thus we should not be surprised, or unduly alarmed, this argu- ment goes, to find the membership predicate modeled by a complex, unin- tuitive number-theoretic relation.‘* This is a little odd, however, in view of the fact that thcrc are unintuitivc Skolem models for m7y formalizable theory, e.g., Newtonian mechanics or the kinetic theory of gases. No one argues in these cases, however, that lack of clarity in the idea of a standard model forces us back upon non-standard models. It is only in set theory that the Skolem models are given any particular importance, and to carry his point that all we can reasonably expect in set theory are unintended models, the Skolcmite will have to come up with some explanation of why set theory is different from other theories in this regard.

Where does this leave us‘! The Skolemite has failed to demonstrate that his denumerablc model conforms to our ordinary notion of membership, and his defense that it need not do so is, so far, unconvincing. We are forced to the view, it seems, that while there are indeed dcnumerable models for set theory, these arc not the intended models, and since we must then turn to other interpretations, the Lowenheim--Skolem theorem remains without ontological force. This conclusion is only reinforced by a consideration of the “upward” version of the Skolem theorem, which tells us that set theory has models of any arbitrary cardinality. To conclude that there arc only denumerable sets because any set-theoretic term ctzi? be interpreted in a denumerable domain is simply to beg the question, since the term can also be interpreted in a non-dcnumerable domain.13 The most sensible con- clusion seems to be that urged by both Resnik and Myhill: formalized set theory admits of models of any infinite cardinality, and what the Lowenhcim-Skolem theorem shows is not that nondenumerable sets do not exist, but only that they cannot be completely characterized by the struc- ture of first-order logic, since any first-order proposition can be construed as a statement about a denumerable collection.

Of course, as Thomas and others have pointed out,14 this conclusion, and the objection to the Skolemite position on which it is based - that the Skolem models are not intuitively adequate representations of the concept of set - rest upon the supposition that there is an intuitively adequate model, and that it can somehow or other be adequately characterized. Unless the Platonist can establish this claim, there is little point to his

480 VIRGINIA KLENK

objection. In the following paragraphs 1 should like to examine the basis of this claim.

Both Myhill and Resnik, in arguing for the existence of nondenumerable standard models for set theory, make an essential appeal to the distinction between formal and informal, or intuitive, mathematics. Myhill claims, for instance, that “there is evidently only one standard model of set theory”, and that the existence of nonstandard models shows us only that we must never “forget completely our intuitions. . . . a formalism . . _ only remains set-theory as long as the intuition of membership has not slipped away from US”.lS And Resnik argues that “the primary purpose of most formal systems is the formalization of informal mathematical theories. It is via these theories that the intended models of the system are furnished, . . .“.16 Thus “the intended model is already at hand when the formalism is introduced”.”

This distinction between formal and informal mathematics is essential to the Platonists’ claims, since it is obviously impossible to characterize the standard model by formal means. Any formal description would again be open to the Skolem interpretation and thus could just as well be taken as a description of a denumerable nonstandard model. The question at this point, then, is what this distinction amounts to, and what the relationship between formal and informal mathematics might be.

Myhill is merely suggestive, noting that formal mathematics depends upon an “informal community of understanding” among mathematicians.18 He says that we can no more assure formally that our interpretation of set- theoretic concepts (and nondenumerability in particular) will conform to other mathematicians’ than we can assure that our “internal” perception of green is the same as others’. But this is a particularly unfortunate analogy for the Platonists’ point, since for this very reason a great many philosophers have concluded that it is futile even to discuss the question of whether your green is the same as mine. It is not as if there is some other, informal , level at which the question could be decided. Perhaps by the same token we should conclude that it is futile to try to decide, even on informal grounds, which is the intuitive, intended model for set theory. Or perhaps this “informal community of understanding”, as in the case of the word “green”, can be taken as something like community use of the word, and indeed, this is how Resnik tries to spell out what is meant by an informal understanding of set theory. But as we shall see, if this is all that informal understanding amounts to, it does little to support the Platonist’s claim that there is an intended, informally specifiable model for set theory.

TIIELijWENHEIM-SKOLEMTHEOREM 481

According to Resnik, set theory as formalized can be considered just a part of number theory, but on the informal level number theory “can be distinguished from set theory and its terms by appealing to the rule each has in the totality of mathematics. We learn the role of set theory when we study it. We learn more than its quantificational structure; we learn to use its terms and statements.“rg (My emphasis.) He goes on to say that we clearly understand the meanings of set-theoretic terms and statements, and do distinguish them from number-theoretic propositions, and that these meanings are probably best elucidated in terms of use.2o

However, it is difficult to see how this appeal to the role of set-theoretic

statements, or their use in mathematics, is going to help in making the dis- tinction between standard set-theoretic interpretations and non-standard number-theoretic interpretations. Certainly the role or use of statements seems to have very little to do with what sorts of models they have. No mat- ter what the use, no matter where a proposition appears in the formal theory, it is still open to various semantic interpretations. And to say that we learn the role of set theory when we study it, learn to use its formulas, sounds very much like saying that we learn to put the right formula in the right place - which is very much like saying we learn to manipulate the formulas - which is very much Like saying we learn a purely formal pro- cedure. The Plato&t cannot get out of this bind by claiming that we learn something more than manipulation, e.g., truths about sets, because the problem in the first place was to show we were doing set theory and not number theory. We cannot appeal to a supposed distinction between num- bers and sets (the objects of our discourse) in trying to explain what it is we are learning, since we were appealing to the learning process to explain the difference between sets and numbers. The attempt to elucidate the notion of informal mathematics in terms of use, then, is of no help to the Platonist in his search for an intended model.

Of course, there are pictures associated with the learning process, and the pictures of sets are certainly different from the “pictures” of numbers, and perhaps it is something like this that the Platonist has in mind when he talks about the informal difference between set theory and number theory. We are aided by graphics; we are given certain representations of finite sets, perhaps, and the results of the union and intersection operations upon them. But the Platonist should not put too much emphasis on this visual aspect, or suppose that the difference between formal and informal mathematics can be described in these terms, for such a move could lead to a conceptual-

482 VIRCINIAKLENK

ism akin to Intuitionism, a position very much opposed to the spirit of Platonism.

In fact, it is not at all clear how we should characterize the difference between formal and informal mathematics. According to Resnik we have a great deal of infonnal knowledge about sets prior to any axiomatization, and in fact, the point of formahzation is simply to codify, and perhaps in the process make more rigorous and precise, an already existing domain of mathematics. Our concept of “set” is to be found at the informal level, and can never be fully captured by the formalization. But it is not clear to what extent this informal set concept is independent of the formalization, for by Resnik’s own admission the formalization may lead to changes in the con- cept by pointing up inconsistencies, or simply by increasing our precision.*’ Thus the formal system is not just a repository for previously acquired con- cepts, but is in some sense creative, and it is difficult to see how the in- tended model could already be “at hand” when the formalism is introduced.

Perhaps a’more adequate view of the relationship between formal and informal mathematics would be the following: informal mathematics con- sists of sentences which may be vague, incomplete, and even inconsistent. Formal mathematics is a more precise (and, one hopes, consistent) language into which the sentences of the informal theory may be transcribed. The relationship, above all, is between two languages, one far more precise and comprehensive than the other. (In fact, it would no doubt be more accurate to say that what we have is a whole series of languages, with a fragment of our ordinary language on one end and the fully formalized theory of sets on the other.) Our intuitive mathematics should be seen as something less, not more, than our formal mathematics: less precise, less consistent, perhaps, and less complete (unless, of course, it is inconsistent, in which case the completeness hardly counts as an advantage.)

But on this view there is little reason to think that pre-formal mathe- matics has any special power to express the intuitive sense of mathemat- ical concepts such as “nondenumerability”. Intuitive mathematics must, after all, be couched in language, and there is no reason to think that the ordinary language of informal mathematics is in any way superior to the formal language of first-order predicate logic. Aside from the fact that in- formal mathematics is temporaZZy prior to formalization, it is difficult to see what the evidence might be for Resnik’s claim that the formalization pro- vides only a partial specification of the informal set concept. Once we have

THE LOWENHEIM-SKOLEM THEOREM 483

formalized the notion, moved from our ordinary language to the formal language, what more could there be in the informal theory, unless one sup- poses that vagueness is a hallmark of comprehensiveness. Where is this “real” concept of set to be found? “Nods, and Becks, and wanton Wiles” (to mangle a metaphor) won’t do in mathematics.

There is one other possibility, of course, and that is that our ordinary language is in fact a second-order language, in which case it would be express ively superior. But aside from the inherent difficulties with this idea, the problem of the intuitive concept of “set” would still be more satisfactorily discussed on the formal level, as a question of the relative expressive powers of first and second-order languages.

It seems, then, that the attempt to make clear the notion of an intended model for set theory in terms of informal mathematics is a failure. Neverthe- less, Resnik declares that there is no more difficulty in interpreting set theory than in interpreting elementary logic, arithmetic, or physics.22 The analogies do not hold up, however. except in the case of arithmetic, and there the similarity points to a conclusion quite different from Resnik’s own. In the first place, it seems a little odd to speak of an interpretation of elementary logic in this context, since absolutely anything will count as an interpretation. This can hardly help us with the issue of distinguishing stan- dard from nonstandard models. Nor is the case of physics analogous to set theory. There are obvious physical interpretations for macroscopic terms, and strong, if not absolute, definitional links between macroscopic and theoretical terms. But there is no sort of definitional link between even the most abstract physical term and the terms of set theory.

This leaves arithmetic, and it is not surprising that the problems of inter- preting set theory should bc similar to those of interpreting arithmetic, since on the one hand numbers can be taken as sets. and on the other, sets can be taken as numbers (the fact which motivated this whole discussion). But as we shall see, the analogy undermines rather than supports the Platonist’s claim that there is an intended model for set theory.

In the first place. there are also nonstandard models for arithmetic, and the same difficulties in distinguishing standard from nonstandard models. But even if WC could make this distinction clearly, there are many different sorts of things which could serve as a basis for standard models. We might take numbers just as indefinable abstract entities, or we could construe them as sets. In the latter case, of course, thcrc are various possibilities: WC could

484 VIRGINIA KLENK

take the number two, for instance, either as {(O)} or as (0, {O)}. We might even take numbers as collections of inscriptions. This does not worry Resnik in the least, and in fact he even suggests that the same thing might be said about sets. It is here that he gives away the whole game.

He wants to claim “ontological neutrality” for his thesis that there is an intended model for set theory, and thus he asserts that it is possible to main- tain that an ontology of sets is “actually an ontology of properties or even that an ontology of sets is not even an ontology of abstract entities.“13 But now what has become of our informal notion of set? If we had anything in- mind before we reached the point of formalization it certainly included the idea that sets are abstract. And even apart from the difficulties noted earlier in distinguishing between formal and informal mathematics, how can it make sense now to say, as Resnik does, that formalized set theory is derived from our informal concept of set, when that concept is so ill-defined as to include non-abstract entities? How can he claim that the intended model is “at hand” when so many different sorts of things could serve as intended models? What on earth is an intended model supposed to be? And finally, what is now the justification for rejecting the Skolem models? The original objection was that they did not provide an intuitively adequate representation of the concept of a set, but surely numerical models arc closer in spirit to sets than any collection of non-abstract entities. It almost looks like a Reductio of the Platonist’s position, and at the least he seems to have given up any serious attempt to say what the intended model for set theory is like.

Perhaps with good reason. We have seen what the difficulties are in saying, apart from the formalization, what sets are like. Because of the tiwenheim-Skolem theorem it is impossible to describe them formally, and because of the problems cited above, we cannot succeed at the informal level either. Perhaps the moral of the story is that sets (and numbers) are just what the axioms tell us they are, and any set of objects which satisfies these axioms will do as a model for set theory.

But now aren’t we jumping too quickly to a Formalist conclusion? Perhaps all Resnik has in mind by “ontological neutrality” is that we take our set theory at face value, as Quine suggests, and simply accept the prop- osition that there are nondenumerable sets. And perhaps what he has in mind by “informal mathematics” is what Quint calls the “background theory” in which we discuss the formalization. This is no help, however.

THELOWENHEIM-SKOLEMTHEOREM 485

For what can it mean to take a theory “at face value” when the theory has so many faces? As Quine points out, there is no ontological hay to be made from accepting the sentence of the formal language; it is only through an interpretation that we can say what model is meant by a theory.24 But interpretations are made in the metalanguage, and in this case our meta- language is precisely the language which tells us a denumerable model will do for set theory. Of course, it also tells us about other models, but how, in the midst of this abundance, are we to make a choice? Quine’s suggestion is that we get the intended reference “by paraphrase in some antecedently familiar vocabulary”,25 but again, there seems to be a choice of vocabularies at this point. Perhaps it is the “background theory”, presumably Resnik’s informal mathematics, which is the vocabulary meant here; Quine does claim that the predicates “set” and “number” can be distinguished against such a background theory. But: he claims that the way they are to be distinguished is in terms of “the roles they play in the laws of that theory.“26 And as we saw earlier, such an appeal to the role of a term or sentence looks very much like a Formalist interpretation.

Leslie Tharp has suggested, in arguments directed as much against Formalism as Pythagoreanism, that standard models can be distinguished from non-standard counterfeits in terms of their epistemological roles. We want to learn about sets, and “arithmetical or numerical models contribute nothing.” 27 This sounds persuasive, and looks like a promising approach to the distinction between standard and nonstandard models that the Platonists have been trying to make, until we ask ourselves what an intended model would contribute. A great deal, according to Tharp. “There can be no question that a natural model provides a unique source of insight into new truths.“*’ But what is it that provides insight? Is it really the abstract model, the universe of sets? And if so, what is the mechanism by which this insight is obtained? Do we have, as Plato suggested, a special faculty, as yet unexplored by scientists, for apprehending abstract objects? I myself am not aware that I have ever learned a set-theoretic truth by being confronted with an army of sets. I do remember confronting new notation, drawing pictures, and working problems, and I suspect that my experience is typical Of course, there can be no definitive answer to this question until we learn more about how we learn, but I doubt very much that the learning process in set theory will be found to be analogous to that of zoology.

Yet there is clearly something in Tharp’s suggestion; the number-theoretic

486 VIRGINIA KLENK

predicates which would correspond to the set-theoretic vocabulary are highly unintuitive, and would be extremely difficult to work with. I submit, however, that this plays into the hands of the Formalist, rather than provid- ing an answer for the Platonist. The problem is that if we spelled out in detail these numerical predicates we would have a formal system that was completely unwieldy; I think what bothers us is our vision of the complex- ity of the resulting formulae.

Having disposed of these arguments, perhaps we are now entitled to jump to our Formalist conclusions. The attempts, and failures, to distinguish between standard and nonstandard models illustrate the problems inherent in taking mathematics as a descriptive science - descriptive of either sets or numbers. In the end, we find that we are describing almost anything -- in other words, nothing. What WC arc left with is a formal structure which may fit any number of different domains, and the only reasonable conclusion to be drawn at this point is that it is the formal structure, the formal language, and not any particular set of objects, which is the important thing in math- ematics. But this should be neither surprising nor disturbing; mathematics, after all, like logic, has traditionally, and correctly, been seen as a discipline which is applicable in almost every situation. It is a formal theory which we use to talk about objects of all sorts, not a theory about objects. (And this, by the way, answers the question which was earlier put to the Skolemite, of why set theory differs from other theories in not requiring a standard model.)

A Formalist account of mathematics is often rejected out of hand, on the grounds that it makes mathematics nothing more than the manipulation of meaningless marks, a view which we all know to be mistaken. But this objection is based upon the supposition that meaning is reference, and there is no reason to suppose that this is an accurate account of meaning in math- ematics. Indeed, in view of the ambiguity of reference for formal systems, we could hardly maintain that meaning is a function of reference for in that case we should have to admit that 1~ means something different when we construct mathematics according to Zenndo’s definitions than when we use von Neumann’s, a conclusion few mathematicians would be willing to draw. It is likely, 1 think, that the most promising account ol‘ meaning in mathematics is to be found in something like the use.

Another objection to the Formalist view might be that if we take it seriously a whole field of mathematical endeavor may go out the window

THE LijWENHEIM-SKOLEM THEOREM 487

Formalism may reasonably be taken to be the view that what counts in mathematics is the derivation of theorems from axioms and rules, and that truth is to be defined in terms of provability rather than satisfaction. It may seem, then, that there is little need for the investigation of various sorts of models. I think, however, that there is little danger that model theory will be abandoned. even if a Formalist view is generally accepted. What does need to be done: perhaps, is to reexamine the assumptions of model theory, and reassess the relationship between model theory and proof theory. IIilary Putnam has even attempted to elucidate the notion of “standard model” in purely formal terms,‘” and perhaps more research along these lines would be rewarding.

Finally, it might be objected (has been objected, in fact) that if the formal system is all that counts, then there is no place left for one very important aspect of mathematics: the analysis of intuitive concepts, which plays such a large role for people like Gijdel and Kreisel: But I set nothing inconsistent in combining a Formalist view with this sort of enterprise. It is not necessary that the meaning of intuitive concepts be a function of objects which they represent, and indeed, both Gijdel and Kreisel are care- ful not to insist on this connection. Intuitive analysis is important, but so is the investigation into the SOUIC~ of our intuitive concepts, upon which little effort has been expended. Perhaps greater effort would yield a satisfactory account of meaning in terms of something other than rcfercncc. In any case, the point we have tried to make here is that intuitive mathematics has nothing to do with ontologyT not that there is no such thing as intuitive mathematics.

It remains to tie up a few loose ends, Has not the Skolemite won his case after all? We have concluded that since it is impossible to specify, even informally, an intended model for set theory, any model will do. including the nonstandard “unintuitive” models of the Skolemite. And we have noted that set theory is not analogous to physics, e.g., or to any other substantive theory, which helps the Skolemite make his point that there is no need for an intended model in set theory. These points may be conceded, but the most important claim: that there are only finite or denumerable sets in the universe, remains open to the same sort of objection raised against the Platonist. We saw that the Platonist cannot establish the existence of non- denumerablc sets because of the difficulty in saying what they are like, in either the object or meta-language. By the same token, the Skolemite must

488 VIRGINIA KLENK

fail in trying to establish that there are only denumerable sets, since the formalism leaves it wide open what sorts of models there are.

West Virginia University

NOTES

* This is a revised version of a paper which 1 read at the University of Colorado in July 1975; I gained a great deal from the lively discussion there. I would like to thank especially William Reinhardt, from the Department of Mathematics, who read the paper and made extensive comments. Many of the changes I made have been the direct result of his very useful remarks. The changes I did not make are, of course, no fault of his.

’ See, for example, the exchange between Michael Resnik and William J. Thomas, in Journal ofPhilosophy, LXHI, 15 (1966);Annlysis, 28.6 (1968) and 31.6 (1971); Nous III, 2 (1969).

* Resnik, “On Skolem’s Paradox”, Journal ofPhilosophy, LXIII, 15, p. 428. 3 Ibid., p. 433. ’ Thomas, “On Behalf of the Skolemite”,Analysis, 31.6, p. 184. ’ Zbid., pp. 184-S. ’ At this point 1 must apologize to Resnik for over-simplifying his position for the

sake of a more coherent discussion. He himself discounts the ontological signifl- cance of his arguments and thus would not want to be labeled a Plato&t in the traditional sense of the word. His arguments, however, are so well-adapted to those a real Platonist might use that I have taken the liberty of construing the debate as one between a Platonist and a Skolemite.

7 I am omitting a great many points which may seem crucial to a discussion of the Lowenheim-Skolem theorem, on the grounds that they have already been covered in sufficient detail in the Resnik and Thomas articles. I shall here raise only those issues which I believe have not been given sufficient attention.

B Resnik, “On Skolem’s Paradox”, p. 428. Qume makes much the same point in “Ontological Reduction and the World of Numbers” (in The Ways ofparadox) where he denies that set theory can be reduced to number theory. Myhill also argues along these lines, claiming that in the Skolem models none of the relations can be taken as membership. (“On the Ontological Significance of the Lliwenheim- Skolem Theorem”, in Contemporary Readings in Logical Theory, eds. Copi and Gould.)

9 William Reinhardt has pointed out to me that it is possible to obtain an elementary submodel without appealing to the Axiom of Choice, but I believe this does not materially affect my argument.

” Resnik, “More on Skolem’s Paradox”, Nous, III, 2, p. 195. ” W. V. Quine, “Ontological Relativity”, Journal ofPhilosophy, LXV, 7 (1968),

p. 206. ” Thomas, op. cit., p. 184. I3 Arthur Fine has suggested that the “upward” version doesn’t really make much

THE LGWENHEIM-SKOLEM THEOREM 489

sense, since it seems to rely on some absolute notion of “uncountability”, to which he pleads “lack of understanding”. I am in sympathy with this point of view, but in the absense of a clear account of meaning in mathematics I prefer not to base my argument on these grounds. (See “Quantification over the Real Numbers”, PhrlosophicaIStltdies, XIX, l-2 (1968), p. 31.)

I4 Thomas, “Platonism and the Skolem Paradox”, Anulysis, 28.6, p. 195. Michael Jubien makes much the same point, though from a different philosophical perspec- tive, when he argues that we reject nonstandard models because we have “an already accepted ontology”. (Set “Two Kinds of Reduction”, Journal OfPhilosophy, LXVI, 17 (1969), p. 540.) And Arthur Fine point out that the objection that the Skolem models only map statements onto statements, not sots onto numbers, rests upon the supposition that there is a set of reals to be mapped, i.e., that there is a senx of mapping other than just interpreting the theory in a countable domain. (“Quantifi- cation over the Real Numbers”, p. 30).

” hlyhifl, op. cit., p. 50. I6 Resnik, “On Skolem’s Paradox”, p. 19 1. I7 Ibid. I8 hlyhill, op. cit., p. 50. I9 Resnik, “On Skolem’s Paradox”, p. 436. =’ Ibid., p. 437. *’ Resnik, “More”, p. 190. == ibid. *3 Ibid., pp. 191-2. 24 Quine, “Ontological Relativity”, p. 204. l5 Ibid. 26 Ibid., p. 207. *’ Leslie Tharp, “Ontological Reduction”, Joumol OfPhilosophy, LXVIII, 6, (1971),

p. 161. 28 Ibid., p. 162. *’ Hilary Putnam, “Mathematics Without Foundations”, Journal OfPhilosophy, LXIV,

1 (1967), especially pp. 20-22.