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CAN WE INTEND AN INTERPRETATION?

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate School

of The Ohio State University

by

Pierluigi Miraglia, MA.

*****

The Ohio State University 1996

Dissertation Committee:

Approved by Professor Stewart Shapiro, Adviser

Professor Neü Tennant Adviser Professor Mark Wilson Department of Philosophy mix Number: 9710625

UMI Microform 9710625 Copyright 1997, by UMI Company. All rights reserved.

This microform edition is protected against unauthorized copying under Title 17, United States Code.

UMI 300 North Zeeb Road Ann Arbor, MI 48103 u ABSTRACT

Some mathematical theories are thought to have intended interpretations: they are thought to be about a reasonably well defined subject matter. For such theories, an intended interpretation is also presumed to encompass the intuitive concepts that the theory represents formally. Thus intended interpretations play a semantic, an ontological and an epistemological role: they give the preferred reference of the terms of the theory; they embody a conception of the objects described by the theory; and they are a source of evidence for judging the adequacy of proposed axiomatizations.

Two series of questions naturally arise: are there intended interpretations?

And if so, how are they determined? In the context of set theory, these questions summarize the “Skolem problem,” for it is most prominently displayed in the considerations relative to the Skolem paradox. According to some, the challenge posed by the Skolem problem is met only by a strong variety of platonism about mathematical objects. It is also thought that one must embrace this variety of platonism in order to account for the epistemological role of intended interpretations. I argue against this conception, with particular regard to the case of set theory. The thesis is that skeptical challenges such as those mentioned can be rejected on the basis of a moderately realist view according to which the intended

lU interpretation of this theory is largely dependent upon the logical and inferential

practices which the theory codifies. The centerpiece of the argument is the role

played by the axiom of Choice in the formation of the notion of set which is at the

basis of the modem understanding of set theory. While Choice, as platonists

typically argue, is essential to a proper understanding of set-theoretical ontology, it cannot be easily justified in platonist terms, as a feature of that ontology. Rather, I argue that Choice is a logical principle: it is intrinsically tied to the laws of quantification, and the notion of choice function itself responds to a Tarskian criterion for the distinction between logical and non-logical notions. This approach also illuminates some of Skolem’s original contributions to the philosophy of logic.

IV ACKNOWLEDGMENTS

This work might never have seen completion without the concerted effort and support of the members of my dissertation committee. I grateftilly acknowledge the contribution of my adviser, Stewart Shapiro. Not only did he guide me through the intellectual development that led to this work, he also supported me in more personal ways in the occasions — more frequent than he may wish to remember — when I most needed it.

Similarly generous with their advice and philosophical wisdom (not to mention their time) were Neil Tennant and Mark Wilson.

For support of a different kind, I am deeply grateful to my wife, Laurie, and my parents, who stayed with me through thick and thin. VITA

1984 ...... B.A., Philosophy University of Milan

1989 ...... M.A., Philosophy The Ohio State University

1987 ' 1992 ...... Graduate Teaching andResearch Associate, Department of Philosophy, The Ohio State University

1993 - present ...... Research Associate, College of Humanities The Ohio State University

PUBLICATIONS

1. “A Note on Truth, Deflation and Irreahsm,” Sorites 3 (Nov. 1995); 48-63.

2. Review of T. Oberdan, “Godel, Camap and the thesis that mathematics is empty,” Jahrb. der Kurt Godel GeseUschaft, 1991. Mathematical Reviews, March 1994.

3. Review of W. Demopoulos and J.L. Bell, “Frege’s theory of concepts and objects and the interpretation of second-order logic,” Philosophia Mathematica 3, 1, 1993. Mathematical Reviews, 1994.

4. Review of R.L. Kirkham, “Tarski’s Physicalism,” Erkenntnis 38, 1993. Mathematical Reviews, August 1994.

5. Review of A. Drago and 0. Vitiello, “La storia dell’introduzione della matematica costruttiva nella flsica teorica: il caso della termodinamica, ” Conferenza sulla Storia della Matematica, 1988. Mathematical Reviews, April 1994.

6. Review of J. Burgess, “Hintikka et Sandu versus Frege in re Arbitrary Functions,” Philosophia Mathematica 1, 1993. Mathematical Reviews, March 1994.

VI 7. Review of I. Jane, “A critical appraisal of second-order logic,” Hist, and Philosophy of Logic 14, 1993. Mathematical Reviews, February 1994 .

8. Review of A. Oberschelp, Logik fUr Philosophen, Bibliographisches Institut, Mannheim 1992. Mathematical Reviews, September 1993.

9. Review of T. Koetsier, Imre Lakatos’ Philosophy of Mathematics, North-HoUand, 1992. Mathematical Reviews, January 1993.

FIELDS OF STUDY

Major Field: Philosophy

vu TABLE OF CONTENTS

ABSTRACT...... i i i

ACKNOWLEDGMENTS...... v

VITA ...... v i

CHAPTER 1 ...... 1

INTRODUCTION ...... 1 The Skolem problem ...... 1 Skepticism, through and th rou gh? ...... 16 The thesis of moderate anti-skepticism ...... 22 Use and practice ...... 24 The role of the axiom of choice ...... 26 A twist of epistemology ...... 32 An o u tlin e ...... 34

CHAPTER 2 ...... 38

SKOLEMITE R ELATIVISM ...... 38 The paradox and the skeptic ...... 41 The Lowenheim-Skolem T h e o r e m ...... 41 The argument for relativism ...... 44 Foundations and axiomatic t h e o r ie s ...... 58 The big p ictu re...... 58 Skolem’s view of axiomatic theories ...... 68 The role of C hoice ...... 70 Conclusion: skepticism a g a i n ...... 87

CHAPTER 3 ...... 91

SKEPTICISM VS RELATIVITY...... 91 Introduction ...... 91 Quinean r e la tiv ity ...... 98

viii How to commit oneself to an ontology ...... 107 Is ontological relativity t r u e ? ...... 118

CHAPTER 4 ...... 138

A “COHERENT SOURCE OF AXIOMS” ...... 138 The iterative con cep tion ...... 143 How to be p ro-C h oice ...... 151 Intrinsic argum ents ...... 155 Proving Choice as a t h e o r e m ...... 164 Extrinsic arguments ...... 168

CHAPTER 5 ...... 171

CHOICE AS A LOGICAL PR IN C IPL E...... 171 Hilbert and the E p silo n ...... 171 Logical n o tio n s ...... 183 The Self-Evidence of Choice, a g a in ...... 193 Indefinitism and C h o ic e ...... 203 Conclusion ...... 216

APPENDIX A ...... 218 A BRIEF OVERVIEW OF PAPERS BY SKOLEM DISCUSSED IN THE CHAPTER ...... 218

APPENDIX B ...... 224 THE AXIOMS OF ZF IN THE ITERATIVE CONCEPTION . . 224

BIBLIOGRAPHY...... 231

IX CHAPTER 1

INTRODUCTION

The Skolem problem

In 1923, the logician Thoralf Skolem published a paper in which he severely criticized the formulation of the central axioms of set theory given in 1908 by Ernst

Zermelo. The primary motive for Skolem’s objections arose from his discovery that axiomatic set theory inevitably led to what was in Skolem’s opinion an intolerably paradoxical situation. The paradox, as Skolem described it in considerable detail, originates from the fact that it is possible to make true the assertions of a theory of sets " axiomatized in the manner proposed by Zermelo — in domains where there is at most a countable multiplicity of objects (i.e., at most as many elements as there are natural numbers) each of which contains at most countably many elements; at the same time, those very axioms of set theory entail the existence of an object that has uncountably many elements, that is, strictly more than there are natural numbers. Therefore the assertion that there is an uncountable object is also satisfied in a domain where there is no uncountable object. Yet if all assertions that are forthcoming from the theory on the subject of uncountabUity can be satisfied even in the absence of an uncountable object, wbat are we to make of the property of uncountability in itself as appears to be describable in Cantorian set theory? One does seem to gain a fairly definite grasp of such a property firom set theory, understood at face value. But wbat is precisely this understanding that one gains?

As Skolem would have it, the Skolem paradox insinuates the suspicion that there is no definite notion of uncountability to be understood by working one’s way through axiomatic set theory. On the other band, bow else can one acquire a grasp of such a central set-theoretical notion but by “working one’s way through set theory”? Wbat additional element can guarantee that the notion of uncountabüity (or other similarly important notions) we seem to understand well is in fact “fully determinate”? This is clearly not a case in which a certain mathematical structure has been described by an insufficient amount of detail: the principles we already have, the most basic principles of Cantor and Zermelo, are enough to commit one to the existence of uncountable infinities. The problem is how to interpret the principles: but how can such an interpretation be constrained in such a way as to yield the desired result? The core of the Skolem paradox hes in the standing opposition between our sense that we know what property of sets we are referring to when we are thinking of “uncountability” and the possibility, foreshadowed by

Skolem’s argument, that someone else may disagree systematically in applying the concept, even while using, ostensibly, the same criteria of application. We would be talking of different things, then, though in the impossibility of determining exactly who is talking about what: we could not specify what we had intended by what we were saying. At least initially, Skolem appears to impute the difficulty to Zermelo’s axiomatization of set theory. The fault would lie, according to this way of seeing it,

with the axiomatic form of the theory of sets: were we to confine ourselves to using the set-theoretical concepts in a naive form, without formahzation — Skolem seems to be saying — we would not run into this problem. This much is, strictly speaking, a fact of logic known as the Lowenheim-Skolem theorem(s): every theory formulated in a first order language containing an at most denumerable quantity of symbols has a model of every infinite cardinality if and only if it has a countably infinite model. It is this fact, a theorem of pure first order logic, that leads to the peculiar situation in axiomatic set theory. Because axiomatic set theory in the style of Zermelo is typically construed as a theory formulated in a first order language, it will have countable models if it is consistent (that is, if it has a model at all). It is worth pointing out, however, that Skolem himself was in fact the first to cast set theory in a first order language in a more or less precise way. On the other hand, it is much less clear that a rigorous first-order presentation was what Zermelo had in store for his set-theoretical axioms.* Thus when criticizing “Zermelo’s axiomatization,” Skolem is attacking, at least in part, a figment of his own analysis; the full extent of Skolem’s conception of the foundations of mathematics will be taken up in the next chapter.

It is elementary to verify that the paradoxical situation Skolem uncovered conceals in fact no genuine contradiction: there is, in actuality, no object or objects

* * See Moore (1980), pp. 112-114, for a discussion of Skolem’s revision of the Zermehan axioms. Taylor (1993) illuminates the complex nature of Zermelo’s notion of‘definite property, which is essential in his 1908 formulation. which set theory genuinely asserts to be simultaneously countable and uncountable.

Skolem himself was quick to point that out. But he thought that the price exacted by axiomatic theory in exchange for preserving consistency was still too high: a

thoroughgoing relativism applied to almost every central notion of set theory, itself the centerpiece of the foundations of mathematics.

What did Skolem’s relativism consist in? His fear seems to have been that the foundations of mathematics, far from being the sohd ground on which the rest of the edifice might rest, would become the stage of incommensurable “perspectives”: for it appeared to him possible that, for instance, the same uncountable set “for” one practitioner of set theory would be a countable set for another. Furthermore, there would appear to be no principled reason to privilege the “point of view” of one practitioner over that of another (short of an imputation of relative incompetence, of course). Thus we would have to conclude that the notion of uncountabüity has turned relative, and there is no determinate notion of uncountability simpliciter at work in the practice of set theory.

But why is this a serious problem? There is a strand of relativist thought which fosters the idea that relativism itself -- our confinement within incommensurable “perspectives” — is how things already are. In ethics, for instance, many relativists argue that the moral values and principles we use are de facto relative to cultural and historical parameters; a consequence of this attitude seems to be that, while our general understanding of what morality is may or may not change, there is no need to change our current moral practices. Taking stock of the fact that our concepts are relativistic may be superficially surprising, but should not call for any significant revision in the way we work with the concepts in question: our practices are already thoroughly permeated and fully compliant with relativism.

There is, in other words, a certain “quietist” attitude that most relativists will try to foster: don’t worry, you may just go on as before. In keeping with this attitude, a set-theoretical quietist might say, for instance: Skolem’s arguments are compelling, but why worry? We just face and accept the fact that talk of uncountability unrelativized, no matter how often or naturally we seem to lapse into it, is inaccurate: whenever we describe a set as uncountable, we are actually omitting a parameter - our universe of discourse — to which such a description must he relativized. The relativity of set-theoretical notions may be hard to realize and harder to accept, but that is only because our horizons tend to become parochial, and we regard the universe of discourse with respect to which our own concepts are relativized as the obvious or more “natural” setting for using and interpreting them.

This line of thought would of course be congenial to a relativist, but would compel no revision of the practice based on the concepts in question: this relativist would he happy to let us go on with our talk of uncountability, the moral law, or whatever else it is about which he takes a relativist stance. But it is a consequence of this view that relativism should be neither surprising nor troublesome: Skolem, on the other hand, seems to think that set-theoretic relativism is somewhat surprising and certainly troublesome. Is he right?

Putting aside for the moment the question of exactly what consequences for set theory this set-theoretic relativism may or may not have, I believe that Skolem is at least correct in considering relativism as being fundamentally in contrast to mathematical practice. There is at least one element of the set-theoretical case that should advise caution against the claims of the “quietist” relativist. An analogy with a form of relativism in a non-mathematical context may turn out to be of help here.

Consider thus a moral relativist who articulates the view that criteria of moral evaluation are inescapably relative to distinct value-systems, “cultures,” ethos, or something of the sort. The relativist’s claim will not be just that people raised in different cultures will tend to apply different criteria of moral evaluation; that much need not be denied by the staunchest advocate of moral absolutism. The relativist argues, instead, that different cultures are ethically incommensurable: it is in principle impossible to provide a compelling moral justification (or, respectively, condemnation) of a given culture’s criteria, if by “compelling” we intend here something hke “a justification such that it would convince people belonging to different cultures that the criteria of the culture in question are morally admirable.”

This sort of relativism, in other words, not only denies that there are neutral, culture-independent criteria for moral evaluation on the basis of which an impartial observer can actually compare the distinct, possibly conflicting moral criteria employed by different cultures. This first denial implies that any intercultural evaluation is itself culture-bound. But the relativist must also deny that it may conceivably be rational for someone to accept (or reject) a judgement about the moral fitness of one’s own criteria formulated by someone belonging to another culture.

The relativist will of course always admit the possibüity of abjuring or converting: someone may just decide to jump ship and change culture (to whatever extent possible). But relativism denies the possibility that someone would ever have good moral reasons to change his ways and adopt those of another culture.

In the case of the concepts of set theory, the relativism suggested by Skolem’s argument seems to me to lack precisely this central feature. Consider the state of

6 affairs depicted by the set-theoretical relativist envisioned by Skolem. Imagine two set theorists, A and B, each describing a respective universe of objects called sets.

On the face of it, A and B hold and accept the same truths about set theory and their respective universes. They will say the same things, and agree on any assertion they may feel warranted to make; in particular, they agree on the correct criterion for applying the notion of uncountability. In other words, they both agree that a set is countable if and only if there is a one-to-one correspondence between its elements and the set of natural numbers. This statement - that there exists a one-to-one correspondence between a given pair of sets — is itself a set-theoretical statement; one-to-one correspondences are (or can be construed as) themselves sets, of a particular ük. Now basic principles of set theory entail that there exists an uncountable set: thus in any domain that may serve as a universe of discourse for set theory there will be a set such that there is no one-to-one correspondence between it and natural numbers. Both A and B, good set theorists that they are, wül obUgingly exhibit objects in their respective domains of discourse (i.e., sets) that are uncountable, by their mutually agreed upon criterion. In fact, they will even agree about the properties of this particular set (they will pick the same, in a sense). But by Skolem’s argument they may be actually talking about different things: one of them, say B, is pointing to a set that actually has countably many elements. In fact, the whole universe of B does not contain any uncountable set. The question is: what kind of universe of discourse does that "actually" subtend? It would be evident to A or to us, upon accessing B’s universe, that everything in there is countable, and therefore A and B cannot be talking about the same thing. The universe of A contains a one-to-one correspondence for the set that B asserts to be uncountable. But, of course, B claims to have “all” sets available in her purview, just as A does.

The relativist suggests that there is no genuine disagreement between A and B: each is right, so to speak, on her own turf

“Actually uncountable” simply means, in the relativist’s eyes, “uncountable even in a universe that contains more stuff,” but that this still does not entail that B is simply living in the wrong universe: who is to say that the set-theoretic universe we, as outside observers, have been working in all along is not also a “diminutive” one, akin to the one inhabited by the erring set-theorist? That is, although it is possible to look into the erring set-theorist’s universe and to judge consequently that it lacks something that other universes have, this is another judgement still, based in turn on notions and criteria that are relative to a particular choice of universe of discourse: thus the possibility of formulating such a judgement no more could tell against the truth of set-theoretic relativism than any other set-theoretical assertion.

The set-theoretic relativist, therefore, follows a path that so far runs parallel to that marked by the moral relativist. Yet the analogy between the two cases is not convincing, in the light of what I said above about the purpose of relativism in general. The point is that in the set-theoretical case it is much more difficult to agree with the relativist about the impossibility, for A and B, to come to a shared assessment of their respective practices. A spontaneous reaction in the case sketched of A and B would be that if B is to claim full competence in set theory she at least ought to see the point of A’s complaint about her, i.e., B’s, universe of discourse. That is, when A exhibits the one-one correspondence (available in her domain but not B’s) relative to the set that B judges uncountable, B ought to know enough set theory to recognize that object as a set by her own hghts. It would be odd

8 if she refused: her own “parochial” principles should enable her, by assumption, to

recognize A’s domain of discourse as a set-theoretic discourse, and A’s exhibited correspondence (which enumerates B’s “uncountable” object) as just that — a perfectly good one-to-one correspondence. Therefore, a refusal on her part to agree with A could not be justified by any principle implicit in her own engagement with set theory. It is just very hard to see what, in her conceptual apparatus, could justify the claim that the additional set — the one-one correspondence — produced by

A should not be considered a set at all B’s knowledge of set theory will enable her to see that A’s universe is also a model of set theory, and is moreover an ontologicaUy richer one. If her mathematical practice really incorporates a complete understanding of set theory, then she wül be able to see that there could be more inclusive set-theoretic universes than her own.

Set-theoretic relativism, therefore, never seems to gain the initial plausibility

(such as it may be) of moral relativism. Not only is it not irrational for B to appreciate the point of view expressed by A it would actually be irrational for B not to appreciate it, given her previous knowledge. The moral of the story is that set- theoretic relativism is by no means a trivial doctrine: for it does run counter some prima facie intuitions about mathematical practice. That was Skolem’s reaction in

1923: the possibility of relativism needs to be rejected. Relativism is, in Skolem’s view, the reason why the axiomatic method cannot he accepted as the methodological centerpiece of the foundations of mathematics. Skolem, we might well say, was actually moved by anti-relativistic passions, and by the presupposition that the more significant concepts of set theory are already understood in a fully determinate way within mathematical practice. So far, however, we only have enough to conclude that set-theoretic relativism comes with a strike against it. The problem is that we have reasoned merely on the basis of as yet unspecified “intuitions” about set theory and set- theoretic practice. We can say that relativism ought to he wrong if these intuitions are to be preserved and substantiated, not that they themselves prove the falsity of relativism. There remains the major hurdle of justifying such intuitions, of explaining why the first reaction of so many is that set-theoretical concepts are to some extent univocal and not relative in the way foreshadowed by Skolem. Skolem feared that the widespread adoption of the “axiomatic viewpoint” would lead to a

“pervasive” relativism, because of the limitations of that approach. But the fact is that this apparently did not happen; no such relativism has pervaded the practice of set theory, let alone that of other mathematical theories, which can be constructed in set-theoretical terms. The practitioners remain largely unmoved by relativism. But why so? I take this to be the main question, and it is my hope that the work that follows makes a contribution toward answering it.

In the philosophy of mathematics it has been somewhat customary to distinguish algebraic theories firom theories that are aimed at the description of a given, self-standing subject matter. To the former belongs, for instance, the theory of groups; to the latter, theories such as arithmetic, analysis and - perhaps — set theory. The distinction pertains not to the formal constitution of the theories themselves, but rather to their semantic and ontological interpretation: it makes little sense, apparently, to wonder whether a certain structure “really is” a group; the question that bears asking is whether a structure satisfies the group axioms, and this question has an unequivocal answer that can be given according to a

1 0 formally unambiguous criterion (at least if we regard the relation of satisfaction as formally unambiguous). Thus there is no room to doubt whether the group axioms for their part correctly capture the notion of group. For numbers or sets, the matter is reversed. Here the axioms are thought of as representing a certain type of objects; thus it is legitimate to ask whether the domain described by certain axioms really is the universe of sets, which amounts to questioning whether the axioms satisfied by the domain are the correct axioms for set theory, whether they correctly represent the intended subject matter. In cases such as these, we say that the formalized theories of sets, integers, etc., have intended models: they have unique (up to isomorphism) interpretations, corresponding somehow to the types of objects and structures they are understood as describing in the common practice of mathematicians. This beUef that some theories have intended models is also reflected in the widespread belief that at least some mathematical statements

(namely statements belonging to just such theories, i.e., statements that are intended to describe particular types of individual mathematical objects) are in some non-trivial sense true. We have, after Tarski, a formal definition of a notion that is usually referred to as ‘truth in a model’, which can be apphed to statements of any formalized theory. Yet it is still possible to ask, say, whether the Goldbach conjecture is true of natural numbers: this, we seem to assume, is an issue concerning the nature of numbers or the structure of the number-theoretic universe.

When one wonders whether such a statement is true, one is not simply looking for a model in which the statement is satisfied: one is wondering whether it is satisfied in one particular model - the domain of all and only the natural numbers. One is wondering, in other words, whether the statement is true under the intended

11 interpretation. Thus, to the extent that it makes sense to inquire about the truth

(simpliciter) of a statement about some area of mathematics, we have to accommodate the presupposition that at least some mathematical theories have intended interpretations.

The Skolem paradox brings to life a character, the Skolemite skeptic, who attacks precisely the idea that our intuitive understanding of set theory gives rise to one determinate intended interpretation of set theory. If every statement of set theory that we can come up with can be satisfied in a “Skolemized interpretation” — one that is based on a countable domain of objects, and is thus markedly different firom the intended one — there seems to be no set-theoretical means available to single out the intended interpretation, to say that there is one model that adequately represents our intuitive understanding of set theory. The main question can therefore be rephrased as one about the intended interpretation of set theory: on what grounds can we stand to repel the challenge of Skolemite skepticism about set- theoretical notions? In fact, what grounds do we have to treat it as a skeptical challenge, rather than simply as a correct assessment of the situation?

If the skeptic is right, we may or may not have to revise mathematical practice. But we would surely need to revise certain “reflective” conceptions of what that practice is ultimately about. The skeptic attacks the presupposition that we can determine, by means available within that practice itself the meaning of the concepts seen as central to that practice. In particular, the Skolemite skeptic undermines the idea that a firm grasp of a specific kind of objects, sets, is available to the competent practitioners of set theory. The Skolemite skeptic is in this respect

12 attuned to other varieties of epistemological skepticism. Like other forms of skepticism, the Skolemite holds that even our most informed judgements ahout a certain subject matter may ultimately fall wide of the mark -- though in a somewhat peculiar sense. The possibility of “error” envisioned by the skeptic consists in the fact that, by Skolem’s arguments, we can never be assured that our concept of an uncountable infinity applies only to sets that are “in reality” uncountably infinite.

Such a possibility, however, is not just due to the ever present possibility of an outright failure on the part of our epistemic faculties: rather, the “error” occurs (if it does) because we are locked into using set-theoretic concepts that cannot be understood in the same way “firom outside” our particular “perspective.” But then if the meaning of uncountability is to be connected to the way uncountability is understood and applied by practicing mathematicians, the relativism of set-theoretic concepts forces the conclusion that there is no fixed meaning o f‘uncountable’. This is also what is expressed by denying that there is an intended interpretation of set theory.

There seems to be one natural rebuttal to the skeptical challenge. It is available, prima facie, to anyone who holds some form or other of realism about set theory. A realist believes that set-theoretical concepts are directly involved in the description of a certain portion of reality: the universe consisting of all objects of the species ‘set’. The set-theoretic universe, which comprises all the objects referred to by the terms of set theory, is in itself fully determinate independently of our cognitive faculties. Thus also the states of affairs described by the propositions we formulate and prove in set theory have a definite truth value, although the truth value of a given proposition of the theory may be unknown to us, or even impossible

13 to know. In essence, the realist believes that the set-theoretic universe is just the

region of reality that we can attempt to describe in set theory; we attempt to glimpse

its features by formulating conjectures and proving theorems. One can easily see

that firom this point of view there really is no significant problem regarding the

existence of an intended interpretation: that portion of reality involving the objects

called ‘sets’ - that is, according to the realist picture, the intended interpretation of set theory, what set theory really is about. There is no “choice” involved, as it were, in picking the right interpretation: it is already done for us, and before we even know it, by the world itself

Someone holding this view of set theory and mathematical objects in general

“ a mathematical reahst (or at least a certain variety thereof) - would have no particular reasons to find the skeptical challenge compelling. This mathematical realist is also a “meaning realist”: the meaning of set-theoretical discourse is not simply to be distilled fi-om the patterns of linguistic usage that predominate in the relevant community of speakers; it is also determined, to some significant extent, by the ontology of mathematical discourse, i.e., the universe of existing sets. Within this universe, some sets are uncountable and others are not, in a completely non- relativistic fashion. The content of the notion of uncountabihty, in other words, is simply the property possessed by certain class of objects in the universe; any set is either determinately uncountable or not. In this perspective, Skolemite relativism is, as it were, a featvire of a particular formal presentation of a given theory: a theory susceptible of being “Skolemized” is simply inadequate as a characterization of the subject matter of the theory itself (thus one would say, in reaction to Skolem’s arguments, that they show that the Zermelo axioms formulated in a first-order

14 language do not adequately characterize sets). The nature of this subject matter, though, does not depend upon the choice of characterization.

It is therefore tempting to respond to the Skolemite skeptic by appealing to a realist conception of set theory, what is usually identified as the platonist conception of sets. There is a ghtch, though. In order to be effective against the skeptic, platonists cannot simply oppose to the Skolemite contention that the set-theoretical notions have no determinate meaning their view that the meaning of such notions is already fully determined by the nature of the existing universe of sets. The contrast between the two pictures is clear, but one would have no reason to privilege the platonist picture solely on account of the contrast. In other words, the platonist needs not only to flesh out his opposition, but also to offer justification for believing that things do in fact stand as he says with respect to the set-theoretical universe.

And the evidence he provides to this purpose must have remained untarnished by the Skolemite objections. Thus it would not do, for instance, to justify a realist attitude toward sets by pointing out that certain statements in the language of set theory are typically judged by mathematicians to be “non-triviaUy true” or such. For the argument made in the Skolem paradox is that exactly the same body of truths about “sets” can be preserved in a world in which the universe of discourse is nevertheless different in respects that are set-theoretically relevant. Thus the truth-conditions of the kind of statements mathematicians utter when they are doing set theory are determined only relative to a choice of universe; moreover,

Skolem argues, the choice of this universe cannot be accomplished by yet another statement, because such a statement, too, would include concepts that are relative to the choice of (another) universe of discourse... and so on. This is the Skolem

15 paradox. So the skeptic attacks precisely the idea that the truth-conditions of

statements made by practising set theorists are determinate. In conclusion, to hold

that the correct understanding of set-theoretical concepts is that which can be

gleaned from the true statements of set theory, however plausible, amounts to

begging the question against the skeptic. That this move is legitimate is what needs

to be shown.

Skepticism, through and through?

The set-theoretical realist, therefore, is in a bit of a bind; he has to explain

how, compatibly with the platonist picture of the set-theoretical universe, our set-

theoretic concepts come to acquire their meanings if not simply by learning what practising mathematicians say about sets — for discourse about sets could have

already been infected with relativism, and so could any further explanation coming in the form of “more set theory.” The predicament of the realist in this regard has been noted by Hilary Putnam," who believes that because of this problem the reahst is pushed into an uncomfortable corner, and forced to resort to implausible assumptions.

Putnam asks the question that I have regarded as central to the Skolem problem: that is, what in the world could lead us to beheve that we have an intended conception of set theory, and that there is a way of interpreting set theory - an intended model — that corresponds to this intended conception? But his answer is

■ Putnam (1980), 464-482. I quote from the reprint in Philosophy of Mathematics. Selected Readings. Benacerraf and Putnam, eds. 2nd ed., Cambridge U.P., 1983. 421-444.

16 essentially that nothing can do the job: the skeptic has won. Putnam’s reasoning toward this pessimistic conclusion is presented in several versions and goes through a number of cases and refinements, but the general form of argument — his Master

Argument — can be identified consistently throughout the discussion in Putnam

(1980): Suppose that we face the situation of the two mathematicians A and B, as described above; we want to be able to teU which one of them “has” the correct (i.e. the intended) conception of sets. Nothing they say will do the trick, as we have already seen. So it must be something that someone else (perhaps even we) says - let us add that to the global theory of sets already available firom either A or B. We obtain in this way an “augmented theory,” with information about the respective models used by A and B which they (supposedly) could not come up with. Include everything you can come up with; in fact, include all possible information about the world (the limit theory of “ideal science”) in the form of “theoretical and operational constraints.” Still, once the thus augmented theory has been formulated, the same skeptical considerations will apply to it just as they did to the original set theory. So the augmented theory, too, can be satisfied in a model that is countably infinite.

Presumably, then, this model is intended if any is. But since the theory satisfied by an intended model represents the theory that we actually regard as true

“simpUciter,” the theory of this countable model is true “simpliciter.” Yet this consequence clashes with the sense, essential to the realist picture, that even in this case we have gotten the “real” nature of the set-theoretic universe wrong.

It follows that the reahst cannot hope to buttress his own case against the skeptic simply by identifying the existing set-theoretic universe as the “truth maker” of a select group of true set-theoretical propositions. The presupposition on the

17 realist’s part that there is such a realm of sets may still stand, of course. The realist’s problem is how to go ûrom this presupposition to the thesis that this universe is in fact the intended interpretation of our set-theoretic discourse -- to show that we can, in our practices, intend to describe this universe. Notice that the skeptic need not deny the platonist thesis that sets are actually existing objects, though perhaps removed hrom causal interaction with the physical universe (the skeptic is not an anti-realist). What he denies is that there can he any assurance that we are in fact referring to that universe (that set-theoretic discourse can be determinately true o/such a universe). Seen in this light, Putnam’s argument is exactly that no construction of a reference scheme or a theory of reference for set- theoretic discourse is sufficient to debunk the skeptic’s denial. Here is Putnam’s own summary:

If we are told, “axiomatic set theory does not capture the intuitive notion of a set,” then it is natural to think that something else — our “understanding” - does capture it. But what can our “understanding” come to, at least for a naturahstically minded philosopher, which is more than the way we use our language? The Skolem argument can he extended... to show that the total use of language (operational plus theoretical constraints) does not “fix” a unique “intended interpretation” any more than axiomatic set theory by itself does. (1980, 424).

Some realists, whom it seems more appropriate to label as platonists, have claimed that certain types of mathematical objects, including sets, are cognitively accessible by a process or faculty akin to the external perception of physical objects.

Supposedly Godel held that we need to appreciate the possibility of acquiring

“intuitive” knowledge of the mathematical properties of sets by some cognitive activity which is more or less “like” direct inspection, i.e., observation, of the universe. If we had such knowledge, the problems connected to the semantic

18 interpretation of our set-theoretical beüefe could presumably be circumvented — thus we would get around the obstacles put up by the skeptic. Many, however, find

Godel’s conjecture to be utterly indigestible to naturalistically minded philosophers, because it requires either that we be able to perceive things that are, even by most realist accounts, incapable of entering into causal chains, or that we exert some entirely peculiar faculty, whose sole capability is to perceive sets and assorted mathematical objects. To be sure, it seems unlikely that Godel would commit himself to either one of these unpalatable alternatives, but then one must be able to explain exactly how his conjecture would have to be “correctly” interpreted.

There have been interesting proposals in this regard by Charles Parsons and

Penelope Maddy. Parsons construes Godel’s claims in this regard as appealing to a faculty of mathematical intuition, closely related to its Kantian forebears. Intuition is substantially like perception firom the phenomenological standpoint. Le., inasmuch as it is fundamentally a passive apprehension of a singular object. It is characteristic of the Kantian conception advocated by Parsons that we can have intuitions even though we are not perceiving anything in particular. The main difference between intuition and perception is thus that there need not be a causal occurrence of some kind for us to intuit something, or that something is the case.

Maddy takes up more directly the standard objection that we cannot perceive mathematical objects because they are causally inert. She argues that finite sets are in fact physical objects, and thus possible objects of sense perceptions; in this way at least those concepts of set theory that concern finite sets may be understood as notions concerning a region of the physical world.

Of course, these approaches deserve careful study in their own right; their

19 primary focus is not a response to the Skolem problem. Here, though, I limit myself to the broadest sketch simply in order to explain why I beheve that these approaches would not ultimately address the Skolemite’s challenge. Putnam for his part does not seriously consider any attempt to salvage direct intuition of sets (in the form suggested by Godel’s observations).’ To him, resorting to appeals to a cognitive faculty that directly intuits the set-theoretical universe amounts to a declaration of defeat on the part of the realist: hence his conclusion that the skeptic is actually right, and that there is no way to make sense of the notion of an intended interpretation of set theory. I beheve, with the two authors cited, that Godel’s comments are not reflective of the “mystical” realism chided by Putnam. Yet, from the perspective that has brought us to this matter both approaches, though promising in terms of Godehan exegesis, have obvious difficulties, which I briefly sketch here. As for the first: to view the platonist’s direct apprehension in terms of

Kantian intuition, as Parsons essentially does, may avert the need to explain how we can have perceptual-like knowledge of causally inert entities, but then it remains to be explained why an intuition of an object A should be taken as evidence for the existence of said A. This problem is further underscored by the fact that Kant himself did not regard mathematical objects of intuition (not even the basic objects of Euclidean geometry) as pointing “outward” to an independently existing

’ It is to be noted that the both of the works referred to here came after Putnam (1980), which was actually written in 1976: Parsons’s “Mathematical Intuition” (1980) and Maddy’s Realism in Mathematics (1990). I should also point out that, although starting with an explicit reference to Godel, this work by Parsons is centered on arithmetic and examines specifically the intuition of natural numbers. Parsons’s views on the notion of set (in “What is the iterative conception of sets?” (1975)) might actually generate doubts about the very intuitabUity of sets under the construal of intuition he proposes (as Parsons himself seems to think).

20 mathematical reality. Thus appeal to this kind of intuition may not help the realist in his feud with the skeptic, since to the extent that such an appeal works it will not seem unequivocally to support realism. As for the second: Maddy acknowledges that perception of finite sets cannot reach very far into the structure of sets in general that is usually taken as the target of modem, post-Zermehan set theory. Thus large and significant parts of the theory, including several characteristic axioms such as the axiom of choice and others, have to be added as “theoretical hypotheses " to the rest of the corpus. But while this may be perfectly acceptable methodology to expand our stock of assumptions about sets, and indeed perfectly consistent with platonism, it does not seem to help the realist much in his fight against the skeptic.

Following Putnam’s script, the skeptic may happily consent to adding whatever further hypotheses about the universe the realist thinks plausible, and still claim that, this being just “more theory,” nothing stands in the way of a skeptical reinterpretation of the additional theses: once again, the problem is that under the

Maddy approach the realist seems to oflfer in the end no theory-independent access to the universe (which seemed to be the promise of the kind of “transcendental” perception suggested by the Godehan remarks).

The upshot of this brief excursus is that a platonistic conception of set theory can be redeemed, but apparently not in such ways as to satisfy Putnam and the skeptic. We are stül not clear of Putnam’s unsavory dilemma: either the

“Skolemization of everything,” or the postulation of a non-natural perception of mathematical objects. Putnam’s own response to this dilemma is to jettison realism about sets and mathematical objects altogether. He beheves that the only way out of the skeptic strictures hes in the adoption of a verificationist semantics: on a

21 verificationist interpretation, the correct understanding of set theoretical concepts is to he found in the verification procedures applicable to statements involving those concepts, Le., in the proof procedures of set theory. In this semantics, the notion of truth and that of the truth conditions of a statement are no longer central. The adoption of such a semantic firamework, however, requires a revision of classical logic, and consequently of large areas of mathematical practice where inferences seem accepted as a matter of course. Much of classical set theory itself needs to be given up.

The thesis of moderate anti-skepticism

I intend to show that one can reject the skeptic’s challenge without retreating to the confines of verificationist semantics. The primary aim of this work is to vindicate the instinctive common-sense reaction that suggests that the Skolemite is mistaken: we do have a determinate, non-relativistic understanding of set- theoretical notions, and the concept of an intended interpretation of set theory is based upon such an understanding. In other words, I want to vindicate the instincts manifested by the early Skolem, who in 1923 believed that relativism was false and took what we have described as the “Skolem problem” as a reductio ad absurdum of the axiomatic approach to set theory introduced by Zermelo. I submit that we can do so while nonetheless evading Putnam’s dilemma. That is, we can hold onto the notion of an intended interpretation of set theory without committing ourselves either to an implausible form of platonism about the objects of set theory or to the thorough revision of mathematical practice which would be demanded by adoption of

29 a verificationist semantics.

It is not my purpose here to describe an intended model of set theory, that is, the true universe of sets. That task, I take it, does not require any special considerations in addition, naturally, to set theory itselh that is, 1 assume that the specification of the intended structure cannot be done outside mathematics. Rather,

1 take the central question in this work to he philosophical: what makes a certain interpretation privileged or more "natural"? Why do we think that the alternative interpretations of set theory the existence of which is exploited by the skeptic are essentially unnaturaP.

The overall thesis for which 1 intend to argue is that the “natural” extension of the domain of set theory is, to a conspicuous extent, built into our logic, and in particular into the modern understanding of (i.e., the way we understand) quantification. There is of course an uncontroversial sense in which set-theoretic concepts are inextricably bound up with logical notions, inasmuch as the basic notions of formal semantics are cast into set-theoretic terms. This fact is made clear by Tarski’s work on truth, but it was no less clear, it seems, well before Tarskian semantics: one of the complaints Skolem lodges against “axiomatic” set theory is that the axiomatic way of thinking (in its semantic aspect, with which he was most concerned) already involves substantial set-theoretic presuppositions, so that the axiomatic approach would be captive to some sort of conceptual circularity. But what 1 have in mind is not quite so broad a connection. What 1 would like to get at is instead something like the following:

1. There are specific principles of set theory which are simultaneously responsible for (i) our most plausible conception of what the universe of sets should look like (the kind of conception that the realist contender sketched in

23 the previous pages tries to oppose to the Skolemite) as well as (ii) our understanding of (some) logical principles of quantdfication.

2. Logical principles of quantification are part of the norms that govern our inferential practices, and in particular the inferential practices licensed in set theory.

3. Thus some principles governing our inferential practices have a direct connection with, and in fact contribute to determine, the concept of an intended interpretation of set theory. Importing some technical vocabulary, we could put this by saying that there is a “natural” construal of the range of quantification in a model of set theory. When considered in these terms, the interpretations of the theory surmised by the Skolemite skeptic import an undue restriction on such a range: that is the origin of their purported “unnaturalness.”

Use and practice

Even making allowance for the vagueness of the expression, it seems plausible to consider the inferential practices within a theory (or, in general, within an area of discourse) as part of the “use” of language of that theory. Then if this Une of thought could be fleshed out, we would perhaps have found an answer to the issue contained in Putnam’s question: “what can our ‘understanding’ come to ... which is more than the way we use our language?” As we have seen, Putnam’s picture is that once we have collected aU data necessary to the interpretation of “use” (relative to a certain language) we still have to face the analogous question about the use of the larger theory - the one describing our “understanding of the former use” - which includes all these new use-descriptive elements: “what does the understanding of the larger theory come to...?” The proposal I am urging runs counter to this picture: it is true that, if “use” is cashed out in terms of a class of statements (no matter how comprehensive) of a formalized language, then many distinct interpretations of it

24 are possible;^ but this observation does not entail that “use” -- or in general

discursive practice in a given area — is just such a body of uninterpreted formalized

theory. This non sequitur seems ubiquitous in the skeptic line that Putnam

embraces. In fact, what I shall try to argue is that “use” or mathematical practice in

the case of set theory already contains the elements of an intuitive interpretation,

that is, already determines the correct understanding of at least some of its central concepts.

The general inspiration for this approach can be traced to a paper by Paul

Benacerraf, the greater part of which is occupied with the reconstruction of the

Skolem paradox. Benacerraf finds the strongest version of the arguments for the paradox (a version which Skolem himself may or may not have contemplated, but is

not completely explicit in his writings) to pose a compeUing challenge, in fact one that is irresistible under the assumptions about meaning and use in mathematics that undergird Putnam’s brief in support of the “skolemization of everything.” Still, he notes, whatever the faults of Putnam’s position, they cannot obscure the fact that the skeptical challenge requires a “metaphysically and epistemologically satisfactory account of the way mathematical practice determines or embodies the meaning of mathematical language.”” In this form, of course, this stands as an appeal rather than an argument. Moreover, to the extent that such an appeal sketches the outline

^ Incidentally, if this is the central element of Putnam’s claims about use and its theory (inclusive of operational constraints, etc.), then there is no need to appeal to the special issues raised by the Skolem paradox: the trivial fact of the multiplicity of models for any theory satisfied by at least one would seem sufiBcient to make Putnam’s point.

” Benacerraf (1985), p. 110.

25 of a philosophical project, the project itself would encompass a very broad field of philosophical themes: for one thing, there is no reason to see it as specifically limited to “mathematical language,” since (as Putnam in effect tried to show) whatever disturbances are generated by the Skolem paradox will reverberate throughout the philosophy of language. I share the thrust of Benacerraf s comments in this regard, and I hope to provide evidence that at least in the limited context of set theory, at least with respect to some (but key) principles, Benacerraf s “program” (or, as the cynics may prefer, Benacerrafs “gesturing”) need not remain mere wishful thinking.

The role of the axiom of choice

Where exactly can we point if we want to establish more concrete connections between the “practices” relevant to set theory and the actual meanings of set- theoretic concepts? The set-theoretic principle upon which I focus is the axiom of choice. The axiom postulates that, given any family (possibly an infinite one) of pairwise disjoint sets, it possible to form a “choice set” by simultaneously choosing an arbitrary element out of each member of the family. I hold that the axiom of choice is one of the fundamental connections between logic in general, which governs our inferential practices, and the ontology of set theory, within which the questions raised by the Skolem problem present themselves. This set of claims encapsulates what we might call the theoretical significance of the axiom of choice.

There is also a historical relevance of the axiom in the present context. I have already observed, while introducing the Skolem problem, that Skolem was in effect the first to reject its skeptical conclusions concerning set theory; I noted that the main argument in his 1923 paper is structured as an application of modus

26 tollens against Zermelian or “axiomatic” set theory. In a sense, then, Skolem was the first to carry toward set-theoretic skepticism the attitude that I think is the proper one. There naturally arises the question of why he would do so, and I think

(unsurprisingly) the answer is that he was in fact sensitive precisely to the connection between principles like the axiom of choice and logic. In the case of

Skolem, there is another element that puts further emphasis on the significance of the connection, and in fact in my opinion helps to bring it to the fore. Skolem wrote the key papers about what has come to be called the paradox at a time when some of the central notions of modern logic were still in the process of acquiring the systematic interpretation with which they are today associated. This is specifically the case of the quantifiers. The contemporary model-theoretic understanding of the quantifiers emerged through a gradual process; Skolem made a central contribution to this process, one that helped crystallize first order logic in its current form.

Before him, neither the logicist program of Frege and Russell nor the algebraic tradition of Boole and Schroder conceived of the meaning of quantification quite in the way that is common today. While working out the difficulties of Lowenheim's theorem, Skolem produced a synthesis, of sorts, of those traditions; the element he introduced was an application of the axiom of choice.

This brings us to the first contention I made about the axiom: that it is a logical principle. There are several parts to this claim. First, the axiom can be stated, with no non-logical vocabulary, as a truth of second order logic; it was so presented by Hilbert and Ackermann, and the point is again illustrated by Shapiro.®

See Hilbert and Ackermann (1938), Shapiro (1991). Second, even if one wants to steer clear of the specific issues relative to second order logic, there is the important connection established hy Hilbert between the

“justification” of quantificational rules over an arbitrary domain and the axiom of choice in the form of what came to he known as the epsilon principle. According to

Hilbert, the universal validity of the classical laws of quantification depend upon the existence of a function constant that “selects” an arbitrary individual for every subset of the domain. Third, there is a certain conception of what “logical notions” are that seems to imply rather naturally that choice functions, at least under a certain construal, are logical notions --1 shall in particular argue that Hühert’s epsilon constant is such a notion. This conception of logic is independently motivated, in that it is conceptually dependent upon the Tarskian model-theoretic definition of logical consequence. As is well-known, Tarski originally thought that his semantic definition of logical consequence, although consistent with the intuitive judgments that we may be inclined to make with respect to the logical validity of arguments, rested on an arbitrary boundary drawn between logical and non-logical expressions. In the original paper on logical consequence, therefore, he did not attempt to draw the line. Later on, he proposed a demarcation criterion, without explicitly tying it to the earlier work. It has been argued, however, that the criterion is in fact in substantial agreement with the desiderata that had guided Tarski in the earlier effort of defining logical consequence; thus the choice of a boundary line may be considerably less arbitrary than Tarski had suspected. In conclusion, it seems to me that the application of the Tarskian criterion to the notion of choice would

' Both claims are made in G. Sher (1991), see especially ch. 3.

28 provide strong support for the claim that the axiom of choice is fundamentally

logical in nature.

The axiom plays also a central role in determining some ontological aspects characteristic of set theory. It is commonly stressed, even in introductory books on set theory, that the axiom of choice is a “purely existential” assumption.® This

means that the axiom simply asserts the conditional existence of an object with a certain property, but provides no indications of “where exactly” such an object could be found, even when given those objects mentioned in the antecedent of the conditional Inevitably, introductory explanations such as these will stress the fact that the legitimacy of such assertions in set theory is not in question: the intuitive picture of sets that the textbooks are intended to convey is that of a domain of things where existence does not depend on available means of verification, and a domain is like that precisely when it licenses assertions of the same sort as the axiom of choice.

Clearly this picture, however naive, is coherent with the purposes of a theory in which one can handle completed transfinite totalities. Constructivist aspirations to a mathematics based on our epistemic faculties and more or less concrete verification abilities seem to be extraneous to very elementary set-theoretical intuitions. (And constructivists such as Brouwer who take epistemic accessibility very seriously tend simply to discount the importance of a theory of “sets” — a fact which confirms such extraneousness.) In fact, firom an ontological point of view sets seem to be independent not only of epistemic accessibUity and mental representation, but also of linguistic representation. It would be seen as a

® For example Fraenkel, Bar-Hillel and Levy (1973).

29 restriction, for instance, if one were to “let into existence” only those sets that correspond to the extensions of predicates of any given language.

On the score of such considerations, it is often said that the conception of set which grounds post-Cantorian set theory is combinatorial: sets are arbitrary collections of elements; they are, in a sense, constituted by their elements.'’ It is not required that there be a unifying factor, such as a shared property, a functional rule of association, a definition, etc., to gather the elem en ts.T h e view opposite to this — the view that a set is a collection of elements single out by a property common to all of them, a concept, a predicate in some chosen language which applies to all and only its members, etc. -- is called definabilism. But if set theory is intrinsically averse to definabilism, what is it that makes it so? So far I have appealed mostly to broad intuitions about sets, but this style of exposition does not amount to an explanation. A much better analysis is devised by P. Maddy:” in her view the demise of definabilism is to be regarded as part of the rise of a new “methodological maxim” in mathematics, a sort of meta-rule to adjudicate disputes involving high- level principles (such as V=L, the axiom of constructibility in set theory) for which

® That such a conception deserves the attribute of “combinatorial” is explained by P. Bernays as follows, in regard to its significance in various areas of mathematics: “These notions [sets, sequences, functions, etc.] are used in a ‘quasi- combinatorial’ sense, by which I mean: in the sense of an analogy of the infinite to the finite” (see Bernays (1935), p. 259).

Actually, some restrictions are necessary to avoid paradoxes: for instance, we may consider only sets formed in “stages” extending into the transfinite (iterative sets), or only those sets that are not as large as the set of all things (limitation of size doctrine). Supporters of the combinatorial conception do not usually regard such conditions as genuine restrictions on set-theoretical ontology, since they claim that the objects excluded are not properly sets.

” Maddy (1993).

30 direct and conclusive evidence (such a proof or disproof is not forthcoming.

Definabilism itself is a maxim, which has been rendered obsolete by a number of developments in analysis and the theory of functions. Based on Madd/s study, it seems that the combinatorial notion of set can be conceived as part of the emergent anti-definabilist maxim, or perhaps as one of its theory-specific epiphenomena

(specific, that is, to the theory of sets). The axiom of choice is also a central component in the rise of anti-definabüism: for the axiom of choice asserts the existence of a certain set in complete independence of the availability of a definition or a construction procedure. The point is even clearer if one takes the axiom in the form of Zermelo’s well-ordering principle, for the latter entails that there is, in particular, a well-ordering of the set of all real numbers. Now not only is there no intuitively natural way to string along all real numbers in a well-ordered fashion, there is also a strong suspicion that such a well-ordering - itself a certain set, after all — is in fact undefinable. In any case, this is just a striking example of the anti- definabilist nature of the axiom of choice: the introduction of the axiom caused controversy precisely because it requires the existence of such things as arbitrary functions. So it seems fair to conclude that the axiom of choice is one of the principles that commit us to an “arbitrary” ontology of sets in the sense of the combinatorial notion, and thus express the dominant anti-definabilist conception.

31 A twist of epistemology

A further point of interest in connection with Maddy’s analysis of the development of methodological maxims can be noted here. Maddy developed an extended argument aimed at showing that difficult open questions concerning set theory could be susceptible of solution, if by “solution” one does not mean exclusively

“proof in some kind of formal system. Thus, she claims, the particular question she considers (whether every set in the universe is constructible in the sense of Godel) is not entirely open in the minds of most set-theorists, who, in keeping with the dominant anti-definabilism, generally consider V=L as false. In other words, she claims that, due in large part to the existence of dominant methodological maxims throughout science, the truth of hypotheses such as V=L, which would not in themselves be susceptible of proof or disproof (except perhaps on the basis of other, even less verifiable hypotheses), can be considered as determined on the basis of indirect evidence.

1 have expounded here on the overall structure of Maddy’s argument to stress a difference between the principles she directly focusses on (principles such as V=L) and the axiom of choice. The difference seems to me relevant with regard to the following objection. Someone might say, in response to the claim made in the previous section about the ontological import of choice: Yes, it is of course true that the axiom of choice is acceptable in the context of the anti-definabilist conceptions that have prevailed in analysis and have crystallized in set theory, but this only means that, to someone entertaining a combinatorial notion of set, the axiom will seem obvious; in other words, there is still no reason to think that the axiom has an active role in determining the extent of a combinatorial ontology: its acceptance is

32 the effect of a prior acquiescence to an anti-deffnabüist presumption.

In reply, I note that the axiom of choice is unlike the highly abstract hypotheses at which Maddj^s argument aims at least in this respect; it was applied even before its explicit introduction by Zermelo for the proof of the well-ordering theorem, that is, before an anti-defînabilist maxim could become dominant with respect to set theory. Given this circumstance, it is natural to suppose that choice possesses some independent evidence of its own. The axiom is certainly consistent with the rejection of definabilism, but there is something more, something that is lacking in the case of high-level theoretical hypotheses for which, it seems, the primary source of evidence is, of necessity, consequence from a methodological maxim. The axiom of choice is a constitutive principle of the anti-definabilist conception. This, 1 think, can be explained better if we conceive of the axiom as a logical principle, and thus tied to principles of inference. The claim could be made, in this regard, that logical principles and inferential practices are as much a component of methodological maxims in science and subject to their expansion and revision as any other mathematical theory. One could say that logic itself underwent dramatic changes of perspective during the same period that saw the rise of anti-definabüism. This seems right, in any case. But it does not conflict with my case: it is not a part of that case that logic he an eternally identical, completely a priori source of evidence. 1 claim that the axiom and logic, and therefore inferential practices, go hand in hand: mutual conceptual interdependence is entirely acceptable. In fact, it is in essence what 1 would like to get at: to establish a constitutive bond between the theory of quantification and the most informed understanding of our notion of set.

33 An outline

This work melds two distinct lines of inquiry. The first is an historical one: we must investigate the conceptions that led to the formulation of the “paradox” within the context of Skolem’s as well as other authors’ work in the foundations of mathematics. The second is more theoretical, and concerns directly the main thesis that our notion of an intended interpretation of set theory is indebted to our understanding of certain logical principles. This twofold approach will guide our study of Skolem’s original arguments, to which Chapter 2 is devoted in its entirety.

It contains, in the first place, a reconstruction of the case for relativism that occupied most of Skolem (1923). In order to understand such arguments, however, we need to explore Skolem’s conception of the foundations of mathematics on a larger and deeper scale. As briefly hinted, Skolem was a central figure in the development of logic as we know it today, and also contributed powerfully to the establishment of set theory on the sound footing provided by the tools of formal logic.

In fact, the key aspect to be explored in the chapter will turn out to be the intertwining of logic and set theory in Skolem’s philosophical understanding of logic, which comes to him firom the algebraic tradition of Boole and Schroder. It seems to me that if we are able to unravel the philosophical underpinnings of Skolemite relativism we shall have achieved results both of historical and theoretical value.

That this study would have historical value needs probably little arguing; but I think that it would also reveal the deep relation between key notions of set theory and some of the central elements of our inferential practice, a relation which in my view is the crucial one in the context of the Skolem problem. With all this, a word of

34 caution is still necessary: though open in his discussion of relativism, Skolem’s

philosophy of mathematics is often lying under the surface of his works, rather than

neatly systematized in familiar ways. From a contemporary perspective, his

philosophy appears to hlend currents and trends of thought that we would consider

nearly incompatible; this applies, for instance, to much of Skolem’s apparent

penchant for constructivism.

The following chapter takes up a different brand of relativism, the Quinean

theory of ontological relativity. , This view has been regarded by many as a direct

philosophical descendant of Skolemite skepticism, and as a formidable hindrance on

the path of any argument to the effect that we can, after all, “intend an

interpretation." There is also a view on linguistic reference espoused by Putnam,

which seems closely allied to the Quinean considerations on ontological relativity.

Keeping in mind that ontological relativity has much broader scope than Skolemite

relativism (a view that directly concerns set theory), I tackle the problem primarily

as it applies — according to Quine’s own presentations — to mathematics, and specifically to mathematical objects. In this context, it seems to me that ontological

relativity does not have the force of the Skolemite’s objections against the determinacy of our understanding of set-theoretical notions. 1 argue that, by Quine’s own lights, a structuralist conception of mathematical objects seems to emerge largely unscathed from problems of ontological relativity, but not necessarily so from

the Skolemite’s objections. In sum, it seems to me that, contrary (perhaps) to

appearances, ontological relativity is not “all there is” to Skolemite relativism.

Chapters 4 and 5 return the focus to the central issue: the connection between the intuitive conception of set that is at the basis of our understanding of

35 set theory and logic, as a representation of the inferential norms that are part of mathematical practice. As I have in part already illustrated, the key connection I explore in this regard is represented by the axiom of Choice and its role in the development of set theory. There has been considerable debate over the true nature of the mathematical concept of set that is at the basis of modern-day set theory — a debate that originated, in effect, with Zermelo’s proposal of his system of axioms.

Zermelo put forth his axioms, after all, as the principles of something, not just as an intriguing “language game.” It seems clear that, to the extent that we do have an intended understanding of set theory (of what this theory is about), it must be embodied in a particular conception of what sets are like. The candidate that seems most suited for the role is the iterative conception of set.

Many have argued that it is the iterative conception that is at the basis of set theory, in the sense that one can develop intuitive arguments to the effect that, given the presumptive nature of iterative sets, this or that axiom of the theory can be seen to be true of such sets. This, as we have seen, is the tack of mathematical realists, and the core of the realist conception of the intended interpretation of set theory. In chapter 4 1 review the iterative notion of set, as well as the sort of intuitive arguments in favor of various set-theoretical principles that are supposedly grounded in the iterative notion. I argue that the attempts made to justify the axiom of Choice on the basis of such a notion are not satisfactory; in particular, the sort of evidence in favor of Choice that realists tend to claim on the basis of the iterative conception seems to me defective. Thus, if Choice is nevertheless involved in some central aspects of what seems to be the intended notion of set (as realists would no doubt maintain), we have that some central aspects of the intended notion

36 are not dependent on the realists’ preferred picture of iterative sets as the objects

“out there.”

The idea is, instead, that some aspects of the intended notion depend not on a

“mathematical reality,” but rather on principles constitutive of our inferential practices — of logic. In chapter 5, consequently, I tackle the claim that Choice is a logical principle which is not, as such, justified by any particular conception of set, but rather governs aU such conceptions and at least in part determines the nature of our intuitive conception of the subject matter of set theory. The thesis that the axiom of Choice is “logical” in nature is not novel, having been voiced early on by

Zermelo and Hilbert. But Zermelo’s argument seems to have been primarily epistemological, on the score of the realization that so many used Choice in mathematical application even while denying its truth. Hilbert, on the other hand, tied more closely Choice and logical laws, but mostly for reasons internal to his foundational program. I argue that Hilbert’s characterization can be sustained on the basis of a systematic conception of logic and logical consequence due to Tarski: this conception issues in a criterion for drawing the distinction between logical and non-logical constants, and on the basis of such a criterion, I argue, one can regard the axiom of Choice as a logical principle.

37 CHAPTER 2

SKOLEMITE RELATIVISM

Many key concepts in the philosophy of mathematics have emerged through an apparent breakdown in standard practice, a crisis of sorts. So is for the notion of an intended interpretation. Its interest arises at the point when it becomes doubtful whether there is such a thing as the intended interpretation of some significant theory or fragment of mathematics, and the uncertainty appears to go against the common wisdom. The locus classicus — the focal point of the crisis — is the so-called

Skolem paradox. It is Skolem who first called attention to the fact that certain results in the then budding discipline of model theory might have interesting philosophical consequences. He thought that the most startling consequences of these results would affect set theory, which he saw as a key element in the foundations of mathematics. This view led him to formulate what he took to be a critical problem for set theory, the series of considerations known as the Skolem paradox. He drew his own philosophical conclusions from the paradox, and they

38 were generally negative: he claimed that the paradox generated a pervasive relativism about key mathematical notions, and that it impelled mathematicians to give up axiomatic set theory in the pursuit of foundational studies.

I believe that an examination of Skolem’s reflections on this matter is still key to understanding the problems associated with the notion of intended interpretation—and not only that: I also believe that an understanding of Skolem’s relativism or skepticism about set theory can point the way to elements of a

“solution.” The aim of this essay, therefore, is to carry out just such an investigation. The plan is, roughly, the following: first, to describe the Skolem paradox; second, to reconstruct and explain the arguments that in Skolem’s own view justified his skeptical contention that relativism concerning set theory was inescapable; third, to understand the nature of his position. Le., what exactly he meant by relativism and why he thought such relativism justified a criticism of

Zermelo’s conception of set theory. This last is an interpretive task that will carry over to the next chapter, where we will compare Skolemite relativism about set theory to the doctrine of ontological relativity expounded by Quine. Quine’s views bear more than just a casual resemblance to the problems discussed by Skolem, but here the attempt will proceed in a reverse direction, in the hope that Quine’s conception of ontological relativity may illuminate Skolemite relativism.

It is generally acknowledged that Skolem did not formulate a genuine paradox: the Lowenheim-Skolem theorem, which provides the starting point of the discussion, has no contradictory consequences. Thus, it does not show that the conjunction of Zermelo’s axioms for set theory and the principles of first-order logic is untenable, in any reasonable sense. Skolem’s paradox belongs not to the realm of

39 knock-down refutations, but to that of philosophical morals. Skolem argued in his well-known paper (1923) that set-theoretical concepts, as they are embodied expressed in an axiomatic theory, are inescapably “relative to an interpretation.” He never renounced this view, which is reiterated in every published work where

Skolem returned for whatever reason to foundational themes. Any attempt at exegesis should address at least two quite natural questions: First,one should try to make clear the exact nature of the skepticism here envisioned. Secondly, and perhaps more importantly, one wonders what brought Skolem to see matters in the terms he describes. A further issue, closely related to the latter, is what change in philosophical perspective first to criticize a skeptical position about set theory, and later to acquiesce in it.

For although he steadfastly maintained the philosophical significance of the skeptical argument based on the Lowenheim-Skolem theorem, in later years Skolem appears to have changed his attitude toward the findings of his 1923 investigations.

In a 1941 paper he claims that the relativity stemming firom considerations related to the Lowenheim-Skolem theorem is but a fact of life in the otherwise vastly successful practice of axiomatic theories and, as such, no more troubling than any other technical feature of such systems. The Skolem of 1941 is not opposed to the deployment of the axiomatic method in the study of the foundations of mathematics: in fact, he advocates it. Of course, the Skolem of 1941 must have been aware of the fact that what he had called in 1923 the “axiomatic method” had become one of the most firuitful fields of mathematical investigations (to a considerable extent, thanks to his own contributions): rejecting the method altogether would have been meant professional insulation-and few people could have, like Brouwer, welcomed that

40 prospect. Nevertheless, one is naturally led to ask: why the reversal? What philosophical position has Skolem changed (or abandoned) in going firom the skeptical conclusions of 1923 to the more optimistic outlook of 1941? As we shall see, what has changed is not Skolem’s arguments for relativism, which he thought conclusively sustained, but rather his evaluation of their significance. To understand the reasons for the reversal should help considerably, then, towards gaining an understanding of the nature of Skolem’s earher skepticism as well.*"

The paradox and the skeptic

The Lowenheim-Skolem Theorem

The Lowenheim-Skolem theorem (LST henceforth) is the well known result according to which any denumerable first-order theory with an infinite model has a model (is satisfiable) over a denumerable domain. The theorem is a general fact about first order logic and can be proved as a consequence of completeness. For

Skolem, the philosophical consequences of LST-what came to be known as the skolem paradox—issue primarily firom its application to set theory; and most of the recent commentators seem to agree that the puzzle, if any is justified, is a puzzle about set theory (though at least one, H. Putnam, thinks that it quickly spreads firom there*®). It seems therefore advisable to fix the basic ideas directly within the

*- Throughout this essay, a certain amount of back-and-forth switching firom one to the other of these papers by Skolem is inevitable; in the interest of readability, I have gathered essential data about what each paper contributes to the discussion in the Appendix.

*® Cfir. Putnam (1980).

41 context of set-theoretical notions.

It is in its application to set theory that the theorem seems to produce something like a conceptual paradox. As preliminary definitions for the non­ technical reader, I shall set out the following:

Elementary equivalence: given a theory T with a usual first order language L(T), two models D and D’ for L(T) are elementarily equivalent ifi) for all closed formulas A of L(T), D satisfies A just in case D’ satisfies A.

Transitivity: a class (in particular a set) C is said to be transitive ifi^ for all x, y, if y belongs to C and x belongs to y, then x belongs to C.

As a property of a set, transitivity entails that no member of the set can contain

“more” elements than the set itself; transitivity is also an important property for models of set theory, since it ensures that a set is in fact a model of several essential axioms of ZFC.

From the set-theoretical point of view, then, a perspicuous version of LST is the following: for any model M of (first order) set theory with certain “standard” features (i.e. certain standard properties of the membership relation) there is a model M’, elementarily equivalent to M, that is transitive and has a denumerable universe. Since M* is transitive, every element of its domain will be at most denumerable, including the selected element “playing the role” of the power set of the integers (of which, naturally, the theory “says” that it is nondenumerably infinite)- This is not quite the theorem proved by Skolem and Lowenheim. The traditional proof of LST, for instance the one found in Skolem (1920), yields a model

M’ whose domain is a subset of M, but does not necessarily produce a transitive such

M’. The existence of a transitive model elementarily equivalent to NT is imphed by the Mostowski Collapsing Lemma. The fact that the domain of NT is included in the

42 domain of M is ensured by application of the axiom of choice, by means of the so- called “Skolem huU” construction (cfr. the Appendix A). LST itself however, does not necessarily depend on the axiom of choice; Skolem also proves a purely numerical, "finitary” version of it which does not make use of the axiom. The versions of LST using the axiom of choice have been at times characterized (for instance by Wang and Benacerraf) as "submodel” versions. But at least from the elements provided in the description of the arguments, it is difficult to see that one can actually ensure that NT he a submodel of M.

A sort of philosophical primacy is accorded to the submodel construction and proof grounded apparently in the belief that the philosophical consequences of the

LST can be seen in sharper relief against the backdrop of the submodel version. In part, at least, this seems correct, for the submodel version preserves in a sense the interpretation of the membership relation sign (the only non-logical constant of set theory); we discuss this further below. I would argue, however, that the submodel construction (that is, the fact that the countable model culled via LST derives from a simple restriction of the interpretation of the non-logical constants in the “full” model) is not strictly indispensable to the philosophical morals to be drawn from

LST: all that seems to matter is the state of affairs described in the “perspicuous” statement formulated above. In fact, the distinction between producing a submodel and producing a transitive submodel via the LST is often overlooked even in the secondary literature. For the sake of clarity I shall keep with tradition and refer to that version of the problem (the existence of a transitive, elementarily equivalent model) as the ‘submodel version’ of LST, to be distinguished from the numerical version’ (which dispenses entirely with the requirement that the non-logical

43 constants not be reinterpreted) given later by Skolem.

Naturally, in the numerical version the resulting denumerable M” is not a submodel. Whether any of these elements bears significantly on Skolem’s interpretation of LST is a matter of dispute (see Benacerraf (1985), McIntosh (1979) and Tennant and McCarty (1987)), and wiU be discussed in what follows.

Apparently, Skolem himself thought that either version would work well to sustain the philosophical point he intended to make; presumably he would not find the version stated at the opening of this section as more “perspicuous” than one omitting the requirement of transitivity. For the time being, I postpone consideration of such problems. However, the transitivity of the M7 resulting firom LST seems particularly important if one is to understand Skolem’s concerns. Since this smaller model M is transitive, no set designated in it, i.e. none of its elements, can be more than denumerably infinite (otherwise by transitivity the domain, i.e. M” itself, would be non-denumerably infinite). On the other hand, M’ is a model of all the consequences of the axioms of set theory—among which is the statement that there exists a non-denumerably infinite set. Thus we have the following “paradoxical” situation: a model of set theory may contain a denumerably infinite set that the axioms “assert” to be non-denumerably infinite.

The argument for relativism

Skolem sets out to criticize Zermelo’s axiomatization of set theory in his

“Some remarks on axiomatized set theory” (1923a). His observations constitute the core of the so-called Skolem’s paradox and the paradox is in turn the basis for what has been labelled as Skolem’s “skepticism” about set theory. The central contention

44 of the paper (pp. 295-296) is that set-theoretic notions as they are expressed in

Zermelo’s axiom system are relative and, of course, Skolem’s argumentation is also essential in order to understand what kind of feature of an axiomatic system this

“relativism” is. Let us start with Skolem’s own explanation:

By virtue of the axioms we can prove the existence of higher cardinalities, of higher number classes, and so forth. How can it be, then, that the entire domain B can already he enumerated by means of the finite positive integers? The explanation is not difficult to find. In the axiomatization, “set” does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradiction at all if a set M of the domain B is nondenumerable in the sense of the axiomatization; for this means merely that within B there occurs no one-to-one mapping $ of M onto Zq [Zermelo’s number sequence]. Nevertheless there exists the possibility of numbering all objects in B, and therefore also the elements of M, by means of the positive integers; of course, such an enumeration too is a collection of certain pairs, but this collection is not a “set” (that is, it does not occur in the domain B). (p. 295)

The argument centers around the existence of more-than-denumerahle sets: the notion which Skolem claims to be relative is uncountability (“non- denumerabUity” in Skolem’s vocabulary), though this is not the only or, arguably, even the most important concept that turns out to be relative. Via the Dedekind definition of finitude, for example, Skolem fears that even the notion of a finite sequence will be “relativized.” The fact is that, because set theory is a foundational theory, it provides the building blocks to reconstruct even the more significant concepts of mathematics, such as that of number: if relativism spread to these, in

Skolem’s view, mathematics would be even more in trouble.

At any rate, the relativity stems firom the fact that we cannot trust an object designated as a non-denumerable set in accordance with an interpretation of the axioms to be a non-denumerable set. A set ought to be non-denumerable in virtue of

45 some of its internal properties. If a set is nondenumerable in this way, then it should be simply impossible to enumerate its elements. But if a set is nondenumerable in a given interpretation satisfying the axioms, this simply means that the interpretation is too narrow to contain the “possibility of numbering all objects” in the set itself Yet this possibility “exists,” according to Skolem: this means, presumably, that we can define a mapping F with the properties described in the preceding quote in a “larger” domain of interpretation also satisfying the axioms of set theory. Thus, it only makes sense to speak of nondenumerabihty relative to an interpretation.

It is worthwhile at this point to summarize and reconstruct the crucial argument of Skolem (1922). This might go as follows:

(1) In Zermelo’s theory, one can prove the existence of a nondenumerably infinite set P(w);

(2) If Zermelo’s theory (including the Axiom of Infinity) is consistent and thus has a model M, it has a denumerable transitive model NT. This is exactly LST.

(3) But in such a model P(w)^r — the object designated as the nondenumerable set -- is denumerable. This claim is to be interpreted as stating the following: there is a function (in the set-theoretical sense) which is an enumeration of P(a>)^r-

(4) Taken literally, however, the existential statements formulated at (1) and (3) form a contradictory pair. In fact, they are not: the apparent contradiction requires that the existential statement (1) proved within the theory be taken as implicitly relativized to the domain of interpretation, i.e., to M’ in this case.

(5) But what about the existential statement (3)? That statement also is implicitly relativized to the point of view of the external observer. That is, (3) needs to be relativized to a domain of reference different firom that of (1), but what grounds do we have to regard the domain of reference of (3) as any more significant than the domain of reference of (1)? It seems that nothing makes the point of view firom which (3) is asserted any more representative than that in which (1) is asserted.

(6) Therefore, Zermelo’s theory justifies claims about the existence of sets and other mathematical objects only in a relative sense. This is Skolem’s relativistic or “skeptical” conclusion.

46 A point worth noting is that, under specific conditions briefly described in the previous paragraphs (that is, assuming the axiom of choice), the AT mentioned at (2) wül actually be a denumerable transitive subset of M, as weU as a model of

Zermelo’s axioms. In this case we can state

(2") If Zermelo’s theory (including the Axiom of Infinity) is consistent and thus has a model M, it has a denumerable transitive submodel NT.

This produces the so-called submodel version of the argument for relativism.

Two observations seem in order. First, we have already remarked that, according to some commentators, the argument for the relativism of set-theoretical notions loses much of its efficacy unless one calls into service the submodel version of LST. Is this true? It turns out that many deep-seated intuitions about Skolemite relativism are influenced by an evaluation of this aspect of Skolem’s argument. The view expressed, for instance, by Benacerraf (1985) and accepted in the main by

Wright (1985) is that the numerical version of LST involves a reinterpretation of the basic predicates of set theory—in particular, of membership. In the numerical model, the terms of set theory denote numbers; LST, in the version proved by Skolem in

(1922), shows that we can pick “enough” numbers and define enough relations between them to satisfy any sentence of first order set theory. Call a numerical model with this property N. But, the reasoning goes, membership'" (the relation denoted by V in N) is not the same relation as membership, because the two are defined among objects of different types. Once the concepts of set theory have been so reinterpreted, the theory itself does not appear to have anything to do with sets.

Thus, what has been shown is not that membership must be relativized to a domain of interpretation, but just that there can be quite different relations, all ambiguously

47 called ‘membership’. But this would be trivial, in the sense that the fact that

“unintended” interpretations are obtained in this way can hardly be taken as evidence justifying the conclusion that the predicate under study admits of no firm interpretation.

A similar point is raised by Tarski in a discussion following the presentation of Skolem (1958) (see Fenstad (1970), 638): what if we were to “fix the meaning” of the predicate expressing the membership relation in set theory and thus treat it in effect as a logical expression (the interpretation of which is held constant throughout domains)? Then, Tarski claims, denumerable models of set theory wül not be in general avaüable. However, as noted by Benacerrafi the adoption of this proposal would not block the relativistic argument: in fact, the existence of a denumerable submodel, proved by Skolem with the help of the Axiom of Choice, shows exactly that simply holding constant the interpretation of‘e’ will not solve the problem. In the situation represented by the denumerable submodel, what happens is this: the sentence affirmingthe nondenumerabihty of P(co) is a universally quantified formula stating that there is no enumeration of the elements of P(w); but the universal quantifier occurring in this sentence of course ranges on the reduced domain. Put in other words, what matters to the relativistic argument — and what should therefore be the target of its critics - is not so much the interpretation of the membership predicate, as the size of the range of quantification; and this size wül in general vary with the domain, even if membership is taken as logical. To be sure, this need not imply that Tarski was wrong: he might have meant that to treat the membership predicate as a logical constant presupposes that the size of the range of quantification must be fixed in advance to be “standard” in the sense of second order

48 logic. Whatever Tarski’s precise intent, the point remains that the strategy laid out in his remark does not block the relativist’s move. Here, the existence of a submodel version of LST allows a stronger response to Tarski’s objection: the charge that the relativist has changed the interpretation of V turns out to be groundless.

Now, this observation, though ostensibly relying on the existence of a submodel version of LST, indirectly sheds light on the numerical version as well: for in the numerical version of the proof found in 1923a, Skolem uses the infinite sequence of positive integers simply as a convenient representation of an arbitrary denumerable domain. An enumeration of such a domain, after all, is a one-one correspondence between the domain itself and the positive integers. The proof of the numerical version of LST presupposes the existence of an enumeration in order to eliminate the application of the Axiom of Choice found in the original proof since the domain can be weU ordered, the choice of individuals can be defined by an explicit criterion. (Skolem also notes that this method of proof need not employ

Dedekind’s impredicative definition of the number system as the intersections of aU

“chains”: but this, he remarks, can be obtained even without choosing the positive integers as the domain of the reduced model.) It is true, on the other hand, that the membership predicates and other set-theoretical relations are “suitably determined”

(Skolem’s words) to hold in the denumerable domain (whether numerical or not), and this in general wül mean that the relation designated as membership wül not be a simple restriction to the reduced domain of the relation in the original model.

Yet one must wonder whether every such modification is aptly described as a reinterpretation. The membership relation of the reduced model wül have the same properties as that of the original one; in a paraUel manner “sets” wül maintain the

49 same characteristics. This is not an epistemological point about the presumed lack of possible evidence that could help one to discern between ‘"living” in a numerical as opposed to a standard model of sets; the question is whether such a distinction would be in order at all, with respect to the set-theoretical consequences. Suppose that, by investigating the numerical model, we came upon a new property of the membership relation, expressed by a statement in the language of set theory.

Benacerraf, for one, acknowledges in passing that countable models of set theory are used in practice as “testing grounds” for set-theoretical conjectures, given their more manageable structure. Now the problem we are discussing is the following: supposedly, the relation in the model we have been investigating is not quite membership — it is a numerical “reinterpretation” of the membership predicate.

But does this restrict the validity of the statement we have found to the class of numerical models? By the LST, the answer is no (at least if that new property is expressed by a first order formula): if the statement about membership is a logical consequence of the axioms of ZFC, we are allowed to “generalize” it to the larger, standard, non-numerical models. To clarify: suppose that M” is a countable model of

ZFC elementarily equivalent to M, a standard (therefore not countable) model We know that, if M exists, then we can find such an M*, by the downward LST. Now suppose that we have an argument showing that M’ >= 0, where $ is a sentence in the language of ZFC. Then we also have M >= . If the statement expressing the property in question is a consequence of the axioms of set theory in all denumerable models, we have learned something new about membership, with no need of further qualification of the interpretation.

Of course, one should take care to notice that, speaking rigorously, the

50 numerical model constructed by Skolem for the purposes of the relativistic argument can only ensure that the first order properties of the membership relation are satisfied (to be sure, this qualification applies to all models constructed by Skolem for this purpose: in this respect, the numerical model is no different firom the submodels obtained via the Axiom of Choice). The restriction is significant, because there are crucial mathematical properties that seem to be formulated most naturally in a second order language. With regard to membership, weU-foundedness is such a property: it is not properly “captured” by a first order axiomatization. More specifically, an argument is available to the effect that any first order formula expressing the weU-foundedness of a relation R (the Axiom of Foundation in ZF set theory is such a formula, asserting the well-foundedness of membership) can be satisfied in a model where R is not well-founded (the argument is an application of the compactness theorem; see Shapiro (1991), sect. 5.1.4).

Applying this observation to the case in point: it might be that the numerical relation interpreting membership in the Skolem model (what we have previously called the “reduced” model) satisfies the axiom of Foundation (expressed by the usual first order formula) without being itself a well-founded relation. Then it could not be said that this model preserved “all” relevant properties of the membership relation. It is not clear (to me, at least) that this need be the case (there might be numerical models where this does not happen), but the possibüity must be considered, and it is clear, on the other hand, that Skolem does not dwell on it. This raises, first, an interpretive question: what motives would prevent Skolem firom addressing the possibility just envisioned - that is, the possibility that key properties of a set theoretical model might be more properly formulated in a second

51 order language? Of course, this is part of a broader and crucial issue concerning the development of logic and the rise to "supremacy of first order logic—a process in which Skolem played a primary role. As noted, however, the papers espousing the cause of relativism do not address the difference between first and second order logic directly. The task here will be, therefore, to reconstruct an argument on Skolem’s behalf to the effect that considerations (in particular, further restrictions) based on the use of second order logic should not be brought to bear upon the relativism problem. There is further the substantive question of whether the use of second order logic should be advocated to provide a philosophical “solution” to Skolem’s skepticism, independently of what Skolem might or might not have thought. This question remains open to this day, and, after the seminal discussions in the first third of the century, has only recently returned to the fore.'^

There may be a different angle on this matter. Perhaps the feeling is that the step (3) in the relativistic argument (see p. 46) is scarcely inteUigible when made in reference to a numerical domain. Obviously, the point urged by the relativist at (3) is that the set designated as nondenumerable is actually denumerable on some other criterion. But it may not make clear sense to say, of the object designated by P(w) in the numerical domain, that it is intrinsically denumerable, since that object is by construction a number, and so not the kind of thing that can be intrinsically enumerated or not (this point is raised by C. McIntosh (1979)). The relativist would be guilty, firom this perspective, of committing a category mistake. The idea is that a set is the kind of thing that can be denumerable or nondenumerable as a matter of

" See Shapiro (1991) both for a compendium and a new treatment of the issue.

52 its internal properties and structure, while a number is not.‘“ The point is not well taken, however. The claim pressed by the relativist at (3) is that there will be an enumeration of P(w) in the case at hand in the sense of‘enumeration’ provided in a more “inclusive” model of the axioms of set theory. The fact is that the object designated by T(to)’ will have elements"'* (in the sense of membership^, above); the question is whether there are in fact denumerably many of these elements*'*. And the answer to this question, crucial to the relativistic argument, is that there are, although of course they cannot be counted by an enumeration'*, which itself cannot be a member of the domain of the numerical model, on pain of contradiction.

(Essentially the same observation is made by McIntosh (1979), 324, although he believes Skolem himself would not have seen the matter in these terms: his relativistic argument, according to McIntosh, would have been open to the objection presently under consideration.) In other words, the relativistic argument does not trade on the presumption that we have available to us, as it were, a notion of essentially denumerable or nondenumerable objects, much less a notion of the ultimate internal structure of an object (or at least of a mathematical object).

Although Skolem does not articulate this point very exphcitly in (1923a), it seems clear firom the text that he does not see any relevant distinction between the numerical version and the submodel version of LST, as far their philosophical consequences are concerned. Either one will serve the philosophical point of the relativistic argument. Hao Wang notes (in Skolem 1970, p. 17)) that Skolem’s

One might object that this is literally false if numbers are regarded as a special sort of objects suppUed by set theory; but Benacerraf argues in another paper that we should not view them that way, and I tend to agree. Anyway, the issue is beside the point.

53 published papers are designed as reports of work in progress, and therefore closely

match the development of his work. If so, upon observing that the relativistic argument is presented in (1923a) rather than in (1920) - where the submodel proof was already given — we should suppose that Skolem saw the new numerical proof as strengthening the case for relativism, if anything. As noted, the difference between the two is primarily that the latter dispenses with the axiom of choice. Although the nature of Skolem’s attitude towards the axiom of choice is a matter of some dispute

(with some, like Fenstad (1970) and McIntosh, maintaining that Skolem was philosophically suspicious of the axiom), 1 think we should not count him among those who would oppose the axiom on philosophical ground (on this subject, see next section). The view 1 attempt to sustain is that Skolem conceived of the axiom of choice in a manner that inextricably linked the axiom to the interpretation of quantification, as suggested by Groldfarb (1979, p. 358; again, see next section). Put into this perspective, the progress intervening between the two proofs could not have been, for Skolem, simply that a problematic set theoretic assumption had been shown unnecessary. Instead, anticipating a theme that wiU deserve a clearer discussion in the next section, Skolem must have looked at the issue rather in the foUowing manner; his first proof, in (1920), shows that expressions containing (what we would caU) first order quantification can be interpreted as involving choice functions over a given domain, and that any such domain need be only at most denumerable; the second proof, in (1923), shows that the choice functions presupposed are in efiect of an “elementary” nature. The point is that the expressive power of first order quantification, so to speak, is always “approximated” by purely numerical constructions (Goldfarb, cit.: “with respect to satisfiabiUty, first-order

54 propositions cannot distinguish their intended objects firom the integers”).

Thus we must register here a substantial difference between the historical

Skolem and the “reconstructed” skeptic Benacerraf calls ‘Skolem*’ (1985). The aim of Benacerraf was to render the strongest and most plausible reconstruction of the relativistic argument(s) of (1923). Skolem* is the result of this strengthening process. He needs the Skolemized model NT to be transitive, because otherwise the set P(w) could after all be in NT, and to preserve the interpretation of the membership symbol, so that RT wül plausibly be seen as “saying” something about sets—Skolem*, therefore, will not favor the numerical reinterpretations proposed by the real Skolem. To be sure, part of the explicit purpose of the reconstruction is to present a position (Skolem*) that was rejected by the actual Skolem in 1923, but embraced later on (as evidenced by the 1941 paper, among others). However, precisely in order to stress this contrast between an earlier and a later phase of his thought, Benacerraf relies upon the methodological assumption that Skolem* would present an argument about the matter in much the same way the actual Skolem might have done, or anyway would in fact have done had he given more careful consideration to the course of the phQosophical argument. It is this assumption that seems questionable (of course, none of this implies that Benacerraf s reconstruction does not provide the Skolemite skeptic with a strong, perhaps even a better argument).

The point of view expressed by Benacerraf is that axiomatic set theory comes to us as the reconstruction of a firagment of pre-existing mathematical practice; the theory answers to pre-theoretical intuitions of what a set should be hke. These intuitions apparently include a rough understanding of sets as a certain type of

55 mathematical object, rather than just as a classificatory notion: we presume that the theorems of set theory wiU be true descriptions of this world, as it were. What this means, precisely, is as yet vague and undetermined. But somehow following from these presuppositions, Benacerraf suggests, is the belief that set theory has an intended interpretation, a privileged model-theoretic domain which is, in some sense, what the theory is ultimately about. Note that the attempt here is to introduce the notion of an interpretation of a (formal) theory as preceding the process of formalization itself In other words, the suggestion is that the relevant interpretation of the formal theory-the intended one-is primarily dependent on an understanding of the concepts articulated in the theory, an understanding that is antecedent to the theory itself

Of course, a skeptic is likely to oppose such a conception of the significance of formalization. According to Benacerraf a skeptic hke Skolem* will tend to treat theories such as axiomatic set theory as “uninterpreted formalisms,” for which there is a genuine open question which of the many possible models is the most adequate; against this, Benacerraf suggests that we question the idea that such theories are uninterpreted formalisms. They are associated with, perhaps even inseparable from, an interpretation. But since an axiomatic theory manifestly can be treated as an uninterpreted formalism after the axioms have been chosen and the language made sufficiently rigorous, the question becomes that of determining at what point in the process from naive intuition to formalization the “natural” interpretation comes into being. Benacerrafs tentative answer is that this natural interpretation must originate somehow before formalization, in what he calls “mathematical practice”; when he says that “mathematical practice determines or embodies the

56 meaning of mathematical language [including the language of formal theories],” he is thinking of “mathematical practice” as that—whatever it is—which come before the formal systematization of a theory.

Thus the antagonistic pair of characters staged by Benacerraf is this: on the one hand, the practising mathematician (Benacerraf*?), with his predetermined understanding of set theory, deriving &om “practice”; on the other, the skeptic

(Skolem*), who attempts to undermine the belief that axiomatic set theory has a determinate intended meaning, and fostering a conception of theories as uninterpreted formalisms. But to show that indeterminacy lurks in the intended intepretation, simply changing intepretation would not do, for the skeptic. For this would, in general, show only that the theory in question may have different meanings, not that the intended meaning is insufficiently understood. Now, the numerical model envisioned by Skolem seems to require precisely such a change in meaning: to start with the numerical model amounts, for Benacerraf, to saying that there is no intended interpretation in the first place, and in such a case the argument results in a standoff, with the skeptic and his opponent clashing on the acceptability of the premises. Therefore, Benacerraf takes great care in formulating the skeptic’s challenge on the basis of the submodel version of LST, where the intended meaning seems to be preserved.

Benacerraf does not really argue in (1985) that an intended interpretation of set theory, with the characteristics mentioned (Le., originating firom mathematical practice), exists and is available for us to examine. The moral of the reconstruction is that Skolemite skepticism (after the sort of refinements proposed in the paper) need not affect our belief that there is an intended interpretation of (among others) axiomatic set theory, but does force us to seek a somewhat novel explanation of

interpretation, a different (different firom model theory, that is) understanding of

what set theory is about. At this stage, then, it is really a presupposition of

Benacerraf s analysis -- not one of its results -- that we have an intended interpretation of axiomatic set theory. Working under that presupposition, the submodel version and the numerical version of LST markedly differ in their philosophical significance. The fact that they do not so differ for Skolem is evidence that his point of view was different: Skolem* is not just a more clever philosopher than Skolem; his philosophy is different. Skolem did not think that axiomatic set theory, or any axiomatic theory for that matter, had an intended interpretation. Yet it would be wrong to think that he regarded axiomatic theories as uninterpreted deductive systems, purely syntactic machines. Rather, I think the correct way to interpret Skolem’s remarks and objections would be this: Skolem thought that all such theories had an intuitive meaning but that their intuitive meaning was to be analyzed in a distinctive way (certainly different firom the one offered to us by present-day model theory). Furthermore, he appeared to consider set theory as a sui generis region of mathematics-a foundational theory that should be treated somewhat differently firom the rest.

Foundations and axiomatic theories

The big picture

I have mentioned that it is generally assumed that Skolem retracted at least part of his objections against axiomatic set theory in his (1941) [Sur la portée du

58 théorème de Lôwenheim-Skolem]. In fact, the positions expressed respectively in

1941 and in 1922 are seen to be much closer in inspiration once we interpret the early critical claims against the background of Skolem’s own views on the foundations of mathematics. These views change little between the two papers.

Moreover, as stated previously by way of promissory note, we must understand what these views are if we are to understand the nature of his skeptical objections to set theory.

Skolem operated in the conceptual tradition of the algebrists of logic (as pointed out by McIntosh (1979)). This school of thought had risen to prominence with the work of George Boole in the mid-nineteenth century; it was founded on the behef that logical operations are in essence algebraic operations of a special kind.

Boole himself attempted to construct an abstract algebra that could be given a logical reading once the operation symbols were interpreted as being defined over classes or propositions—in other words, as, respectively, intersection (logical product, conjunction), union (logical sum, disjunction) and complementation (negation). The idea was to characterize a domain the elements of which could be taken to be, indifferently, numbers, classes or propositions as the operations defined over such elements were interpreted correspondingly. Boole’s work exemplifies the efforts spent in order to preserve this abstract generality in the face of what seemed clear disanalogies in the behavior of his operators under the different interpretations he envisioned for his formalism. (See Kneale and Kneale for some details on this subject). At any rate, a peculiar conception of logic emerges from Boole’s pioneering work. Logic, as the domain of the “laws of thought,” is on a par with arithmetic, conceived as the domain of the laws of number and counting. Logic is a calculus or

59 an algebra, defined as the range of application of its constitutive operations: logical truths can be expressed as equations, and logical rules of inference are then regarded as rules for calculating equations, on a par with the numerical cases.

Logical operations, whether on propositions or on “concepts” (it is characteristic of this approach that the distinction between these two be somewhat obfuscated), should be thought of as no different in nature from operations over numbers, or areas, or vectors, etc.

In the German-speaking world, the tradition inaugurated by Boole was represented by Schroder, author of a textbook treating logic algebraically,

Lowenheim, who first proved a version of the LST, and others. It is useful to compare, however briefly, the conception of logic fostered by the algebraic tradition with that espoused by Frege and the logicist tradition, for the differences are remarkable and shed an instructive fight on the way Skolem must have approached, later on, the foundations of logic and mathematics. For Frege, logic consisted of the most general truths; the truths of logic, expressed in a special language, are descriptive, in the sense that they have (what Frege considered as) a well defined range of application: everything there is. Since the subject matter of logic is the most inclusive, it does not make sense to consider a variety of domains of application: any other domain would be a restriction of the proper one. Notions such as satisfiability and validity (in the modern, model-theoretic sense) make no clear sense in such a perspective. A certain symbolic apparatus, that of the Begriffschrift, was needed to express such truths in a perspicuous and rigorous way, but it would not be entirely apt to regard the symbolism as a formal language, that is, as a language the interpretation of which is yet to be determined and can be changed to

60 suit needs and purposes. Thus, the nature and degree of “rigor” afforded by the

Begriffschrift was, from Frege’s point of view, quite different from formal rigor as the

“hygiene of mathematics.”'® For Frege, rigorous, gap-free derivations were necessary to exhibit the logical nature of mathematical propositions, i.e., the fact that they depend solely on truths of logic. For the “higienist,” the concern is just to avoid linguistic entanglements that might conceal contradiction or engender ambiguity; the formalism aids in communicating things more effectively, but the mathematical content of the proof remains independent (this is the sense, for instance, in which one feels the need to “clean up” a proof). In designing the pure notation of concepts, Frege was after univocity and clarity of meaning - not the universal applicability of a contentless formalism.'' Or, perhaps more accurately: for Frege, logic was universally applicable, not, however, because of its “indifference” to the many different contexts to which it could be applied, but rather in virtue of the fact that it consisted of the most general truths - truths about everything, as it were.

The salient characteristic of algebraic structures, on the other hand, is that the axioms defining such structures are open to quite a large variety of interpretation. The variability of interpretation is the characteristically non-

'® This quip is by Kreisel, who uses it in his (1967) to disparage the philosophical prejudices exhibited by those he calls “positivists” (close relatives, for what I understand, of the formalists).

'' This may partly explain the rather astonishing fact that Frege was able to invent his technical “instrument” and put it at the same time to such intensive and far-reaching use. For Frege, there was no conceptual hiatus between the instrument itself and its application to the reduction of mathematics to logic: he must have believed that someone reading the Begriffschrift would grasp its philosophical point seamlessly, as it were. This kind of reader turned out to be exceedingly rare.

61 Fregean element. Viewing logic as algebra leads one to countenance several,

potentially very different domains in which the principles of logic are satisfied. Of

course, this fact does not by itself make the algebraic approach incompatible with

Fregean logicism; the logicists would agree that logical truths are, characteristically,

true everywhere, but "everywhere" for the logicist is the designation of a specific

domain — the one domain of logic, in fact. The algebraist, on the contrary, does not

single out a particular domain as the proper one for logic. It would not make sense

to do so, just as it would not make much sense to single out the proper domain for

the group axioms (there is none more proper than any other satisfying the axioms).

This explains why, in the perspective of Schroder and the algebraists, the “universe

of discourse” is given contextually: in particular, the meaning of the symbol ‘1' in the

Boolean algebra (i.e. the symbol denoting the domain of interpretation in its

entirety) will thus vary according to the domain of interpretation, though always

respecting the laws of the “identical calculus” (Schroder’s designation of logic).

Skolem, preceded in this perhaps by Lowenheim, uses the designation “identical calculus" to refer to logic. But Schroder (who presumably introduced the term) made a distinction, as the very choice of name -- ’identical’, rather than ‘logical’, calculus — may indicate. The identical calculus is “an auxiliary discipline that precedes logic proper... and is of purely mathematical nature” (see the editors’ note in Van Heijenoort (1967), fii. I, p. 515). We see here stated in the clearest terms a deep-seated motive of disagreement between the algebraists on the one hand and Frege on the other: according to the former, there are parts of mathematics that are prior to logic - -that which Frege would vehemently deny. One may conjecture that the concepts constituting what is called “intuitive model theory” (see below in the text) would in fact embody those parts of mathematics that are prior to logic proper. A different question, which I do not pursue, concerns the motivation of Schroder’s correction of Boole in this regard (i.e. in the way to construe the notion of a universe of discourse). From a quote reported by Frege in his critical review, it looks as if Schroder saw risks of a paradox involved in conceiving (as Boole had done) the universe of discourse as one all-inclusive domain. Perhaps Schroder thought that a restriction against the all-inclusive universe of discourse would be

62 Frege finds this very idea clearly distastefid, and the absence of a universal domain

(as per Schroder’s proposal) its most distressing aspect: “[w]hereas elsewhere logic

may claim to have laws of unrestricted validity, we are here required to begin by

delimiting a manifold with careful tests... Hence in a rigorously scientific statement

one would always have to give an exact specification of the manifold within which

the inquiry is being carried out.” (1895, p. 215). From the Booleans’ point of view,

the “generality^ of logical laws. Le., the fact that they could be interpreted equally

well as being about, e.g., classes, propositions or integers, was of paramount

importance. So this kind of generality had to be preserved, in spite of Boole’s early

difficulties in accommodating the various interpretations (as evidenced by the efibrt

he made to find plausible logical counterparts for certain numerical operations, e.g.

division). From Frege’s point of view, instead, this concern for generality is an

indication that conceptual confusion festers in the Booleans’ systems of logic; the

variability of interpretations of the algebra is akin to semantic ambiguity, and logic

properly so called cannot even start unless the content of the expressions in the

logical vocabulary is completely determined and disambiguated. Indeed, it was not

until Skolem’s work that the logical interpretation, spelled out in terms of

prepositional functions, was singled out as a well-defined subject of study in its own

right. So Frege had good reason to believe his approach to be superior at least in

clarity and intelligibility to that of the algebraists.

On the other hand, this fundamental idea of the algebraic conception - variability of interpretation, the validity of logic over a plurality of domains - has

necesssary to preserve the soundness of the fundamental notion of logic as algebraic operations.

63 shaped the understanding of logical semantics which one sees embodied in contemporary model theory. However, it should be noticed that the conception of logic we find in the followers of the algebraic school does not include a concern for logic as a deductive system, or for attendant notions such as proof and provabüity.‘®

There is consequently no real grasp of the interplay between the deductive notion of theoremhood and the semantic notion of satisfiabüity which is central to contemporary treatments of formal logic. The principles of logic apply to every domain, but do not constitute a “formalism” in the modern sense (after Hilbert), that is, a system amenable to purely syntactic consideration. For the same reason there is little attention paid to axiomatics, as a way to formalize and treat logic. In fact, from some of the observations Skolem makes about axiomatic theories in general in

(1923a), it sounds as though he would have taken issue with the idea that logic could be treated axiomatically (more on this later).

Anyway, since the deductive aspect is by and large neglected, the typical problem is posed by the algebraists in the form of a question about the satisfîabüity of an expression (or of expressions of a given kind). Lowenheim’s paper containing the first intimation (or the first version) of LST is a good case in point. Proofe (such as Lowenheim’s or Skolem’s, for example) are what we would today call semantic

A cautionary note here: ‘deductive’ is used in this paragraph in the contemporary sense — a sense to which Skolem (at the time we are considering, i.e., the early Twenties) would not have subscribed. 1 am not saying that logicians of the algebraic school had no working concept of ‘deduction’, or had no concern for proving things. It would be correct to say that they held to a notion of proof and deduction which is considerably different from what is usually regarded as a proof today. We make a distinction (we think we can make a clear distinction) between a proof (an object in a formal system) and a model-theoretic argument (such a semantic counter­ example). For Ldwenheim and the early Skolem, on the other hand, proofe were all model-theoretic arguments.

64 arguments (or counter-arguments): it is shown that from the satisfiability of

something there follows the satisfiability, respectively the unsatisfiability, of

something else. An important consequence of the difference between the

algebra-of-logic school and (later) formalism seems to be the following. As noted, the

algebraists do not regard the expressions to be evaluated as uninterpreted, that is, in itself devoid of meaning. Today we routinely think of the expressions of a calculus

(such as expressions in the Schroderian ‘calculus of relatives’ considered, e.g., by

Ldwenheim) as having whatever meaning is supplied by the model-theoretic interpretation. We start, or we proceed as if we started, with an uninterpreted expression which is part of a formal system; the interpretation is grafted onto an expression by the semantic apparatus — assignment functions, the relations of denotation, satisfaction, etc. The uninterpreted expression is regarded as an object with no presupposed meaning. In the algebraic tradition, however, the starting step of this process (the expression as a syntactic object, completely devoid of meaning) is not clearly delineated. The situation that must have presented itself to these authors is rather like the following: formulas express their own satisfiability in whatever domain is under consideration—they “say,” in other words, that certain relations hold in the domain and/or that the domain is closed with respect to certain operations. To put the matter in thoroughly contemporary, and perhaps more perspicuous terms: expressions have an intuitive meaning (contra certain stripes of formalism), but not a unique meaning - unless, of course, they constrain their own interpretation in such ways as to be satisfiable only in domains of a certain kind.

The intuitive meaning associated with a formula is determined by the relations that have to obtain in the given domain in order for the formula to be true; notions such

65 as ‘relation’, ‘obtaining', ‘individual’, ‘domain’ are taken as understood: they are the intuitive basis in terms of which any formula is to be interpreted. Since the domain can vary, the meaning of a formula is not really fixed, though. Similarly, many different structures could aU be groups, though, in a sense, for very different reasons: typically, then, group theory is not considered one of those mathematical theories that have an intended, privileged interpretation.

It would be fair to say, then, as McIntosh does (1979, 320-321), that the algebraists of logic took themselves and other mathematicians to be working in an

“intuitive model theory,” the basic concepts of which are the ones just mentioned.

What would later be (and is today) called ‘model theory’ is based on the same concepts, but after Tarski’s make-over these are spelled out in set-theoretical terms

" on the implicit assumption, it seems, that set-theoretical notions are more basic.

But this assumption in itself did not unequivocally imply, from the point of view of the algebraists, a committment to a set-theoretical foundationalism. Set-theoretic concepts are deployed in a background “theory” which is rather undistinguishable from logic itself; at least some set-theoretic notions are just needed to formulate even the simplest mathematical questions. Most obviously, the basic notion of domain employed in these early semantic investigations is a set-theoretic notion. In the contemporary way of framing these issues set theory enjoys, at the very least, an important explanatory priority with respect to model theory: the intuitive plausibihty (whatever it may be) of model-theoretical arguments rests mainly on the plausibility that the reformulations of such arguments have when couched in terms of the preferred notion of set. On the other hand, from the standpoint of the

Boole-Schroder approach, the situation would have looked rather differently:

66 Model theory was an independent branch of mathematics. It had its own intuitively acceptable notions of domain, model and satisfaction. These had for the most part developed out of investigations in geometry and algebra, especially the latter. (McIntosh (1979), ibid.)

In other words: within the conceptual perspective of the algebraists, there is a certain separation between two distinct strains of thought associated with set- theoretic notions, which we may perhaps label “set theory as model theory” and “set theory as the theory of sets.” To the first variety belong the set-theoretic foundations of (what we would today call) model theory: domain, model, satisfaction etc. The nature of such concepts is intuitively the same as that of set; the intuitive idea of collection “in general,” of putting together a set of things sharing some arbitrary characteristic, is prima facie at the roots of these notions (domain etc.) as much as of the Cantorian notion of set. But, for the algebraists, such notions as that of domain are necessary to even just formulate basic mathematical and logical questions about the satisfiability of given statements. Therefore, these notions are thoroughly enmeshed in the “background theory” in which mathematics takes place, and are essentially undistinguishable from logic itself - logic being in essence a kind of generalized model theory. There is a second way to conceive of set-theoretic notions: there is set theory as the theory of a specific variety of mathematical objects, as a “theory of sets,” so to speak. Such a theory simply concerns a particular variety of algebraic structures satisfying certain relations, and can be investigated through the ordinary means of mathematics: we can ask what domains satisfy it, etc. This kind of treatment had already been applied to theories in geometry and algebra, and was in eSect a driving intuition behind the axiomatization of many such theories. Axiomatic theories, in other words, seemed especially suited to this

67 kind of early model-theoretic investigation.

The distinction does not become apparent, however, until a clear formulation

of set theory as a theory of sets is available. In order for this to be the case, it is not

sufficient to have an explanation of the concept or concepts of set deemed relevant to

mathematical practice. No matter how accurate a conception of set one could find in

Cantorian set theory, for example, it was perfectly legitimate to take work on sets in

the Cantorian spirit as an investigation, conducted on a purely intuitive level, into

the ultimate constituents of mathematics. Naturally, some problems - such as perhaps the independence of the continuum hypothesis — could not be fully

understood and developed in the Cantorian (non-axiomatic) firamework. But an impressive amount of set theory, after all, had already been mined before Zermelo’s

axiomatization (a fact that Zermelo, of course, has no qualms acknowledging). The axiomatization, though, does something diSerent: it enables one to deal with set- theoretic questions in the same way as one deals with questions about axiomatic

Euclidean geometry or group theory — suppose that a domain satisfies these axioms, is closed under these operations, etc. So it is that an axiomatization of set theory can not only offer a more rigorous method to study sets, but also suggest a new conception of the proper place of set theory itself.

Skolem’s view of axiomatic theories

Skolem is more keenly aware of the distinction of the previous paragraphs than his predecessors because he has before his eyes Zermelo’s set theory, which he regards as a typical instance of the second variety. Le., a theory of sets: it is an axiomatic theory, therefore, firom Skolem’s point of view, “just another”

68 mathematical theory. But Skolem thinks that Zermelo’s theory is also expected to

do the work that he would assign to “real” set theory, Le., to a theory of models. This

is the conviction he expresses when he claims that set theory is, considered

according to its “real” conception, a logical theory. His thought on this point can be

found in a short and somewhat neglected passage that opens his anti-Zermelo paper

(1923a, p. 292). The proper question to ask of an axiomatic theory, Skolem imphes,

is in what domains the axioms are satisfied. But when the axiomatic theory is set

theory, this question takes on a strange flavor, because set theory also concerns the

“general” notion of domain that is presupposed. Outright circularity can be avoided,

if we simply treat set theory as any other axiomatic theory, by selecting an arbitrary set as the given domain of interpretation, calling its elements “sets” and thus satisfying set theory in a “merely verbal sense,” as Skolem says later. But then, he

concludes, “when founded in such an axiomatic way, set theory cannot remain a privileged logical theory; it is then placed on the same level as other axiomatic

theories” (emphasis mine).

I think that Skolem here has in mind an argument that might go as follows:

The “real” set theory is the general, all-encompassing foundation of mathematics. It is presupposed in mathematical practice by the nature itself of “logic,” Le. by the basic operations that cannot be avoided in mathematical thought. The study of any

mathematical theory involves these notions prominently, because the typical

mathematical questions a theory poses are things hke: what domain(s) satisfy^ it?

What structure do such domains exhibit, and what variety or varieties of structures

are suitable for this? And so on. The model-theoretic treatment is the basic approach for any mathematical theory. But real set theory clearly encompasses a

69 notion of domain, and (almost automatically, it seems) a notion of a domain’s satisfying certain relations, etc. So this kind of conceptualization cannot really be treated itself “just like any other theory,” on pain of “circularity.”

There is, naturally, a significant issue concerning the exact purview of the intuitive model theory countenanced by the algebraists of logic: what sort of notions were part of this framework? McIntosh correctly believes that a proper evaluation of

Skolem’s arguments in favor of relativism hinges on just such question; however,

McIntosh’s answer to the question is objectionable. In order to see exactly why, we must now consider the peculiar ways in which the philosophical underpinnings of the Boole-Schroder school are incorporated into Skolem’s work.

The role of Choice

From a technical standpoint, Skolem greatly clarifies the logical notions involved in the argument laid out by Lowenheim. In the first place, although he adopts the Schroderian terminology of the “identical calculus,” Skolem does not get bogged down in the ambiguities deriving from the early algebraists’ preoccupation that their notation be seen as suitable to a great variety of interpretations-logical, numerical, set-theoretical, etc. From the start he resolves to deal with propositions, not with “equations” of relation terms. Although ‘proposition’ itself is defined in terms of “relative coefGcients in the sense of Schroder,” he states all relevant results in (1920) unequivocally in terms of “first-order propositions,” according to the translation in Van Heijenoort (1967). (On the other hand, the identity sign is still used in the double role of “equivalence ”-a logical relation—between relative coefficients and numerical equality.) This allows him to bypass the complexities of

70 the Schroderian notation of relatives with subscripts in favor of one that can be

easily interpreted into the more perspicuous functional notation introduced by the

logicists. (Wang notes, in his (1970) introduction, that Skolem’s main papers came

after a period of careful study of Principia Mathematica.) The reward is a deeper

philosophical insight about the same result. In particular, Skolem establishes a firm

link between the set-theoretical principle of choice and the semantic interpretation

of quantification, and this connection in turns sheds light on Skolem’s journey

toward relativism.

Within the algebra of logic perspective, expressions of quantification are interpreted as shorthand for (possibly) infinitely long formulas. A universally

quantified formula corresponds to a (possibly infinite) conjunction or “logical product” of its instances, and analogously an existentiaUy quantified formula corresponds to a disjunction or “logical sum” of its instances. The quantifiers can then be eliminated by expanding, i.e. spelling out, the appropriate conjunction or disjunction. Obvious complications will arise in the general case in which a formula has more than one quantifier or sequences of alternating quantifiers (as is the case for a prenex formula): then every instance in the first expansion (which eliminates the outermost quantifier) wiH in turn have to be expanded, and so on.

The proof of LST in Lowenheim’s original paper (1918) proceeds by expanding an arbitrary quantified formula into actual disjunctions and conjunctions, and then showing that the resulting expansion (which is purely “truth-functional,” we might say) is satisfied in a numerical domain just in case the compressed (i.e. non-expanded) quantified version of the formula is. The project, though, was hampered by the author’s adherence to Schroder’s unwieldy notation, in which

71 atomic sentences are symbolized by relation symbols and subscripts play the role of variable places as well as that of individual names. An instance of an n-ary atomic formula is, in this notation, a relation symbol (a capital letter) with n subscripts. As mentioned in the previous paragraph, things become rather complex in the case of formulas in prenex form with dependent quantifiers (ie. multiple blocks of difierent quantifiers). A formula of the form 'For all x, there is a y such that F requires the existence of a particular y for each given x: the choice of such a y, in other words,

“depends” in a certain way on the particular x at hand. In order to express this dependence, Lowenheim resorted to using multiple layers of subscripts, i.e. to subscripts of subscripts. And in the general case the expansion will produce infinitely long conjunctions and disjunctions (it certainly did under the assumptions of the theorem Lowenheim set out to prove, in which we consider a formula satisfiable in an infinite domain). Skolem dispenses wdth such complications by using choice functions to encode the information contained in the existential quantifiers that are within the scope of universal quantifiers. Each such function selects an individual for which the property F holds; the value of the function depends upon the individuals assigned to the variables bound by the preceding quantifiers (Le., the quantifiers having wider scope). Every existential quantifier can thus be replaced by a choice function of this kind. Once this is done, the rest of the argument (i.e. the proof of the LST) seems remarkably natural

Of course, as Skolem explicitly notes, the stipulation that all existentiaUy quantified variables in an arbitrary formula can be replaced by Skolem functions of this kind presupposes in general the application of the axiom of choice: Skolem functions are in effect choice functions, and their existence is not always guaranteed unless one assumes the axiom of choice.

As W. Goldfarb (1979) documents, the emendation to Lowenheim’s method proposed by Skolem represents a significant step toward the now familiar understanding of first-order quantification. Goldfarb shows that the understanding of quantifiers that is second-nature in today's model theory was arrived at gradually.

The logicists were able to formulate the logical rules of quantification in much the same form as they appear today, but, as we have seen, quantifiers are interpreted by the logicists as ranging over a fixed domain: model-theoretic investigations, which involve varying the range of quantification, are not contemplated. The algebraists, on the other hand, see quantifiers as abbreviations for non-quantified expressions.

Quantifiers are always in principle eliminable, so there is no appreciation of the expressive power of quantificational logic in itself The possibility to vary the domain of “interpretation” is built-in, so to speak, but the conception of quantifier logic as a uniform deductive system that can be used to formalize several mathematical theories remains foreign to the algebraic approach.^ Skolem’s contribution bridges to some extent the gap between the two approaches.

Quantifiers are treated model-theoretically, preserving the basic intuition of the algebraists. But instead of eliminating them, Skolem assigns a semantic value to quantifiers directly, by using set-theoretic constructs. The result is that one can use the logicists’ sophisticated notation while interpreting it in models much as the algebraists would have done, and the notion of model-theoretic interpretation of a formalized axiomatic theory comes into full view.

See Goldfarb (1979), p. 356.

73 So much for the broader historical significance of Skolem’s insight. Now two

questions that concern us more closely are: first, what did Skolem himself read into

his analysis of quantification? Second, how do the technical aspects of the result bear upon Skolem’s set-theoretic relativism?

To begin with, for Skolem the improvement achieved over Lowenheim’s proof must have represented a vindication of sorts of the algebraic viewpoint: the lesson was that, while the crude idea of quantifiers as abbreviations of infinitely long sequences of logical sums and products had to be abandoned (on pain of running into

Lowenheim’s complications-what Skolem called his “detour through the infinite"), the idea that the logical import of quantification could be “reduced” to that of the basic Boolean operations (in a language enriched with names for choice or Skolem’ functions) could in efiect be maintained. Naturally, this required that choice functions be seen as a rather “ordinary” kind of functions, and Skolem does appear to regard them in that way.’^ The appeal to the principle of choice (see (1920), p.

By comparison, one may note David Hilbert’s attitude toward choice principles. The strict connection between choice on one hand and logical principles of quantification on the other was emphasized by Hilbert, too, who in fact wrote that “it turns out in the logical analysis carried out in my proof theory that the essential thought on which the principle of choice is based is a general logical principle, which is necessary and indispensable for even the most initial elements of mathematical inference” (Hilbert (1923)). But for Hilbert the main issue is to bridge the gap between two separate classes of mathematical statements, the finitist and the transfinite; he believes that the “bridge principles” serving this purpose are the laws of formal logic, and the axiom of choice is one of them. Formalization is essential to the project. One may conjecture that it is because of this that the value of a second-order logic is upheld in Hilbert-Ackermann (1938): a strong form of the axiom of choice, equivalent to the epsilon-axiom (or tau-axiom) of (1923), can be naturally formalized in second order language using only the logical vocabulary. Note, furthermore, that Hilbert intends his claim that choice is a logical principle as providing a handy retort to those who had challenged Zermelo’s assumption of choice in his axiomatization of set theory. As I note in the text, Skolem never entered this firay—not surprisingly, since he was not especially

74 257, for instance) and the introduction of choice functions are not accompanied by a broader discussion of the principle itself or of the more general philosophical issues surrounding it, and no hint is given to the fact that the principle had indeed been the subject of heated controversy.

Significantly, notions such as those of fimction, operation, etc., central as they are to what appears to be Skolem’s foundational enterprise, are not discussed firom a general point of view in the papers under examination. They are clearly presupposed as understood by Skolem: as such, they should be seen as an integral part of the intuitive background theory (the “intuitive model theory") in which

Skolem works. In the foundational scheme, such notions are the explanans, while more “abstract” set-theoretical (or even logical) notions are the explananda. This conception is peculiarly intertwined with the principle of the “priority” of mathematics upon logic, which is upheld within the algebra-of-logic tradition (see previous footnote on the “identical calculus”). Of course, it is an accepted tenet for this school of thought that logical operations are algebraic operations. However, we are not told exactly what algebraic operations are in turn, and what their nature is;

Skolem is no exception in this respect, and he is no more forthcoming with explanations. In light of his attitudes, though, it seems plausible to conjecture that, for Skolem, elementary arithmetical operations, and in particular recursive operations, are the prototypes of the general concept of algebraic operation. The basis for all mathematical as well as logical reasoning is what he caUs the “recursive interested in the status of axiomatized set theory. It seems to me an interesting fact that he would arrive at his conception of the principle of choice as a primary component of our understanding of logic “directly,” so to speak, that is, by considering the principle of choice solely on its own merits.

75 mode of thought.” Thus, he regards his own logical work on quantification as aimed at showing the pre-eminence of this mode of thought. He writes in (1923b)

Now what I wish to show in the present work is the following: If we consider the general theorems of arithmetic to be functional assertions" and take the recursive mode of thought as a basis, then that science can be founded in a rigorous way without use of Russell and Whitehead’s notions “always” and “sometimes.” This can also be expressed as follows: A logical foundation can be provided for arithmetic without the use of apparent logical variables, (p. 304)

It is certainly worthy of notice that the project described in this passage is, in

Skolem’s own assessment, similar in spirit to the approach of intuitionists such as

Brouwer and Weyl.^ This point deserves some careful thought, for a correct interpretation of the relativistic argument hinges upon it. Passages hke this might suggest that Skolem embraced the philosophical principles of constructivism advocated by those authors, and that therefore his misgivings about axiomatized set theory - hence his set-theoretical relativism - arise at least in part firom such constructivist attitudes. After all, a special concern with the existence of uncountably infinite collections is a constructivist mainstay, and of necessity: the

“ By ‘functional assertion’ Skolem means here a formula (built up firom atomic ones and truth-functional connectives) possibly containing free variables. Such a formula expresses that “a proposition holds for the indeterminate case” (ibid.). Later in this quote: ‘apparent logical variables’ are bound variables: the attribute ‘apparent’ indicates that, since they are quantified, they are no longer genuine “variables."

“ See Skolem (1929a), p. 217 (in Skolem (1970)): “I, too, wrote some years ago an essay [Le. 1923b] in which, independently from Brouwer and without knowledge of his works, developed similar thoughts, limiting myself to elementary arithmetic.” Note that even after becoming directly acquainted with Brouwer’s works, Skolem finds them difficult and “hard to read” (ibid.), and attributes his own understanding of intuitionism to Weyl and (the characterization given by) Fraenkel.

76 procedures that intuitively deserve to be classified as constructive do not yield more than a countable (and not necessarily infinite) collection of mathematical “things.”

The essential element of any constructivist account is a boundary to be drawn between those mathematical notions, objects, and methods that are acceptable (that is, can be explained in terms of constructive methods) and those that are not so (a constructivist account is essentially “non-promiscuous,” in other words). A given variety of constructivism may be more or less strict: the class of acceptable notions might include, say, only those natural numbers that are not “too large,” or all natural numbers and even the set of all natural numbers. No matter how generous the boundary, however, no constructivist account can include the “fiill” continuum, conceived as the completed totality of all real numbers. Yet Cantorian set theory, whether in its “naive” or axiomatized form, characteristically sustains the claim that such totalities as the set of all real numbers exist: the continuum is such a set. On this point, classical set theory and constructivism must then collide head-on. So, if

Skolem was in fact a constructivist, this would certainly explain his hostility toward set theory.

This interpretation is advocated by McIntosh (1979). Accordingly, Skolem harbored the conviction (characteristic of constructivism) that the existence of uncountable infinities, as borne out of Cantorian set theory, was itself dubious and problematic. Traditionally, this kind of generalized distrust toward set theory could be traced back to at least two distinct sources: (i) the awareness of paradoxes emerging from what seemed to be the “naive” Cantorian conception, which pointed to the need for careful formulation of set theoretical principles; (ü) the debate surrounding issues of definability, and the disinclination to accept such notions as

11 that of in principle M/idefinable functions and classes -- the latter being a strand of the more general philosophical debate on the “effectiveness,” the constructive character, of mathematical methods. These two sources of concern seem to be present, variously and originally blended and disguised, in many of the important authors of the first two decades of the century; the logicists, for instance, tend to manifest concern especially for problems relating to (i). Some of those trying to solve problems with regard to (ii) (for example, Weyl) may have been moved by concern for the paradoxes; others by more typically philosophical attitudes ahout the meaning of mathematics (Hilbert seems to exhibit both kinds of concern).

McIntosh sees Skolem’s worries as originating primarily firom the second line of thought. (They certainly had Little to do with the first kind of problem: nowhere does Skolem mention the paradoxes as a special source of difficulties for set theory.)

Skolem would then be a predicativist in the mold of Weyl, moved to his critique primarily by concern about the legitimacy of the powerful impredicative concepts that are at the basis of classical set theory. Note that this interpretation attributes to the claim of set-theoretical relativism a slightly different import than the one which is apparent on the “face-value” reading of the relativistic arguments. For, according to the more straightforward reading, the problem is not with the very existence of uncountable infinities, but only with the possibility of making good on this existence claim by means of axiomatized set theory. The gist of Skolem’s criticism, when the argument is viewed in this light, is that axiomatized set theory cannot deliver on its own foundational promises, so to speak. The main argument fi"om (1923a) as reconstructed in the previous section was of course leading to just such a conclusion (that reconstruction, moreover, seems to be essentially the same

78 as that on which Benacerraf (1985) elaborates.) According to McIntosh, on the other

hand, Skolem is a predicativist and a constructivist bent upon demonstrating the

illegitimacy of such Cantorian and Zermehan notions as that of an uncountable

infinity. The “intuitive model theory" to which he avails himself lacks the resources

necessary to justify the introduction of uncountable infinities:

Proving that a domain is denumerable, that there is a 1-1 function fiom it onto the natural numbers, falls easily within these [the intuitive model theory's] resources. Proving the non-denumerability of a domain. Le., the non-existence of such a function, does not. (McIntosh (1979), p. 321)

Consequently, Skolem would focus on the axiomatization of set theory because-in the manner of Hilbert—he views a formal proof given within an axiomatic system as the “last best hope” of set theory: it would be, given the constructivist prejudices and the looming presence of the paradoxes, the only way left to ensure (or at least make sense of) the existence of uncountably infinite totalities. McIntosh ascribes to Skolem a view of axioms as “implicit definitions” of a concept, capable of warranting claims about the existence of objects falling under a certain concept by an “indirect” route that avoids explicit postulation." ‘ Hence, the relativistic argument, by showing the “futility” of the axiomatic approach, impugns the very notion of uncountable infinities (or so Skolem must have thought).

Skolem’s writings (at least those of which I am aware) do not elucidate in any detail his conception of the role and significance of axiomatization-except, of course, for the oft-repeated claim that it leads to relativism. McIntosh’s interpretation on this point is therefore of necessity a “rational reconstruction” of (i.e. inferred firom) what sparse remarks are available. McIntosh’s observation on the implicit definition view poses an interesting question, but it is especially hard to accept owing to this fact: there is really no acknowledgment, even in later writings, that axiomatization and formalization can be evaluated under any other respect but satisfiability in a domain, which is what Skolem consistently worries about.

79 At times, Skolem does display an affinity for certain constructivist ideas.

However, the critical evidence of his writings counsels against the interpretation proposed by McIntosh; as a consequence, the constructivist tendencies manifested by the preference for what Skolem calls the “recursive mode of thought” must be interpreted in some other way. The view I propose (already briefly sketched above) is that for Skolem the recursive mode of thought is simply the realm where we find the clearest examples (the “role models,” in a way) of algebraic operations and methods. In other words, Skolem’s own contribution to the algebraic tradition in logic is based upon his conviction that when we talk of “operations” in algebra we have in mind, at best, something like the recursive operations of elementary arithmetic. Skolem, at least around 1923, views this as a foundation for mathematical thought than which nothing better can be found: all others-set theory, axiomatized formal systems—will fall short on one count or another

(relativism, for instance). Furthermore, any other attempt at providing a foundation for mathematics needs to presuppose, Skolem thinks, the notions and methods constituting the recursive mode of thought. According to a justly famous classification of Van Heijenoort, the algebraic tradition in logic is the bearer of a conception of “logic as calculus” that goes back to Leibniz’s exhortations to devise a calculus ratiocinator, by which complex conceptual problems could be solved in a completely determined and non-arbitrary manner. One might then say that Skolem appears to have interpreted ‘calculus’ in a rather strict fashion (though not implausibly), as including first and foremost effective rules.'®

^ It should be noted that this is not as trivial as might seem at first. Indeed, Skolem may have been in this regard at some variance with earlier algebraists, in

80 Independently from this particular question, anyway, Skolem does not exhibit particular sympathy toward the predicativists’ concerns until a number of years after his relativist views have formed. The clearest piece of evidence in favor of this assessment is his free application of the axiom of choice: if one were to worry about such things as impredicative definitions (in set theory and elsewhere), the axiom of choice would have to be one’s primary source of grief, for it clearly is on the face of it a “purely existential” principle - that is, it implies the existence of collections for which no conceivable definition is at hand. Yet if he was aware of this fact (as he should well be, for it had been the focus of heated and certainly public controversy by the time he approached the subject), Skolem did not mention it in the papers under consideration here.

Thus, it is difficult to subscribe to McIntosh’s claim that “[t]he Axiom of

Choice is also not available to intuitive model theory.” It certainly appears to be

particolar Schroder. Sluga (1987, SSffl) points out that, when Frege and Schroder exchanged attacks on their respective approaches to logic, each discounted the other’s work for supplying a “mere calculus,” but no true lingua characteristica — which would require, it was assumed, an exhaustive analysis of concepts. Each took himself to be engaged really in the latter, loftier pursuit. Thus, Schroder would probably not have accepted the characterization of his work as embodying the conception of logic as calculus (at least iif the conception is presented in the terms made familiar by Van Heijenoort). It appears that on this point Frege had indeed the better understanding of what was going on, but it is somewhat curious to see that even the standard-bearer of “logic as calculus” held a dim view of logic as calculus. Now, from all appearances Skolem did not share this view of Schroder’s: he has a more flattering opinion of calculuses, and if my view is correct he seems to think that logic is at most a calculus. A further point of interest is that in Skolem’s work we see a specific concern with recursive operations (so far only intuitively defined) to arise from what would today be classified as typical model-theoretical problems, such as the satisfiability of certain classes of formulas, rather than from syntactical investigations into formal systems and languages.

81 available in the intuitive model theory which provides the background to Skolem

(1920) and (1923).

In somewhat later papers published in 1929, Skolem did express himself more definitely in support of generally constructivist views about the foundations of mathematics. Even there, however, an outright rejection of the axiom of choice is not in evidence. Skolem’s constructivist leanings, however, hardly constitute a clear­ eyed adhesion to a constructivist philosophy. Rather, his observations on these and related foundational matters often portray a thinker straddling conflicting philosophical perspectives without quite realizing the extent of the conflict. His position seems to be that of an independent “transitional” figure in the debate on the foundations of mathematics: he breaks new ground, but does not necessarily conceive of the emerging issues in the ways that have become familiar with the benefit of hindsight. Thus, consider the remarks in (1929a) — the text of a conference - just in regard to Hilbert’s Vertreteraxiom^^ Summarizing the results of his (1929b), he notes, first, that the addition of the Vertreteraxiom and the principle of excluded middle to a consistent theory based on “numerical statements”

(Zahlaussagen: first order formulae), where excluded middle is restricted to

“elementary” or quantifier-free formulae, will also result in a consistent theory.

Then he adds:

I conceive of the... axioms of the theory intuitionistically or constructively, that is, in such a way that an axiom of the form “there is an X such that A(x) holds” is understood as: “An individual a is available (vorliegt) such that A(a) is the case;” and an axiom of the

^ It should be noted that Hilbert saw this axiom as the logical twin of the axiom of choice in set theory. The important relation between the two is asserted in Hilbert (1923).

82 form “for every x there is a y such that A(x, y) holds” is understood as: “An individual function f(x) is available such that A(x, f(x)) holds for arbitrary x.” (p. 224)

Here Skolem comes as close as he ever does to an exphcit endorsement of intuitionistic logic, but instead runs into potential trouble: be endorses the intuitionistic reading of the existential quantifier, but the reason be gives for this endorsement amounts to a stipulation of a principle of choice. His point is apparently that the Skolem functions, and therefore the existential quantifier itselfi are to be conceived of as constructive functions. And yet this cannot be quite what be means, for if this were the case be would have to acknowledge that the resulting logic would be entirely different: the classical inference rules would not be preserved. But be does not: in fact, be goes on to note that the theory where existential quantification is interpreted in the way just described (the theory, that is, which includes the Vertreteraxiom) avoids “the unpleasant consequences of the rejection of excluded middle for deduction.” This is certainly justified if the

Vertreteraxiom is interpreted classically, because the axiom of choice in set theory entails the principle of excluded middle,"' so that a theory including the former will support classical logic. Even more to the point, a rejection of the axiom of choice in favor of intuitionistic logic would partially obstruct the path to relativism, since the proof of LST, as we have seen, rehes on the axiom.

While it is surely possible that these papers (when compared to the original

“relativism papers”) mark a change or a development in Skolem’s overall philosophy toward constructivism, it seems to me that his viewpoint remains atypical His

^ The argument is spelled out in M. Beeson (1985). Cfr. p. 162.

83 comments on impredicativity, for instance, are not motivated by anything like a wholehearted endorsement of the philosophies of either Brouwer^ or WeyL In

(1929b), for example, he devotes a section to contrasting predicative and impredicative “productive conditions” [Reproduktionsforderungen], i.e. the operations, defined over a given domain, by which new elements of the domain are generated from other elements. The difficulty with impredicative “productive conditions” is cast once again in model-theoretical terms:

Whereas it is very easy to inspect the satisfiability of predicative productive conditions—one need only gather all objects that can be formed from the originally given objects by finitely many iterations of the operations—, this is not true in the impredicative case. It is however natural to attempt, even here, to satisfy the the requirement of closure [of the domain] through a gradual process of extension. But 1 shall soon demonstrate that this attempt is often a failure and can lead to errors. (1929b, p. 242).

Further down the same page, the objection to the use of impredicative conditions is summarized in rather unusual terms: the problem is, it would appear, that when constructing a domain (e.g. for set theory) by impredicative method we may lose sight of the fact that the conditions formulated could have been already satisfied, in virtue of LST, “in the countable” - as Skolem puts it. Thus, it seems that the problem with impredicativity is that one could be led to regard impredicative definitions as an attempt — misleading, because doomed to failure — to “get around” the relativism forced upon us by LST. Of course, if this is the criticism, it is rather weak - only someone already committed to Skolem’s views on relativism would be apt to find any binding force in this objection. Nevertheless, the fact that Skolem

“ There is in (1929a) a brief statement to the efiect that in spite of the current difficulties in the foundations of mathematics there is no need to abandon classical logic altogether (as Brouwer did).

84 pointed to this as the significant drawback of impredicativity tells us that either

Skolem had not quite understood what the issue really was (and this seems to me very unlikely) or else that his peculiar stance was not driven by constructivist concerns.

It is time to take stock and reflect on the relations between Skolem’s thought about general foundational matters and his relativistic theses. Let us begin by summarizing as clearly and briefly as possible the findings of our excursus into

Skolem’s conception of the principle of choice. The upshot seems to me essentially the following: Skolem treated the principle of choice -- classically construed and understood -- as of one piece with logical operations, in particular quantification. He exploited this very insight in his own work to handle quantifier dependence at a time when the understanding of quantification had not yet crystallized and competing conceptions (well, at least one: the logicists’) vied for primacy. It would not be unwarranted to claim, then, that choice is for Skolem a logical axiom, were it not the case that Skolem’s conception takes shape within a tradition of thought — logic as calculus -- to which a sharp demarcation between logic and mathematics

(algebra, in particular) seems foreign. At any rate, the axiom of choice comes to

Skolem firom the same conceptual stock as logical notions, and is equally available within what has been called the “intuitive model theory” of the algebra of logic.

Consider now the differences between the two versions of LST, the original one, which we have called the ‘submodel version’, and the later one, dubbed the numerical version. The critical distinction was the use of the axiom of choice; it should now be possible to see more clearly the significance that Skolem himself would attribute to the two different proofs. The point of showing LST on a purely

85 numerical interpretation was not simply to prove the independence of the argument firom the axiom of choice. This would have been a concern, of course, if Skolem had had independent philosophical misgivings toward the axiom of choice, for instance if he had thought that appeal to the axiom required a basic committment to Zermelian set theory, as opposed to the intuitive principles fully available already within his intuitive model theory. The considerations developed in the preceding paragraph, however, seem to indicate that this kind of resistance to the axiom of choice was not congenial to Skolem’s way of thinking ahout these issues, in spite of his own objections against Zermelo. If this is correct, we conclude that his 1923 proof does not really manifest any such concern. Rather, Skolem must have reflected that, if even with the assistance of a choice principle the quantifiers of an arbitrary formula of set theory cannot take us out of the denumerably infinite, so to speak, then the alleged relativism of set-theoretical notions is more deeply rooted than at first thought. Thus, Skolem must have drawn firom his own analysis a moral that is nicely summarized as follows:

The most that quantifier-language can do—with respect to quantification in a domain—is to require a choice of one value for an existential variable for each system of values for the universal variables governing it. The countably many choice functions involved in any countable set of propositions cannot therefore lead us out of the countable. (Goldfarb (1979), 357).

The numerical model version of LST suggests to him that the work of choice functions can be mimicked by “concrete” arithmetical functions - the safe, uncontroversial archetypes of aU algebraic operations, including the logical ones. It is therefore apparent that, if there is a difierence in “strength” between the two supporting arguments for relativism in Skolem’s perspective, it goes in a direction

86 opposite to that suggested by most contemporary interpreters. A typical claim, in this regard, is once again by McIntosh, according to which the submodel argument for relativity is

no help to Skolem. The submodel version relies on AC and on the existence of a standard model [of set theory], neither of which is available to intuitive model theory. (1979, p. 323)

But in fact AC is available, primarily because, via Skolem’s reading of the quantifiers, it (AC) is a centerpiece of logic within the intuitive model theory. Given

Skolem’s conception of the quantifiers, in fact, the submodel version of LST makes the weaker claim about relativity, and thus the resulting argument would be in

Skolem’s evaluation stronger than that issuing from the numerical version.

Conclusion: skepticism again

In reconstructing the relativistic argument in the first section of this chapter, we saw that the crux of the matter was the existence of certain infinite totalities asserted within Cantorian set theory. The axiomatization of set theory attempts to provide these assertions of existence with a rigorous basis. From the vantage point of Skolem’s foundational philosophy, however, aU mathematical assertions are cast in terms of their satisfiability within a suitably general “interpretation"—this interpretation is not itself formalized or axiomatized, and it is in turn based upon a stock of algebraic and arithmetical notions which Skolem had reason to think fundamental in an absolute way. This is the intuitive model theory, and this is the realm of mathematical logic, for Skolem. Satisfiability is thus the basic, and in fact the only, concern of mathematical logic.

87 Skolem saw set theory as the basis of logic; in his conception, he was a sort of abstract version of intuitive model theory, one in which instead of considering only structures of a certain type and claims made about such structures one takes into consideration structures simpUciter. But doing so is a bit tnclqr, from the standpoint to which Skolem apparently subscribed. For one had to evaluate a claim made ahout structures in general by constructing models — appealing, metalinguistically, to other principles about structures! Simply to overlook this predicament, as in Skolem’s mind the axiomatic set theorist were predisposed to do, would amount to treating set theory in a way that was wholly unfit for foundational purposes. This is what he laments as the “merely verbal sense” in which set- theoretical propositions are “made to hold” in models of the axiomatic theory.

For Skolem, what is appropriate to the study of groups is not appropriate to the study of sets. It is assumed that anything that satisfies the group axioms counts as a group: there is no sense to the question of whether something that satisfies those axioms “really has” a certain group-theoretical property, i.e. a property that is derivable firom the axioms. Sets, on the contrary, must have some intrinsic properties, in Skolem’s view. One of these, it seems, is the cardinahty of a set. This is a metaphysical claim that Skolem does not discuss explicitly, but it seems to me to be plausible even in contemporary terms: a set is constituted by its elements, so the number of its elements ought to be a constitutive property of a set, a property determining its identity as a mathematical object. This view seems to me intuitively confirmed by the extensional nature of sets, and the fact that we still accept as an axiom the principle of extensional identity. In fact, this view of sets as constituted by their elements, applied in the context of a theory hke ZFC without urelemente in

88 which, we only consider “pure” or hereditary sets, leads to the idea that cardinahty and order are the only constitutive properties of sets.

Leaving further speculation aside, this shows in any case that the point of view from which Skolem criticized axiomatic set theory has not been outlived.

Skolem feared a treatment of sets that we might dub as “naive structuralism,” in which one says: anything that satisfies the axioms of first order Zermelo set theory is a set (again, in analogy with the case of groups, rings, etc.). This approach is exposed to relativism, and therefore cannot account for the constitutive properties of sets. It is superficially similar to structuralist views, which are typically committed, for instance, to counting anything that can “play the role” of a number in the appropriate number-theoretic structure as a number (a position articulated in

Benacerraf (1965)). The difference is that non-naive structuralists individuate number-theoretic structures by stronger criteria than those allowed by the first order Peano axioms: they regard ‘number-theoretic structure’ as a concept denoting a class of isomorphic structures. Such a class can be characterized only in a second order language. Skolem, beginning with the 1923 paper, seems to divide up things differently. He simply identifies axiomatic theories with first order theories

(something that Zermelo had not done); the role of modern-day second order logic he assigns to a purely model-theoretic “background” set theory, which cannot be satisfactorily formalized.

Here, I think, also hes an interesting difference between Skolem — in particular the Skolem of 1923, the critic of axiomatic theories — and those contemporary thinkers that have closely echoed his themes. Unlike Quine (see chapter III) or Putnam (see chapter I) Skolem, there really is a rock-bottom

89 terminus to the regress of interpretation initiated by the skeptic: it is the background set theory, or intuitive model theory. It is a substantive theory, including such notions as those of cardinal arithmetic and, as argued, the axiom of choice. For Putnam, as I understand it, there is no end to the regress, unless we radically overhaul the notions understanding, interpretation, etc. and revise the inferential norms governing our language. For Quine, the terminus seems to be the

"face-value interpretation,” but that is eternally fleeting: as long as we fix on it and try to understand it, we have lost its sustenance. The only thing we can truthfully say about it is that we cannot say anything substantial about it.

90 CHAPTER 3

SKEPTICISM VS RELATIVITY

Introduction

Quine’s doctrine of ontological relativity stands in the way of any attempt to kindle confidence in the idea that this or that theory has an intended subject matter or an intended meaning. The central claim of this doctrine is that subject matter, meaning, interpretation and cognate notions are of necessity indeterminate; if this is true, the distinction between the intended interpretation of a theory and other

“spurious” ones will he impossible to draw. Ontological relativity bears a striking family resemblance to the view described in the previous chapter - the view that I have attributed to the Skolemite skeptic, and in some measure to Skolem himself— mostly because Quine and others attracted to his conception, notably H. Putnam, seem to support their views with Skolemite considerations. To be sure, there are important variations on this theme. While both Quine and Putnam, for instance, acknowledge a debt to Skolem, the mere realization that a “theory” or body of beUefe

91 could have multiple interpretations — that it could be construed as referring to something else than it is actually supposed to — often seems prominent among the things that drive their concerns. Thus Putnam in “A problem about reference”^ thinks that the very possibility of reinterpreting term reference systematically in a truth-preserving way is sufficient to throw the concept of reference into inescapable chaos.

I shall a i^ e , however, that if this kind of problems may look troublesome in explaining the semantic properties of ordinary language in the world of everyday experience, it is doubtful that Skolem could have taken it very seriously. In the context of mathematical theories, the multiplicity of interpretations modelling a certain theory is taken for granted: certainly given any model satisfying a theory we can give very many that are isomorphic to the given one, and that wül he mathematically inditinguishable firom it. Clearly it could not have been this what

Skolem meant by relativism: this would have been trivial, and would have appeared so to Skolem. Furthermore, it is in Quine’s own view virtually inevitable, as we shall see. Skolem thought, on the other hand, that set-theoretical relativism was properly about set theory, that it was a special problem for set theory, and finally, at least for a period of time around 1923, that the difficulty was not inevitable: he beUeved, as we have seen, that it could be avoided by “avoiding” a conception of set theory too closely alllied with the axiomatic method. He thought there was some more to set theory than an axiomatic system could illustrate.

One might think- that the “Skolem paradox” really betrays a problem of

^ See chapter 2 and the appendix in Putnam (1981).

92 ontological relativity just in the sense of revealing the obvious, inevitable emergence of different interpretations. But then, if we must agree with proponents of the doctrine that issues of ontological relativity are genuinely insoluble, there will be nothing else to find underneath the shifty surface of set theory. Unlike the early

Skolem, the ontological relativist will tend toward a sort of aggressive agnosticism when it comes to the “true understanding” of set-theoretical concepts, the intended meaning of set theory, or other such; there is no such thing to worry about, the agnostic personality suggests; it is the source of aU evil to worry about it, the feisty personality intimates. Since I want to vindicate the early Skolem’s intuition that there is such a thing as the intended conception of set theory, I believe that we must confront the ontological relativist. So, I propose that we take up here two main issues among the many raised by ontological relativity:

First, if ontological relativity “works,” really there is not much hope of resisting the charges of the Skolemite, and thus of addressing Skolem’s worries about set theory.

Moreover, one ought to appreciate that if Quine’s assault is on target there is not even a clear sense that one could give to those worries: Skolem was mistaken to think of the “problem” as a genuine problem. So the first order of the day is to evaluate the robustness of Quine’s overall argument. Unsurprisingly, I shall argue that support for certain key aspects of ontological relativity is weak.

Second, does Skolemite relativism describe the same syndrome of languages that

Quine diagnosed as ontological relativity? I believe that it does not: not only is

Skolem’s concern different firom Quine’s, but it is also hard to represent the scope of that concern within the Quinean firamework. What worried Skolem (and, I think, should still worry us) seems to be not ontology, but ideology (in Quine’s sense), that

93 is, concepts: Skolemite relativism is ideological,^ not ontological relativity. These two issues are not independent: to see where Quine’s attack fails can be instructive, if we are to understand the focus of Skolem’s concern, and what kind of considerations -- if any -- can address it. A further point of interest in this sense is that Quine himself has offered a “solution” of sorts to the Skolem paradox: but just as ontological relativity does not seem to me to carry quite the sting of Skolemite scepticism, so does the Quinean reply fail to satisfy the anti-sceptic.

Let me first expand a bit on the latter point. Ontological relativity is a problem concerning reference. According to the analysis Quine has provided, the ontology of a theory consists of the things to which the singular terms of the theory must be construed as referring in order for the issuances of the theory (i.e., the theorems) to be interpreted as true. The question is how to determine which things are in fact referred to. The issue, simply enough, is:^‘ How do we know that we are using ‘cat’ to talk about cats? Could we mean dogs by ‘cats’? To determine this is in effect to determine the ontology of the theory. But the Skolem who writes in 1922 against axiomatic set theory repeatedly expresses his worries about the proper rendering of the concepts of set theory in the axiomatic format: he fears that the apparent impossibility of fully characterizing uncountably infinite sets would cast a paU on the very notion of uncountable infinity. That is, the relativity of interpretations shows a sort of conceptual “instability” within set theory and this is what Skolem worries about. The relativity shows that fundamental mathematical

^ Royalties for the title phrase should go to Neil Tennant.

The definitive discussion of these questions, and the possibility of meaning cherries by ‘cat’ and cats by ‘cherries,’ is in Putnam (1981).

94 notions are not properly characterized by axiomatic means: even finiteness comes into question, a fact Skolem regards as unquestionable proof that axiomatic set theory is not worth bothering with as a foundation of mathematics. Is Skolem worrying about referential indeterminacy?

Of course, the differences, such as they are, may he just terminologicaL

Perhaps Skolem did have in mind something like Quinean referential indeterminacy, in the end, or perhaps (more likely) Skolem himself would have been unable, if pressed, to distinguish with appreciable clarity the ontological problem about reference and the problem he was worried about, whatever that was. But the issue is not merely exegeticaL The reason for this, I think, is that the referential problem portended by the ontological relativity considerations does not affect mathematical theories in the way Quine thinks. This is a preview of further developments, at this point, but I think it worthwhile to clarify it now. In short, on closer inspection it is hard to make sense of ontological relativity in mathematics.

The problem seems to be that the conception of reference which presents itself most naturally in mathematics is a structuralist conception, according to which, in a nutshell, the exact individual nature of the objects referred to is irrelevant: for the referential relation within the theory involves directly not the objects themselves, but their structural roles, and these can be played by objects which are, by the lights of everyday language, distinct. In other words, if a given theory has an intended interpretation, it will be “about” a certain structure. As long as we have the same structure present, so to speak, we tend to think of the reference of the terms of the theory as remaining the same. Thus the thrust of the structuralist’s position on mathematical theories is that reference within them should be specified by giving

95 reference schemes that are “structurally invariant”: if the structure is the same, what the terms of the theory refer to has not changed. There is no re-interpretation without a change in structure. The Quinean puts forth alternative interpretations,

“incompatible” translation manuals, diverging reference schemes, and so on; but the question one should ask when considering the case of mathematical theories is: do such alternatives involve a change in the intended structure? For unless they do, the stronger intuition is that the kind of referential indeterminacy advertised by

Quine and Putnam does not arise.

I shall argue that the notion of term reference advocated by Quine in relation to ontological relativity is not structurally invariant: it is apparent that distinct reference schemes (in Quine's sense) are admissible within an identical structure.

This does not mean that questions of reference in general can be simply expunged, once (or if) a structuralist view of mathematics is established - only that the indeterminacy one can extract with arguments of the sort inspired by Quine’s work on ontological relativity does not seem seriously problematic in the mathematical context, so the onset of relativity is at least delayed. On the other hand, I argue that

Skolemite scepticism ought to trouble the structuralist as well: there ought to be a genuine problem about whether, by formulating a certain theory more or less explicitly, we have correctly represented the mathematical structure we had in mind. But here we see the difference: the Skolemite sceptic can still ask nasty questions, because his “alternative reference schema” is in fact supported by a different structure. The challenge of the ontological relativist in comparison is toothless.

On the way toward a justification of a notion of intended interpretation,

96 therefore, one of the goals should be to provide a structuralist understanding of reference and related notions that can resist the Skolemite’s objections. A word of caution about structuralism is in order, though. In the previous chapter I sketched a position, labelled “naive” structuralism, which would not fit the bill. As I argued there, in combating axiomatic set theory Skolem is attacking precisely the sort of position represented by naive structuralism. I further argued there that structuralism and naive structuralism differ in their imphed attitudes toward the issue dramatized the Skolem paradox. Naive structuralism is perfectly compatible with the skeptic’s position. On the contrary the non-naive variety need not, and typically wül not, acquiesce to the claims of the skeptic. One thing I hope to make clear is that the pitfalls of naive structuralism are not avoided trivially, i.e., at no cost in phüosophical efforts and commitment. Quine’s case seems to me an excellent one to see why, for throughout his discussion of ontological relativity he seems to be looking hard for ways of subscribing to a structuralist view of reference in mathematics whüe maintaining (what seem to me) incompatible conceptions about linguistic reference and meaning; the thrill of ontological relativity derives in large part firom the fact that Quine paints a picture in which reference and ontology are construed in a naive-structuralist way; but he successively introduces elements — notably his account of proxy functions in reduction -- that assimüate his position to that of a non-naive structuralist. But once these further elements are introduced, one should question whether there remains an interesting sense in which ontology is relative.

97 Quinean relativity

Ontological relativity is the doctrine according to which the determination of the ontology of a theory, and therefore in general the treatment of ontological issues concerning a given theory, is a task that can be made sense of only within the frame of reference provided by another theory. The notion is introduced by Quine as a

“solution,” of sorts, to the problem of the so-called inscrutability of reference. It is a peculiar solution, for the problem is not explained away. It is a solution in the

Quinean sense that it vindicates a more modest “naturalized” role for philosophical talk of ontology: ontological issues are not reduced to utter nonsense, unless one yields to the presumption that they can be settled “absolutely”; there is room for significant discussion of such issues, although within the strictures of ontological relativity.

The route to ontological relativity begins with the considerations that led

Quine to assert the indeterminacy of translation. The analysis of the imaginary task of a radical translator in Word and Object has shown us, according to Quine, that translation is indeterminate—that, in other words,

manuals for translating one language into another can be set up in divergent ways, all compatible with the totality of speech dispositions, yet incompatible with one another. (1960, 27)

Since the purpose of translation is to associate sentences in a language with sentences having the same or substantially similar meanings in another language, the indeterminacy of translation demonstrates the flimsiness of the very notion of likeness or sameness of meaning. The indeterminacy, moreover, betrays the more general fact that

98 the infinite totality of sentences of any given speaker’s language can be so permuted, or mapped onto itself that (a) the totality of the speaker’s dispositions to verbal behavior remains invariant, and yet (b) the mapping is no mere correlation of sentences with equivalent sentences, in any plausible sense of equivalence however loose, (ibid.)

The radical interpreter has no predetermined understanding of any parts of the

language under study—the native language. All that is available to the radical interpreter and translator is the overt behavior of the natives in response to concomitant linguistic and sensorial stimuli, through the observation of which he must reconstruct the array of speech dispositions that can be attributed to the native speakers. Speech dispositions on the part of the natives -- the patterns of verbal response on the part of the natives elicited by either external circumstances or direct querying from the radical translator — are a coarse substitute for meaning in the task of radical translation: that is, in order to understand the meanings of the natives’ linguistic constructs the translator can do no better than associate them to the patterns of stimuli that seem to prompt native utterance of such constructs.

What the radical interpreter has initially access to, in other words, is the stimulus meaning of native utterances. The observed stimuli can be connected to translation via a system of analytical hypotheses, higher-order claims not usually open to direct empirical confirmation. Since the “fit” of a manual with the empirical evidence available to the radical interpreter depends upon any number of analytical hypotheses, two or more different translation manuals can be, as it seems, equally correct: that is, equally confirmed by the evidence. This is the indeterminacy of translation.

Quine’s findings in (1960) can be summarized thus: if one translation manual can be so construed by the radical interpreter as to do justice to the empirical

99 evidence (that is, the complex of speech dispositions exhibited by the natives in the presence of the interpreter), then many distinct such manuals can be found. Thus meaning is indeterminate, for the many manuals are distinct precisely in that they assign different meanings (Le., sentences different in meaning) in the translator’s mother tongue to the same identifiable sentence in the natives’ language.

The emendations required by the indeterminacy are radical. It is often said in connection with this thesis that since there is no fact of the matter concerning the determination that a given translation manual is the “correct” one, then there is no conceivable fact about semantic predicates to which locutions such as ‘S means that p’ may answer. '” Quine clearly holds that any putative “fact about meaning” would have to be the sort of fact that constitutes evidence available to carry out radical translator. Hence, the indeterminacy of translation suggests that there are simply no facts about meaning at aU. Thus Quine in a recent précis, for example:

Considerations of the sort we have been surveying are all that the radical translator has to go on. This is not because the meanings of sentences are elusive or inscrutable; it is because there is nothing to them, beyond what these fumbling procedures can come up with.“

Yet the ontological thesis that there are no facts of the matter about meaning is not a mere fallout of the indeterminacy arguments. Quine’s quote makes it exphcit that the ontological thesis is intimately connected to his avowed behaviorism, or what he calls “naturalism”; but the behaviorist prejudice in itself is not a result of the indeterminacy arguments. It is rather one of the prerequisites on which one must

Gibson (1986) effectively explains the role of this thesis within Quine’s picture of indeterminacy.

“ Quine (1992), p. 47.

1 0 0 suspend judgment, as it were, in order to find some initial cogency in the arguments evinced firom the predicament of the radical translator. As Gibson (1986) explains clearly, Quine’s attitudes toward the fiictuality of meaning and assorted semantic notions is an ontological stance whose motives are somewhat deeper than, and largely independent firom, the indeterminacy of translation. Even if, per impossibile, one were to determine beyond reasonable doubt the correctness of one and only one translation manual, it might still be the case that there were no facts of the matter about meaning—at least under the interpretation of this claim described by Gibson.

This ontological stance is motivated by other considerations, such as those articulated in “Two Dogmas.” Semantic holism is the primary reason why there are no and there could be no facts about meaning, and why we are ultimately led to conclude that sentences just do not come attached to these meaning entities.^

The indeterminacy manifests itself in the impossibility of choosing between manuals of translation that, though incompatible, accord equally well with the totality of speech dispositions observed in the native population. And yet, the incompatibility of two sentences (and by extension two translation manuals) usually portends a difference in meaning (following pre-Quinean naivety about semantics, one would say that incompatibility is a difference in meaning). But if one is limited to a notion of meaning as consisting of “surface irritations” of sensory receptors, it may turn out to be difficult to flesh out a different interpretation for sentences that admittedly share a set of relevant stimuli. The vexed “gavagai” cases, for example,

^ Again, Quine in a comment to Gibson (1986): “I submit that if sentences in general had meanings, their meanings would be just that [Le., it would be what counts as their evidence, as verificationists hold]. It is only holism itself that te Us us that in general they do not have them.”

101 show according to Quine the impossibility of discerning a native comment about (Le.,

prompted by) rabbits from a native comment about (Le., prompted by) undetached

rabbit parts. That is because the physical presence of rabbits and the physical presence of undetached rabbit parts are the same physical occurrence; no sensory receptor can be expected to discriminate such subtleties of individuation. So the surface irritation is the same in both cases. This granted, the question now seems to be: assuming for the moment that the sets of speech dispositions (the sets of pairs stimulus-response, if I understand correctly) associated with either comment by competing translation manuals are the same, where exactly can one locate the

Quinean (as opposed to the naive) difference in meaning?*® Again, I mention this problem only because it has, in my opinion, an interesting analogue in the ontological relativity case (an analogue that is relevant to our central concerns).

In (1992), Quine explains that “gavagai” never was meant to carry the burden of exemplifying (as it did) the indeterminacy of translation: it was merely an example of the phenomenon of indeterminacy of reference—the uncontroversial realization, as described by Quine, that any theory can have a plurality of models. This realization only impinges upon the interpretation of the language “analytically,” that is, only when sentences are considered as semantic compounds resulting compositionaUy from the interpretation of their component words. According to Quine, the analytical indeterminacy of reference is ineliminable, but not as troubling as the difficulties relative to translation. The indeterminacy of translation, on the other hand, concerns the language at the “holophrastic” level, that is, when sentences are considered as unarticulated wholes, drawing semantic significance from their own rough but more or less direct association with the respective prompting stimuli. Observation sentences of the simplest sort live at this level That is why presumably indeterminacy of translation is a more serious problem. This holophrastic indeterminacy affects our (or a translator’s) capacity to associate sentences (as wholes) with univocal uniquely determined, sets of prompting stimuli What of gavagai, then? That sort of example demonstrates the possibility to construe the reference of a term according to alternative reference schemes (by providing different but “isomorphic” models, as it were); it does not demonstrate an indeterminacy in the empirical meaning o f‘gavagai’, the translation is determinately (under the circumstance described in the example) lo, a rabbit!’

102 The indeterminacy of translation concerns likeness of meaning, as we saw.

Now, intensional notions such as meaning are even preliminarly suspect if we embrace the Quinean conception of language understanding as grounded in publicly observable behavioral response. Extensional notions such as denotation and reference, on the other band, prima facie do not suffer from these problems. The notion of sameness of reference does not seem to be plagued by the sort of problems that Quine discusses in "Two Dogmas” with regard to the notion of synonymy, or sameness of meaning. At least in cases where the context of evaluation is properly restricted, there seems to be no difficulty in determining that two expressions refer to the same thing. Yet in radical interpretation the trouble extends to extensional notions. This is, indeed, only mildly surprising because the broad framework of indeterminacy we have sketched in the previous paragraph seems to have been devised without especially targetting intensional notions. In other words, not much in the arguments relative to the indeterminacy of translation seems to rely upon features peculiar of intensional, as opposed to extensional notions. The fact is that in order to explain the reference of terms in the native language the translator must rely on his understanding of an “apparatus of individuation,” a “cluster of interrelated grammatical particles and constructions: plural endings, pronouns, numerals, the ‘is’ of identity, and its adaptations same and ‘other’” (1969, 32). In fact, such apparatus is needed even prior to the determination of the reference of terms, is needed just in order to determine which expressions and positions in sentences count as referring terms. Of course, the translator will develop a translation of the natives’ own apparatus of individuation, in the usual ways: but it seems possible, then, that the same piece of native idiom would be construed as

103 referring to different things if we chose a different way of translating what is

ostensibly a piece of the native apparatus of individuation. Quine’s classic examples

(rabbit, rabbit stages, rabbit parts, etc.) aim to making the reader receptive to just

such a possibility, of course. And this is to say nothing of the translator’s apparatus

of individuation, which he must perforce take and use “as is” in his parsing of native

utterances. As a result, we are caught in the plight of the inscrutabüity of reference:

[wjithin the parochial limits of our own language, we can continue as always to find extensional talk clearer than intensional For the indeterminacy between “rabbit,” “rabbit stage,” and the rest depended only on a correlative indeterminacy of translation of the English apparatus of individuation.... Given this apparatus, there is no mystery about extension; terms have the same extension when true of the same things. At the level of radical translation, on the other hand, extension itself [hence reference, denotation and their ük] goes inscrutable. (1969, 35)

Radical translation reveals reference to be inscrutable, but the inscrutability does not depend upon the need for a translation, radical or otherwise. It does not presuppose that there is an unfamiliar language to be understood. We specify the reference of terms in our home language only by assuming the familiar apparatus of individuation and background ontology, or universe of discourse. We can say with good reason that ‘rabbit’ refers to rabbits, but just as we say that we are assuming a familiar background ontology and criteria of individuation. Our talk could be construed as being about undetached rabbit parts, if only we would change the individuation schema while translating English into itself. So even in the same language, we could make different assumptions and obtain different construals of reference, different reference schemes. The upshot is this: it makes sense to determine the reference of the terms of a language only once certain anchors for interpretation have been fixed: “reference is nonsense except relative to a coordinate

104 system” (ibid., 48). Such interpretation schemes are ever flimsy: if there is one that seems to fit, there certainly are more that would fit; they can be obtained by mutually compensating adjustments in the apparatus of individuation, grammatical analysis of a sentence, and such. But interpretation according to Quine always is a species of translation: to interpret is to reconstruct a piece of language in (perhaps) different, more fam iliar terms. The superficial impression that reference is determinate in the home language is due to the fact that the translation scheme applied in this context is usually presupposed, and is the homophonie translation.

But even in the same language, this scheme is not mandated by any of the relevant evidence—not even by one’s subjective understanding of the mother tongue—, except perhaps to simplify things in daily matters.

In one of Quine’s examples, talk of formulas could be interpreted (within the common background of mathematical English) as talk of Godel numbers. Le. as being about Godel numbers instead of formulas. There are well-known one-one mappings of formulas onto Godel numbers, a fact which allows one to go back an forth between formula-talk and number-talk. So there is no question whether the speaker of the formula-language referred really, in some absolute way, to formulas or to Godel numbers. Posed in these terms, the question has no principled answer, and therefore is nonsensical The legitimate question must be posed relative to a fram e of reference in which the speaker’s talk of formulae is interpreted. It is only in such a frame of reference that we can specify the domain of the original discourse; the face-value interpretation, which is the one yielded by the homophonie translation scheme, is thus merely one among many, equally plausible, interpretations.

105 For Quine, notoriously, the ontological import of a theory is displayed in the interpretation of quantification. We should note, as a preliminary remark, that it is a seamless transition, in the Quinean firamework, firom theories' to languages’, and viceversa. We assess the ontological import of theories and languages in the same way as a matter of course. Quantification is a feature of language, and a body of theory has the ontological import of the language in which it is expressed. In order to make the structure of quantification transparent any firagment of discourse must be properly regimented by translating it into the language of first order logic in familiar ways. As one can see, the methodology applies indifferently to set theory and to everyday talk of medium-sized enduring physical objects.

The ontology of a theory is the domain over which assignments to its bound variables must range in such a way as to make the statements of the theory true.

The idea at the core of the doctrine of ontological relativity is that the domain of quantification for a theory T cannot be specified except within another theory T which interprets T. To “specify" a domain of quantification may involve several different tasks, but certainly requires (i) that we indicate a generic class of objects such that they — or sequences of them — may take the place of bound variables in closed sentences of T, and (ü) that we specify which properties and relations between such objects are denoted by, respectively, predicate and relation terms in the language of T. But the description of these objects and of their properties and relations is itself a “theory,” couched in more or less regimented language that can be in its turn subjected to the same kind of interpretation. This is the picture painted by Quine: every description of ontology must be formulated in language; but then it encapsulates a new or anyway a different theory, susceptible in turn of

106 interpretation. The regress of interpretation cannot be avoided; this entails that the question “what is the correct interpretation of T?” has no answer - to the extent that it calls for selecting one interpretation among the many that make the given theory true on the basis of some a priori criterion (as opposed to merely pragmatic ones, such as “simplicity^. This is the source of the relativity of ontology. Ontology, says

Quine,

is relative to a manual of translation. To say that ‘gavagai’ denotes rabbit is to opt for a manual of translation in which ‘gavagai’ is translated as ‘rabbit,’ instead of opting for any of the alternative manuals (1992, 52).

The only ontological question that can be intelligibly asked is about the choice of a background language and an interpretation schema of the old T into the new language, and about whether such a schema can preserve the truth of the original statements under the mapping induced by translation:

[wjhat makes sense is to say not what the objects of a theory are, absolutely speaking, but how one theory of objects is interpretable or reinterpretable in another (1969, 50).

How to commit oneself to an ontology

There is a significant distinction to he recognized, in Quine’s thought about ontological matters, between the “ontic commitment” of a theory and its ontology.

The ontic commitment of a theory is made explicit by the existential assertions that the theory contains or entails: a theory that entails the statement ‘There is a cat on the mat’ is committed to the existence of cats, as well as (if the definite description therein is unpacked according to the Russellian reading) of mats. This is the view,

107 often expressed by Quine, that

existence is what existential quantification expresses. There are things of kind F if and only if 3xFx.^

The notion of the ontology of a theory is slightly different, however ~ and not nearly as clear-cut. Some trivial examples should suffice to show that unpacking the explicit existential pronouncements of a theory cannot be all there is to ontological analysis. A first order theory such as Peano arithmetic is no doubt committed to the existence of prime numbers; this is obviously true, but the explicit commitment one finds in arithmetic is in the form ‘there is something that is prime’, not in the form

‘There is something that is a prime numhef, for the predicate ‘is a number’ is not in fact a predicate of arithmetic (unlike ‘is greater than’, ‘is odd’, ‘is prime’, etc.). But if we want to know why someone working in arithmetic would be somehow committed to the existence of numbers, we should probably point out to some sentence like

“There is something that is equal to itseff” This, however, is a sentence which is valid in every non-empty domain: it can express a commitment to numbers, rather than, say, dogs, only in conjunction with other sentences - in particular, the true sentences of arithmetic. The committment it carries to the existence of at least a number derives firom its being vahd in a theory that is interpreted in a domain consisting of numbers.

Arithmetical commitments to numbers, therefore, are to be understood in a different manner: regardless of exphcit assertions of existence, numbers are objects the existence of which we need to presuppose in order to model arithmetic itself

“ Quine (1969b), p. 95. See also his (1953). The distinction outlined in this paragraph is drawn by Mark Wilson (1981), pp. 41 Iff and also by C. Chihara (1973), ch. Ill, sec. 3.

108 (that is, in order to interpret the theory in such a way as to make arithmetical statements true). What we can say is that numbers are the domain of arithmetic, and within the domain the existence of prime numbers is required by arithmetic.

Sim ilarly, ZF and ZFC are two theories of sets, only one of which is committed to the existence of a particular set or type of sets (choice functions); their commitments are different, but is their ontology? Not necessarily (intuitively, they both are about an ontology of sets).

The ontology of a theory (or “the objects of a theory,” as Quine also says) is the domain of interpretation of the theory, the range of its bound variables (this is the way we have treated the concept of ontology in the paragraphs immediately preceding). This raises an obvious question, however. Tracking down the existential assertions of a theory provides a reasonably clear criterion for gauging its ontological commitments. A theory wears its ontological commitment on its sleeve, so to speak. But as we have seen in the trivial arithmetical example the same criterion that works for ontological commitment seems inadequate to specify the ontology of a theory. The range of quantification (i.e., the ontology) is not necessarily made explicit within the theory. The question then is: how can we specify the ontology of a theory?

While we can point out the precise place, so to speak, where arithmetic commits itself to prime numbers, the commitment of arithmetic to numbers as such is more diffuse. It seems to be a matter of semantic involvement: it has entirely to do with the interpretation of the theory, with the meaning that is assigned to the theory in a given context -- with its intended meaning. This distinction is similar in spirit to one made famous by R. Carnap between “internal” and “external” questions

109 of ontology. Carnap classiGes queries such as “are there prime numbers?” among the former: these questions have a theoretical answer, that is, they have a sense within an appropriate theory (elementary arithmetic, in this case). Queries such as

“are there numbers?,” on the other hand, do not have according to Carnap a definite theoretical resolution. Therefore, they admit not so much of theoretical investigation as of pragmatic decision: arithmetic does not determine whether there are numbers; instead, the issue is clinched by the decision to use a theory of numbers. The problem is that for Quine questions of interpretation are always preliminarily suspect, while the notion of an existential commitment arising firom the decision to adopt some kind of theoretical or lingpiistin convention -- the notion one might glean firom Carnap's distinction -- is even worse. To say that the ontology of a theory can be determined firom its intended interpretation would be therefore of no help, for, in discussing ontological relativity, we are in the process of questioning precisely the idea that such an interpretation is ever to be determined. So what strategies are left open to the Quinean to determine the ontology of a theory?

In my judgment, there are basically two possibilities. Both of these strategies are, as far as I can teU, consistent with basic Quinean principles - yet they apparently lead to incompatible results. The problem is, as I see it, that Quine seems in relevant places to be determined to appeal to both; and this position, if 1 am correct, is untenable. 1 hope to make this clear in the remaining of this chapter.

The two strategies can be described approximately as follows.

First Strategy: Someone holding Quine’s view may want to treat ontology and ontic commitment in, as much as possible, the same way. It seems likely that a

Quinean might want to construe the ontology of a theory in terms of explicit

110 existential commitments. As noted, however, this is in general not feasible within one and the same theory. The trick is, then, that in order to say what the ontology of

T1 is we appeal to the (explicit) ontic commitments of another theory, T2, one that provides an “interpretation” of T1 in the sense that we have sketched in those previous paragraphs devoted to the explanation of ontological relativity. So, for example, using Zermelo-Fraenkel set theory as a background theory we could couch the ontology of arithmetic in terms of finite ordinals. That is, using set theory as the background theory, we can explicitly assert the existence of numbers: for the existence of finite ordinal.^ is part of the ontic commitment of such a background theory in the same way that the existence of a prime greater than 20 is part of the ontic commitment of arithmetic.^'

Second Strategy: Alternatively, we could pursue a structuralist tack.

Certainly the strong intuition that arithmetic (or using arithmetic) “imports” an ontology of numbers originates at least partly firom the thought that any domain of which the arithmetical laws are true must contain things that play the role of, or are

“as good as,” numbers. This suggests that we might see the ontology of - in this particular case — arithmetic as determined in some way by the structure that a collection of objects must exhibit in order for it to support a true interpretation of

The possibility of this strategy is clearly contemplated in Chihara (1973) as part of what Quine would allow in addition to a “straightforward” criterion of ontic commitment such as that put forth by A. Church. Church’s criterion simply states that “the assertion of (3x)M carries ontological commitment to entities x such that M” (Chihara (1973), 97). However: Unlike Church's criterion, Quine’s allows us to go “outside” the target theory in determining ontological commitments.[...] By Quine’s criterion, the standard mathematical theories are ontologically committed to abstract entities even though they do not explicitly say there are such things. (Chihara (1973), 101)

111 arithmetic.'^ More generally, we would say that the ontology of a given theory can be individuated up to a structural identity of some sort, and reckon that two domains structurally identical with respect to a theory T contain (copies oQ the ontology of T, even though their respective “native” languages (corresponding to the respective “backgrounds” in which T is interpreted) carry superficially different ontic commitments.

The thrust of the first strategy is an aspect of Quine’s doctrine of ontic commitment that M. Wüson dubbed the “univocality of there is’.”** This is the thesis according to which the ontological import of a theory must be read off its existential assertions, and that a refusal to apply this criterion can only be due to the sort of misconceptions about different “types” of existence which are thoroughly dispatched in the pages of “On What There Is”: but there is clearly no ambiguity in the meaning of the existential quantifier of the predicate calculus. In the presence of this desideratum, the first strategy leads straightforwardly to the doctrine that ontology is relative — that it can only be determined relatively to a choice of interpreting theory. For consider, once again, that the basic trigger of ontological

“ I have here tried to sidestep, if at all possible, a central question arising naturally in connection with a structuralist strategy: whether we should say that things “playing the roles ” of numbers are numbers, or rather that the numbers are the “roles,’’or something on that fine. I cannot hope to confiront here this difficult problem, which is extensively treated by others (Dieterle 1994, Shapiro 1983, 1990). Suffice it to say that Quine himself denies the first proposed identification (“..is aU there is to number. But it would be a confusion to express this point by saying... that numbers are things fulfilling arithmetic...” (1969a), p. 45), and I suspect he would not be sympathetic to the second. At any rate, Quine (as I try to explain) exhibits structuralist leanings but does not follow a structuralist approach consistently: overall, he favors criteria of ontological commitment patterned after the First Strategy.

** See Wilson (1981), p. 411.

112 relativity is the incontrovertible fact that any consistent theory will have a

multiplicity of truth-preserving interpretations: undetached rabbit parts and rabbit

time-slices, Zermelo ordinals and Von Neumann ordinals loom here. By the lights of

the univocality criterion, these are all examples of background theories with different (explicit) ontic commitments. But according to the strategy we are discussing the ontic commitments of the background theories sped out the ontology of the interpreted theory (rabbit discourse in one case, elementary arithmetic in the other). Hence different background theories may lead us to assign different ontologies to the same theory. It is clear then that we are forced to regard a non- relative determination of ontology as impossible, because the determinations to which we arrive within each background theory singularly taken are, by virtue of univocality, simply incompatible.

On the other hand, the line of thought charted out in the second strategy points in a direction opposite to that of ontological relativity. Mathematical structuralism is a doctrine consisting of at least the following two tenets:

1) The mathematical objects characterized by a certain theory T are exhaustively characterized in such a theory. Thus, for instance, numbers are the objects described by arithmetic; to the extent that we conceive of them as the objects of arithmetic (the subject matter of arithmetic), numbers have no other features than those characterized in the theory.

2) Mathematical theories characterize directly structures, and individual objects only inasmuch as they are part of a certain structure. Arithmetic provides a characterization of the number 3, for instance, only in terms of its relations to other objects in the structure of natural numbers. That is, 3 is the number marking a

113 certain place in the sequence of natural numbers.

If it makes sense to regard the structure of natural numbers as the subject matter of arithmetic, then we must admit that thxit same structure is found in every isomorphic interpretation of arithmetic. Moreover, the intended interpretation of arithmetic would be such that its domain consists of exactly those objects that arithmetic exhaustively characterizes -- objects, that is, that have only arithmetical features. Natural numbers are such objects; in the intended interpretation, the variables of arithmetical statements range over such objects. Therefore, if it makes sense to regard arithmetic as a theory that is about natural numbers, a structuralist would consider any interpretation of arithmetic isomorphic to the intended one as importing the same ontological commitment of the intended one: the commitment to an ontology of natural numbers.

In other words, from a structuralist standpoint isomorphic interpretations are structure-preserving: they represent the same structure. But the structuralist also regards the structure that a theory described as the true mathematical subject matter of the theory; this structure is the ontological region described, more or less adequately, by the theory. Hence, according to this structuralist conception, changes of interpretations portending the substitution of a domain of discourse with an isomorphic one do not reflect a change in ontology. That by itself would lead us to deny that just any switch in interpreting theory marks a distinct and incompatible rendering of the ontology of a given theory. Of course, the doctrine of ontic commitment must undergo considerable modification to accommodate this strategy.

To begin with, this line of thought requires that the thesis of the univocality of‘there is’ be abandoned, because it allows for cases in which structural identity, whatever

114 that will turn out to be, overrides differences in explicit existential commitment. To be sure, ontic commitment in the traditional sense of Quine or Church (see 61 . 37) is retained. However, its scope is somewhat limited: surely theories that entail logically incompatible existential assertions must be counted as having different ontologies. What is put into question is the connection between a criterion of ontic commitment and issues concerning the ontology of a theory.

The difference between the notions of ontic commitment and that of ontology, as they are configured in Quine’s writings (at least in those cited here), has emerged as one of, very broadly, syntactic versus semantic categories. The ontic commitment of a theory is traced back to its theorems -- to (typically) existential statements that are part of the theory. It is true that Quine often puts the matter in terms of “values of the variables,” which of course would indicate a concept with semantic roots.

Nevertheless, the view that the existential commitments retrieved firom a theory are to be taken at face value (the univocality o f‘there is") makes it clear that the source of the commitment are the existential theorems. Variables are introduced to mark genuinely referential expressions, as opposed to expressions which appear to be referential but may actually be “eliminated” by contextual definitions reflecting actual use: descriptions are of this sort, and even a proper name is genuinely referential “only if the existentially quantified identity built on that [name] is true according to the theory.”^® In contrast, the ontology of a theory -- the range of the variables -- is clearly a semantic notion. Now the second strategy depends upon the possibility of giving what we have designated as a criterion of “structural identity,” a

This is the way the issue is presented in (1969b), p. 94.

115 criterion, that is, by which one would determine whether two interpretations

(supplying superficially distinct ranges of variables to a theory) count as ontologically equivalent. Naturally then the criterion will be semantic; it seems clear that, particularly as far as mathematical theories are concerned, a criterion of structural identity could be no stronger than isomorphism of models.

Note, however, that to say that two theories have the same ontology just in case their intended interpretations are isomorphic would in fact he too strong a requirement. One might consider in this connection one of the examples in Wilson

(1981): let ZP he a set theory with the natural numbers as urelements, consisting of

Zermelo-Fraenkel axioms plus the usual Peano axioms restricted to the urelements; let ZF be the standard Zermelo-Fraenkel set theory without urelements. The intended structures of these theories are apparently not isomorphic, under the ordinary membership relation: the “natural” morphism between the two models — inclusion - is not one-one. Wilson shows, however, that it is most natural to regard the membership relations on the two interpretations as different, though interdefinable; furthermore, the interdefinahility is such that an intended model for

ZF with the usual membership relation can be extended via the definitions to one which is isomorphic to the “intended model” of ZP. Thus even in cases such as this, as Wüson proposes, it would he difficult to discern a contrast in the ontology of these two theories. (This also makes us wary of simple identification of the ontology of a theory with its intended interpretation.)

What are, then, the affinities between Skolemite relativism and Quinean ontological relativity? They seem to reside in the shared notion that to interpret a formal theory, and to explain the concepts an understanding of which we can

1 1 6 legitimately attribute to someone using that theory, amounts basically to introducing another theory as the interpreting framework in which the explanation will be couched. Then, the interpreting theory will itself be in some need of interpretation, and the cycle of reinterpretation is thus primed again. The interpreting theory assumed as background is the relativizing parameter. This is the picture that is of great concern to the early Skolem, for whom an understanding of such fundamental mathematical notions such as those of set theory should not suffer of this degree of “instability.”

There are differences, however, though not necessarily easy to gauge.

Skolem appears to be most concerned with the relativity infecting set-theoretical concepts. What Skolem worried about is that the practice of set theory, contrary to appearances (or at least expectations of the practitioners), may turn out to incorporate no well-defined mathematical concepts. The skeptic’s relativism, Skolem thinks, unsettles the usual criteria that we would use for the correct application of those concepts. The worry is ultimately that we might be applying set-theoretical concepts in systematically “deviant” ways (we would be doing so if we were hving in a Skolemized universe). In Quinean terms, this suggests that Skolem is worried about indeterminacy in the ideology of set theory. Such a concern is not immediately translated into the Quinean framework. Presumably, to apply a concept consists for

Quine in applying a common noun or other general term; it is questionable whether

Quine would have much use for a theory of concepts as “implicit in a practice,” or such, aside or beyond the usual behavioristic model of stimulus-response correlations. But it is doubtful whether Skolem would be satisfied with a theory of concepts such as can be found in Quine, or would instead tend toward less

117 logocentric views.

However, the chief difference between Skolemite relativism and Quine's ontological relativity can be seen firom the fact that, as I shall presently argue, the latter fails with respect to mathematical theories, if these answer to a structuralist conception of reference and ontology. The challenge of Skolemite relativism, on the other hand, does not spare the structuralist; the Skolemite casts doubt on the idea that a theory might precisely characterize a unique structure in a non-relative way.

Is ontological relativity true?

The thesis I advocate in this section is that ontological relativity, as a challenge to the concept itself of an intended interpretation, fails with regard to mathematical theories, mostly on the score of the considerations we have articulated in the previous section. The claim that ontology is inescapably relative founders, with respect to mathematical theories, on the kind of structuralist intuition which we have seen embodied in the “second strategy.” The Quinean response to this sort of intuitions is of necessity ambivalent: I argue that the Quinean cannot simply reject such intuitions, for the rejection would in the end make the notion of a theory being “committed” to an ontology utterly unintelligible. Then it follows that, if we recognize the Skolem problem as posing a non-trivial challenge to the notion of intended interpretation as it applies to these theories, we must conclude that the

Skolem problem — Skolemite relativism - is not a problem of ontological relativity

(at least in Quine’s terms).

A further point to be considered is that Quine himself thinks that ontological

118 relativity does not justify Skolemite skepticism; he rejects “neo-pythagoreanism,” by which label he means the result of Skolemism about set theory. But his reasons for doing so appear to be insufScient — and, more interestingly, at odds with stated principles of the doctrine of ontic commitment which constitute the underpinning of ontological relativity. The constraints imported by Quine in the attempt to limit, in effect, the scope of ontological relativity must be regarded as somewhat ad hoc within the boundaries of a Quinean environment: in other words, Quine’s own proposal on this count conflicts with other important Quinean notions that we have seen deployed in setting up ontological relativity.

The point we have urged is familiar. It is based on the structuralist refrain that the ontology of mathematical theories is individuated only up to isomorphism.

Suppose that, in keeping with the style of the previous discussion of ontological relativity, we have two interpretations of a mathematical theory (say number theory) over two distinct ontologies in Quine’s sense: following a famous example of

Benacerraf (1965), these ontologies might consist of, respectively, the Zermelo numerals and the Von Neumann finite ordinals. By the criteria appropriate to ontological relativity, these two count as distinct ontologies - they would support, by

Quine’s count, two distinct interpretations. In fact, this would be an apt example of ontological relativity: it may be used to show that the ontology of number theories.

Le., numbers, can only be specified up to the choice of background theory (together with either a theory of Zermelo ordinals or one of Von Neumann ordinals).^' But the

“ In fact, Harman (1969) presents just this case as paradigmatic of the kind of indeterminacy one gets in translation. Although this is clearly not an instance of radical translation, the Quinean moral is, as Harman argues, that it makes no sense to ask for a specification of the ontology of number theory “unless one asks relative

119 two models underwriting these “distinct” interpretations are isomorphic: that means that firom the mathematical point of view there is no distinction of importance between them. The structuralist intuition is that we should not count two distinct ontologies as being foreshadowed by a case Uke this. On the score of isomorphism, there should really be no significant difference in the ontology of numbers, which is what the theory ostensibly is about.

Since we have set up as a foil the much-discussed case devised by Benacerraf, a comment related to what directly concerned him might be in order here. According to Benacerrafi the seeming impossibility to identify numbers straightforwardly with a given set of objects (chosen among the Ukehest candidates) demonstrates that numbers cannot be “objects” at all:

Therefore, numbers are not objects at all, because in giving the properties (necessary and sufficient) of numbers you merely characterize an abstract structure -- and the distinction lies in the fact that the “elements” of the structure have no properties other than those relating them to other “elements” of the same structure. (1965, p. 291)

This passage may be taken as a challenge for structuralists to clarify, in a nutshell, what they take number-theoretical names to be names of If Benacerraf is right, one cannot address the issue by selecting one collection of things, among the many, to serve as interpretation for number theory. But what should we make of an ontology without objects? As I have indicated, there is a deep problem about the individuation of structural entities, places in a structure, and the Uke (see the works cited in fii. 38). Whether or not such a problem can be solved, Benacerraf s conclusion does not affect the (weaker) point we are trying to score against Quine. to some envisioned general schema of translation.”

120 In fact, the conclusion itself looks remarkably un-Quinean, and would in no way buttress the case for ontological relativity. For Quine’s notion of an object is a logico- linguistic one, on loan firom the apparatus of quantificational logic; to be an object is to be the (possible) value of a variable. On Quine’s understanding of the notion, therefore, it would be nearly unintelligible to claim that numbers, taken as the range of the variables of first-order arithmetic, cannot be objects. The “abstract” nature of numbers, which, Benacerraf suggests, is to be attributed to their being in some sense semantically “general,” presents firom this point of view no additional difficulty (if it did, anyway, a reinterpretation in terms of Zermeban finite ordinals, or pluralities of Julius Caesar clones, or whatnot, would solve the problem, by

“eliminating” the problematic range of quantification). Thus there are prima facie three responses to the Benacerraf problem that have emerged:

(a) since numbers cannot be identified with any particular range of individual objects, they are not objects at all (Benacerrafi;

(b) since numbers cannot be identified with any particular range of objects rather than with any other such range fulfilling arithmetic, we can say what kind of objects they are only relative to a chosen interpretation of arithmetic (ontological relativity);

(c) there is no significant ontological difference between the various interpretations considered: either one is committed to an ontology of numbers, or anyway to the same ontology as arithmetic (structuralism).

To interpret a language amounts to (i) indicating a domain of objects as the referents of individual constants in the language to be interpreted and (ii) specifying what properties and relations of such objects are designated by, respectively, the predicate and relations terms of the language. When we refer to the interpretation of a theory, we can understand this to mean an interpretation which is also a model of the theory, i.e., an interpretation under the stipulations of which all statements of

121 the theory come out true. Consider two interpretations A and B of a certain theory, with respective domains D'^ and D®. We can say that A and B are isomorphic when the following obtains; there is a one-one mapping f of D''^ onto D® such that, for every n-tuple a, ...a, of elements of and predicate R, R'(ai,...,a„) if and only

R®(f(ai),...,f(a„))-where for each n-ary predicate letter R in the language of the theory

R'^ (R®) is the property or relation interpreting R in A (respectively, in B). The main property of isomorphic interpretations is that they satisfy exactly the same formulas: the theory which they interpret cannot distinguish between them. Thus, the structuralist intuition naturally suggests the following idea for comparing ontologies: if there is one-one mapping from the ontology of A onto the ontology of B such that it preserves the truth of everything the theory says about A (or, symmetrically, about B), then A and B represent the same ontology.

According to basic set-theoretic notions, there is a one-one mapping from A onto B just in case we can consider A and B as containing the same number of objects. On the other hand, if A has the same number of elements of an interpretation of T, the equinumerosity can be exploited to induce an isomorphic interpretation of T over A. We see then that ‘structure’ is a concept encompassing at least these two components: ‘how many things’ are there and how they are interrelated; in this sense, a theory of A’ can he regarded as supplying a description of how the elements of A are interrelated (i.e. as supplying the second component of structure).

There is another subtlety lurking here, noted in Wilson (1981). Consider as an example a “diminutive” arithmetic aimed at systematizing only facts about, say, addition; such a theory would have none of the usual Peano axioms defining

122 multiplication between natural numbers. What is the ontology of this theory, and how does it compare to the ontology of Peano arithmetic? We should presumably assign to it the same ontology we attribute to full arithmetic — certainly that seems to be the intended ontology of the theory -- in spite of the fact that there is not structural identity between the two theories: in a fairly clear sense arithmetic describes “more structure” and therefore requires more structure of its ontology. Yet it is doubtful that this kind of case can be turned into grist for the mill of ontological relativity. For a natural way of understanding the comparison between these two theories would be to view diminished arithmetic as a partial description of arithmetical structures. A theory of addition, in other words, can be regarded as capturing only part of the structure that full arithmetic is intended to describe completely. Clearly, the cases that are relevant to assessing the status of ontological relativity will concern theories that are intended to characterize a domain or a structure more or less fully. The problem is not merely that some theories are incomplete and therefore faü to pick out a structure unambiguously: of course, most theories of algebraic structures are Uke that, but significantly Quine never complains that the ontology of group theory is always absolutely indeterminate; that would be true, but is not the point. The bite of ontological relativity must be felt in the contexts where one should reasonably expect that one structure has been characterized, cases in which one (if not the) measure of theoretical success would be to characterize exactly one structure.

It is in cases such as these that the divergence between the structuralist and the ontological relativist can be seen. To the Quinean observer, there is stiU available a multiplicity of distinct, though isomorphic, interpretations, and we must

123 choose one among them whenever we need to “specify^ an interpretation. For the structuralist, there is no ontological relativity at work; under the circumstances we are imagining, the ontology of the theory has been determined; remaining differences, such as they may be, are cosmetic: we can choose several ways of saying the same thing. This suggests in turn that either the Skolem problem, as we have called it, can in essence be “solved” by the same structuralist observations that got in the way of ontological relativity or else that the Skolem problem is not a variant of ontological relativity after aU.

In point of principle Quine is not opposed to structuralist considerations at alL In fact, what he explicitly advocates in. regard to the ontological problems presented by mathematical theories amounts to a variant of the structuralist approach. Although he does not phrase his support in precisely this fashion, he would in all likelihood subscribe with few reservations to the idea that isomorphic interpretations and models of a theory exhibit the same structure, however one may want ultimately to cash out the general notion of‘structure’. But he seems to consider the possibility of isomorphic interpretations as evidence that the determination of which interpretation is operative within a certain scientific practice can only be a somewhat arbitrary affair. Suppose, in other words, that two interpretations of arithmetic are isomorphic: a Quinean might reckon that, precisely because they are, arithmetic can yield no ultimate verdict about which one is correctly assumed.^" The structuralist invitation to consider arithmetically

See Quine (1969a), p. 44: “Always, if the structure is there, the applications will fall into place. As paradigm it is sufficient to recall again this reflection on expressions and Godel numbers: that even the pointing out of an inscription is no final evidence that our talk is of expressions and not of Godel

124 indiscernible interpretations as embodying the same arithmetical ontology is declined with this piece of reasoning:

[i]t is in this sense true to say, as mathematicians often do, that arithmetic is all there is to number. But it would be a confusion to express this point by saying... that numbers are any things fulfilling arithmetic. This formulation is wrong because distinct domains of objects yield distinct models of arithmetic. Any progression can be made to serve; and to identify aU progressions with one another, e.g., to identify the progression of odd numbers with the progression of evens, would contradict arithmetic after alL

However, this argument is surely mistaken. To identify “progressions” of diverse objects to serve as interpretation for the laws of arithmetic may turn out to be a difficult matter, for to make sense of such identity statements as would result from this kind of identification is not at aU obvious. But whatever the sense of these identity statements, it is clear that they would not contradict arithmetic, because no formula of arithmetic is evaluated across interpretations, as it were, and the identification of distinct progressions that Quine mentions here would have to be made across interpretations. The situation is a strict replica of that which we have discussed in regard to the Lowenheim-Skolem theorem: in that case, though we have a set that is in a sense both denumerable and non-denumerable, no contradiction can be produced within set theory, because a statement expressing the contradictory properties of that set can never be evaluated in any one interpretation of set theory.

There is something like an engine of Ontological Relativity, a fact about semantics that Quine has skillfully woven into a web of behef about how reference is determined, and a fact that still impresses many. It is Putnam, however, that states numbers.”

Ibidem, p. 45.

125 it most clearly and attempts to build a case for referential indeterminacy entirely on

this basis (let us call this “the FactT):"

let Ml be a model of theory T, with domain Di. Let D, be a domain of the same cardinality as Di_ Then there is a one-one mapping G firom Di onto Do. We can “induce” a model M, over D, by using G: assign to the predicate P of T the image under G of the interpretation of P in Ml- So G is also an isomorphism between Mj and M, -- by construction of the model

This is pretty much what Quine, quoted approvingly by Putnam, is saying: any

range of the same size can be made to serve as an “alternative” though truth- preserving model by a suitable reinterpretation of the predicates. Putnam, puts it hke this: “ if the number of cats happens to be equal to the number of cherries, then it follows that firom theorems in the theory of models [Le., the Fact]... that there is a reinterpretation of the entire language that leaves all sentences unchanged in truth value while permuting the extensions o f‘cat’ and ‘cherry’.”

Take for example arithmetic (PA). Its standard model is the set of all natural numbers, and that is of order type co. If we are talking about first order PA, there are non-standard models. But many of these non-standard models are still countable.^°

Let NS be a countable non-standard model of arithmetic. Though non­ isomorphic, 0) and NS have the same cardinality. Le., aleph nuU. So by the Fact it is certainly possible to use the domain of NS to produce a model that will be isomorphic to the standard one. This much is uncontroversiaL This state of affairs

This is contained in the Appendix to Putnam (1981).

For a description of the structure of non-standard models see Boolos and Jeffrey (1974), ch. 17.

126 should fît into the Putnam-Quine picture of referential indeterminacy. Suppose we say, for example, that this shows that, even though virtually all arithmetical practice takes place under the presumption that the practitioners are talking about the natural numbers, the same practice and the same discourse are perfectly consistent with the hypothesis that we are in fact referring to (we live in a world consisting of) the “unnatural numbers” -- the unnatural numbers being the ones that make up the domain of the non-standard model in question.

The problem for the Putnam-Quine view derives firom the following situation:

(1) since the individuation of a “structure” requires something more than just counting the number of things, structure is not preserved by the kind of construction of an isomorphic model that the Fact makes available. In the particular example, we have used the “construction” of a isomorphic model changes the relevant structure of one of the domains. Thus it would be absurd to say, on the basis of the fact alone, that we have shown that arithmetic can be interpreted on an indistinguishable reference scheme in a domain with the structure of NS: after the reinterpretation, the structure is no longer there. But (2) Putnam and Quine must sustain the thesis that the construction of precisely that model (the isomorphic one) actuates a change in the reference scheme. But if reference to mathematical objects is reference to objects occupying certain definite structural roles, and isomorphic models embody the same structure, then “replacing” an isomorphic model with another cannot plausibly he construed as affecting whatever reference relation was in place. So in the mathematical case Quine’s statement that “any range of the same size can serve for reinterpretation” is true, but offers no support to the

Putnam-Quine view: ontological relativity, it seems to me, has not gotten off the

127 ground.

It may be objected, however, that the structuralist’s reaction against ontological relativity produces at best a stando& In the sort of cases we have described (cases in which we have multiple but isomorphic interpretations), structuralist criteria yield one verdict. Le., that the ontology has not changed, while

Quinean criteria yield a different one, Le., that there has been a change in ontology.

But the Quinean has no compelling reason to embrace the competing criteria. Thus the structuralist can only oppose a different intuitive picture of the situation, but not argue that his picture is better than Quine’s.

Yet things are not quite so simple. For to apply the Quinean criteria to

“count” ontologies in this way would tear up the notion of an ontological commitment implicit in the use of a theory. Consider once again a Benacerrafian example: Susan is doing arithmetic and engaging in talk of numbers, yet she could be interpreted equally well as referring to Zermelo finite ordinals or as referring to Von Neumann ordinals. Her arithmetic competence, though, need not extend to the theories in which one formulates these two interpretations. If we adopt a Quinean line, we will say that the two interpretations portend distinct ontologies for Susan’s theory. But now the question will be, what is the ontological commitment of Susanl Since she does not need to know set theory to do arithmetic, she may just ignore what a

Zermelo, respectively a Von Neumann, ordinal is. Moreover, since either interpretation is by hypothesis perfectly compatible with her arithmetical practice, nothing in the practice itself can be said to commit her to one or the other view about numbers. But then the only reasonable conclusion is that Susan is not committed to either one or the other ontology. If those were the only ways in which

128 the ontology of arithmetic can be cashed out, it would follow that Susan is not properly committed to anything, in her arithmetical practice. But this, certainly, is an undesirable conclusion for Quine: it would have the consequence that Susan is committed to the existence of prime numbers, but not of numbers. Clearly, if the notion of ontological commitment has any traction, it would have to be the case that someone doing arithmetic is committed to the ontology presupposed by arithmetic.

But this cannot be, as we have seen, the same as either Zermelo or Von Neumann ordinals. The ontology of arithmetic ought to consist of the kind of objects that arithmetic deals with. Le., numbers. This conclusion, however, does not seem within the reach of the Quinean, unless he drops the criteria of ontological evaluation by which the Zermelo and Von Neumann interpretations described by Benacerraf count as importing two distinct ontologies.

It is not surprising, therefore, that Quine himself has shown in several places a propensity to claim the advantages of structuralism. As we pointed out, the distinction of a notion of ontology from that of ontic committment is entirely Quine’s.

Now the structuralist strategy that emerges from that distinction is not his; but it turns out that, in a pinch, Quine appeals to something very much like it. This happens just with regard to his somewhat pecuKar dismissal of Skolemite scepticism, especially in (1969) and (1976).^

Quine’s discussion of the Skolem problem is framed in terms of reduction, and more specifically of ontological reduction. In contexts of ontological reduction, the

Quine (1976).

129 question is whether a certain ontology can be eliminated in favor of another.

Although the notion entertained seems prima facie distinct &om that of intertheoretical reduction. Le., reduction of a theory to another (whatever that comes to), given the way ontological problems are understood in Quine the question becomes one concerning theories: when can we dispense with the ontological import of a theory in favor of another? Once put in these terms, reduction is a species of interpretation (which is, as noted, a species of translation), and a natural criterion emerges, parasitic on interpretation. To show that an ontology of F-things can be eliminated in favor of an ontology of G-things, we need to show that the theory presupposing Fs can be (re-)construed as a theory presupposing only Gs; i.e., if the range of variables consisting of Fs is replaced by the range consisting of Gs and the predicates of the F-theory are suitably redefined as needed, the result is the G- theory. A simple paradigmatic example would be, according to Quine, the reduction of the ontology of a theory consisting of theorems of the form The temperature of x is n-degrees-Celsius’ to that of a theory translating the above statement into The

Celsius-temperature of x is n’: the former presupposes “impure” numbers-of-degrees

(by standard Quinean criteria), the latter only the standard natural numbers. The distinctive feature of ontological reduction is that, as in this case, the predicates of the reduced theory are translated into the predicates of the reducing theory in such a way that coextensiveness is not preserved: it cannot he said, prima facie, that the two statements are true of the same things.^’ Yet the mapping is truth-preserving.

It is clear that the structuralist of our narrative will depart from Quine on this point. Following Wilson, we should say that the two theories mentioned in this paragraph are interdefinable in such a way that, given a model of the latter, one can construe from the definitions a model isomorphic to a model of the latter. But this

130 Clearly, even in this simple case the fact that there is a truth-preserving correlation between the statements of one theory and those of the other is not the whole story: we would not say that the theory presupposing only natural numbers does away with the need for impure degree-numbers if some true statement involving degree-numbers were not uniquely mapped onto a statement about natural numbers. But this requisite can be secured when we reflect that degree- numbers are mapped one-one onto numbers; such a condition genuinely shows (if anything can) that the presupposition of degree-numbers was “unnecessary.” This corresponds to the Quinean requirement that to achieve reduction of the ontology of a theory to the ontology of another theory we “define” proxy functions firom the one to the other: “a function which assigns one of the new things, in this example a pure number, to each of the old things-each of the impure numbers of temperature”

(1976, p. 217). Now, this requirement seems rather intuitive, if our purpose is to show that certain ontological presuppositions are not necessary (are, in effect, not presuppositions!).

But this picture is comphcated by two orders of considerations that Quine introduces in regard to reduction. The first is that sometimes we want to argue not that some of the ontological presuppositions of a theory are not necessary, but that they are wrong, or anyway undesirable. This is the crux of the matter in many philosophical disputes: for example, this is basically what monists about mental states oppose to the dualists, or what nominalists, who reject universals, oppose to the platonists. In these cases, the “reduction” of one ontology to the other (e.g., one does not concern us at the moment: we are trying to show that Quine ends with a requirement (for reduction) which committs him to structuralism.

131 with universals to one without) implies a theoretical revision: the point is not exhausted by mapping true statements of the theory to he reduced onto true statements of the theory doing the reducing, since we want to show that more or less sizable chunks of the original theory are simply false. In the former kind of cases, on the other hand, reduction is basically conservative (as it is basically aimed to show that one theory is an extension by definitions of the other). What Quine says about reduction suggests that the concept of ontological reduction should he applicable in both sorts of cases: where the reduction is fundamentally conservative, and where it requires (or rather purports) revision. It is perhaps characteristic of

Quine’s approach that he would consider the “elimination” of impure degree- numbers firom the presupposed ontology in favor of pure natural numbers, along the lines sketched above, as somehow on a par with the elimination of say, universals firom a nominalistic ontology.^ Whatever the reasons for this attitude, we must now ask: what sense can be made, in the re visionary cases, of the requirement that proxy functions be available? The defining characteristic of proxy functions is that they establish a one-one correspondence, but the point of revisionary reduction seems to be exactly opposite at least in spirit: we want to be able to show that some of the entities postulated originally need not (and do not) have counterparts in the ontology of the reducing theory. Quine notes, almost in the same paragraph, that “what justifies an ontological reduction is, vaguely speaking, preservation of relevant

^ Of course, it is not at all obvious how exactly an account of the intuitive diSerence between these two kinds of cases might go. Further, it is a fair question whether “reduction,” once properly defined, can be revisionary at all. However, we need not concern with this right now: the issue at hand is how proxy functions would fare in the re visionary contexts (assuming that they more or less “work” in non­ revisionary cases, such as those envisioned by Quine at the beginning of (1976)).

132 structure”-and then that he “prefer[s] to let different things have the same proxy...

Relieving such inflation is a respectable brand of ontological reduction.”’*® But it is rather mysterious how even the mildest form of ontological inflation can be relieved by preserving structure via one-one correspondence!

Another problem is that it is rather unclear how this mapping can be defined.

To be sure, if one’s instincts are to refer to model-theoretical instances where such mappings are established between models, one would probably not find this issue critical: all that is needed in such cases is to appeal to some robust metatheory, typically set theory, and to give an argument for the existence of the mapping that can be (though rarely is) formalizable in that metatheory. But in Quine’s case some puzzlement is justified, because the focus is on the ontological presuppositions of the theories involved. It seems legitimate to ask, for example: if we can show that T can be reduced to T via proxy functions defined in set theory, exactly how much are we presupposing? One might say that this is just another reincarnation of good old ontological relativity: not only is ontology relative to a choice of interpretation, so is ontological reduction. Again, though, this response should give pause to someone who, Uke Quine, harbors general antipathy toward set theory of a straightforward

ZermeUan sort, because it quickly leads to the conclusion that a sufficiently robust set theory, complete with “staggering ontology,” has to be presupposed even just to compare ontologies of much weaker theories. What is more, such a committment would not even be relative (to a choice of translation or interpretation schema), strictly speaking: for under the hypothesis we are entertaining ontological analysis

*® Quine (1976), pp. 219-220.

133 presupposes a uniform background theory, namely set theory itself, in order to justify the application of such key notions as that of proxy function; it is not just the

choice of a determinate translation schema that carries the presupposition of set

theory, it is rather translation in itself that requires it (whatever the particular schema). This leaves no room to think of such a presupposition as “relative”-- at least not in the sense of ontological relativity—since there is no evidence (even at the level of thought experiment) that might suggest the availability of schemes not presupposing set theory.

These problems make it rather difficult to beheve that proxy functions may be smoothly integrated in the Quinean picture of ontological analysis (as Quine himself seems to beheve). But the primary reason we briefly discussed the concept is that Quine regards proxy functions as his motivation for rejecting the challenge of

Skolemite scepticism. Quine apparently construes the Skolemite challenge as a brief for what he calls “Pythagoreanism,” that is, the thesis that all mathematical ontology can be reduced to that of arithmetic—natural numbers. Pythagoreanism, as such, is StiU ambiguous: it could mean that every (mathematical) theory-in particular set theory—is in the end reducible to arithmetic; or it could mean that the ontology of every theory is reducible to an ontology of natural numbers. In the first case one needs to prove that every truth in (the language ofi the “bigger” theory can be reinterpreted as a truth of arithmetic. Le. a truth expressible in terms of arithmetical predicates. In the second case, it is required that we provide a model of the bigger theory based on the natural numbers, but not necessarily characterizable in terms of the non-logical constants of arithmetic. Quine (1976) does not completely dispel traces of this ambiguity. But, as Bonevac (1982) indicates, the first possibility

134 is excluded by considerations about the respective deductive strength of theories: for

example, in set theory one can prove statements asserting the consistency of

arithmetic (at any rate, some such statement will be a set-theoretical truth); but

there is no arithmetical statement that can “map” a similar theorem of set theory.^

One could then argue that the first interpretation of Pythagoreanism is simply blocked by proof-theoretical or “linguistic” considerations.

The second interpretation is certainly closer to what Skolem thought could be accomplished by following the sceptic’s argument against Zermelian set theory.

Now, the stipulation that ontological reduction requires the definability of suitable proxy functions scuttles this version of Pythagoreanism in a straightforward manner: since the ontology of set theory and the ontology of arithmetic do not have the same cardinality, there cannot be proxy functions, which “assign one of the new things [numbers] to each of the old things [sets]” (1976, p. 217). It is clear that the proof of the Lowenheim-Skolem theorem does not introduce proxy functions, either.

Thus, no ontological reduction of sets to numbers, such as Pythagoreanism would presume, is forthcoming on that basis.

To be sure, Pythagoreanism is not really what Skolem had in mind, and

Skolemite scepticism is not affected by the demise of Pythagoreanism. Skolem was clearly not concerned with the possibility of reducing or eliminating the ontology of sets in favor of a spare universe of integers—much less with the possibility, foreshadowed by the fiirst version of Pythagoreanism, that “everything is number.”

Skolem’s concern is integral to set theory: it is that the understanding of central set-

^ See Bonevac (1982), pp. 115-118.

135 theoretical concepts does not seem fully determined by the “correct use” of a theory which supposedly embodies precisely those concepts. This concern is at root a typically semantic one. Skolem clearly did not think that a one-one correspondence could be “defined” between a standard universe of set and a standard universe of integers, providing proxy functions for ontological reduction (really, no one familiar with a basic cardinality argument could think that!). But Skolem’s problem was that the concept of correspondence could be not blithely relied upon to solve the problem, for it is itself a set-theoretical concept exposed to relativism. Someone sensitive to Skolem’s argument could not therefore be contented with Quine’s assurance that proxy functions, though not typically available either in the reduced or in the reducing theory, may he definable “in the metatheory.” Not that this is an unreasonable assumption, on Quine’s part: it is just that it would not assuage the

Skolemite’s worries.

It is time to take stock. We see now that Quine’s contribution to relativism in ontology is stretched between two opposite poles. On the one hand, we have ontological relativity, a view borne out of the combination of Quine’s investigations on indeterminacy and the doctrine of ontic commitment. According to this view, essentially any linguistic reformulation (any translation, in fact) must be tallied up as a change in ontology; there is nothing more fundamental to ontological analysis than these linguistic shifts. On the other hand, we have the oft-repeated demand, in

Quine, for “preservation of structure”; and we have seen that there is a very strong pull toward the intuition that in mathematics sameness of structure should he identified with sameness of ontology. What Quine’s reflections on ontological reduction show is that preservation of structure is not achieved cheaply-certainly

136 not at the mere price of supplying an alternative translation schema. This is where proxy functions come into play, and indeed where they have to come into play if preservation of structure is to be attained.

In a sense, the existence of proxy functions will come as no surprise to a structuralist, for they are another incarnation of the idea-basic to the structuralist approach—that isomorphic domains have the same structure. But in the Quinean approach the requirement of their existence appears unmotivated: it is a model- theoretic requirement, inserted in a theory of ontology that has been predicated, up to that point, upon the notion that in order to eliminate a certain ontological commitment it suffices to provide a truth-preserving translation of the theory. This is also the source of the difficulty, briefly described above, concerning the linguistic means at hand for the specification of proxy functions. Since they are set-theoretical objects, it is clear that in many cases of ontological reduction neither of the theories involved (the reduced and the reducing one) will have the means to define the relevant proxy functions-hence Quine’s appeal to whatever metatheory may result convenient here. The question is: are they necessary for Quine’s purposes? As

Bonevac (1982) comments, if reduction is viewed chiefly as the elimination of a certain firagment of discourse, why worry about the introduction of model-theoretic constraints such as proxy functions? The answer is, I believe, that without some such constraint on ontological reduction there could be no claim of preservation of structure, except perhaps in the sense of the mock structuralist view that in the previous chapter I labelled “naive structuralism.”

137 CHAPTER 4

A ‘‘COHERENT SOURCE OF AXIOMS59

The aim of this chapter is mainly to provide some justification for the interpretive thesis we have oSered in regard to Skolem’s thought about set theory.

My contention in that respect was that Skolem conceived of the proper interpretation of the existential quantifier (and therefore of the range of quantification) in set theory as given in an a priori manner by logical principles; this, I beheve, was the idea that led him to reject as “unnatural” the restricted interpretations that are made possible when set theory is formulated as an axiomatic first order theory. Now it is time to verify whether this conjecture has any theoretical plausibüity. According to the interpretive hypothesis, in Skolem’s conception logic and set theory are thoroughly enmeshed. In the conception of logic

“as calculus,” Skolem’s allegiance to which we have there argued for, logical analysis is for the most part carried out in the firame of a metatheoretical set theory; we have tried to show how this conception of set theory would motivate Skolem’s reaction against what he took to be the axiomatic approach of Zermelo. Now it is time to

138 show that this position has some initial plausibility. That is, we intend to show in this chapter that there are indeed key components of what is even to this day considered the “standard” notion of set that are of a logical origin. In particular, we will focus on the role of the axiom of choice in the so-called iterative conception of set, and try to make a case for the the logical nature of the concept of arbitrary choice. Within the general economy of our project, the axiom of choice does not provide merely “anecdotal” evidence for the broader claim sketched above; its deeper significance lies also with the peculiar notion of existence it suggests vis-à-vis mathematical objects. The hope is that it wül be possihile to exhibit logical notions.

Le., notions that have to do with the inferential practices warranted by mathematicians, as intimately connected to notions that are central to the basic understanding of the concept of set; if we can do this, then, I think, we can plausibly claim that the intended interpretation of set theory to which Skolem originally appealed was rooted, so to speak, in inferential practice.

Many today believe that the conception of sets embodied in Zermelo’s axiomatization, on this view, was not formulated in response to lingering doubts about the viability of the notion of set. The immediate purpose on Zermelo’s part certainly was to shore up his proof of the well-ordering theorem; but the substance of his analysis derived firom an informal notion that most set theorists understood as perfectly safe. Kreisel (1967) remarked, for instance, that the development of set theoretical notions, not unlike that of other more “traditional” mathematical notions, is rooted in the analysis of an informal but not unreliable notion (or, perhaps, of several such informal notions). This makes Zermelo’s axiomatization an enterprise of a diffèrent sort, much more like a philosophical analysis of a concept already

139 embedded in mathematical practice. According to Kreisel, Zermelo’s analysis focusses on a notion of multiplicity grounded in mathematical practice: the notion of set of previously obtained objects, or iterative sets. Zermelo noted that the sets customarily introduced in mathematical practice are collections of elements of other sets previously defined. This notion must not he confused (and typically was not confused, according to Kreisel) with two other ideas that also present themselves in the unrefined concept of collection or multitude in general: sets as the extensions of predicates, and the notion of finite set, or intuitionistically completahle collection. It must have been clear firom the outset that the iterative notion is immune firom paradoxes like Russell’s, since it is not assumed that the formation of new sets can be carried out by application of a principle of unrestricted comprehension. This notion, therefore, implicitly supports a separation principle of the sort Zermelo proposed in his axiomatization.

Kreisel asserts that the notion of set analyzed by Zermelo is a “coherent source of axioms,” in that the basic postulates of Zermelo’s axiomatization are — in an informal yet no less rigorous way -- derived firom it. This would show that the formalization of set theory, starting with its presentation in axiomatic form, rested on a preceding intuitive theory, which was not flawed or unreliable. We concentrate now on the previous idea, according to which the axioms of set theory can be derived firom the particular intuitive notion that gives rise to Zermelo’s cumulative hierarchy.

Cantor calls a set “any collection into a whole M of definite and separate

140 objects m of our intuition or our thought.”®' Taken at face value (‘any collection’), this explanation of the concept gives rise to the difficulty uncovered in Russell’s paradox as well as in other paradoxes. While writing these words, in 1895, Cantor himself was not unaware of those difficulties, which he thought would not be fatal to the very notion of set. In the 1899 letter to R. Dedekind, Cantor tackles problems of the sort uncovered by Burali-Forti about the existence of “iR-behaved” multiplicity.

The paradox of Burali-Forti is about one such multiplicity, the collection of all ordinals. Burali-Forti takes it to have shown that “it is therefore impossible to order the order types in general, or even the ordinal numbers in particular; that is to say, the order types cannot provide a standard class for the ordered classes, as the class of integers ... does for the finite classes and the denumerable class”.®"

WTiile Burali-Forti regards the fact as a serious defect of a theory of ordinals.

Cantor takes it to reveal a distinction in the notion of‘multiphcity’. The collection of all ordinal numbers is an “inconsistent multiphcity” in which “the assumption that all of its elements ‘are together’ leads to a contradiction:” in this case, then, “it is impossible to conceive of the multiphcity as a unity.” These multiphcities are not sets. Sets are coUections of things that can be “gathered together into ‘one thing’.”®'’

Thus, a Cantorian set is something that can be considered as a “one”. Le. as an object onto itself, formed when a number of other things are taken together. The operation of collecting these things into one - of forming a one out of many, or

®‘ Cantor (1895), p. 86.

®" Burah Forti (1897), p. 111.

Cantor (1899), p. 114.

141 perhaps, as more traditional philosophers would have put it, of abstracting their differenœs and associating them -- “for what could be more elementary than associating a number of individuals into a class,” Frege — that operation does not guarantee that the resulting object is a set. What does? What must he added to that?

The additional condition is the one embodied in Zermelo’s conception of the cumulative hierarchy of sets.^ Sets are collections of things already available. They are formed in stages: collecting things into a whole is possible and results in the formation of a set, a new “unity,” provided that the things collected have already been formed at a previous stage of this process. Problematic collections result instead when this proviso is violated; in that case the collection includes something that could not be have been formed at a stage preceding that at which the collection itself is formed. The limitation is not linguistic, as it would be if we had constrained the type of definitions that we allow of a collection. The constraint is on the construction process, not on the possibihty of defining new multiplicities. This is the iterative conception of sets that appears in the seminal work of Zermelo (and

Russell, although saddled by further comphcations) and seems motivated by rather straightforward reflection on Cantor’s idea.

Zermelo’s idea is to make explicit the mathematical “laws” of a notion that

^ See Zermelo (1930). According to Zermelo, the conception of the cumulative hierarchy espoused in that paper was a natural outgrowth of his earlier work on axiomatization, and particularly on the Axiom of Separation: the axiom is part of the analysis of such a conception of how sets come into being, rather than just a response to the paradoxes. The idea of proceeding “in stages” of some sort was also present in Russell’s conception of types (as Kreisel notes), although types are not cumulative.

142 seems a reliable vehicle of mathematical practice, that of‘set of something else or iterative set. The presumed unreliability of the concept of set may be blamed more plausibly on the confusion of two distinct concepts, that of (general, abstract) collection according to a predicate and that of set of elements of another collection defined previously. Kreisel’s suggestion seems to be that this distinction would not have been apparent to Frege — for whom sets were extensions of propositional functions, and every propositional function defines an extension -- but may already be in Cantor, as we have seen.®" We shall investigate how far the iterative notion can go. The evidence is that this notion cannot actually carry the weight of the entire edifice of set theory: there are principles that are not justified simply appeal to the iterative conception. Furthermore, I shall try to investigate in what ways an

“intuitive conception” (of set, in this case, though the problem is not specific to set theory) can be said to provide evidence for a given principle.

The iterative conception

According to the iterative conception, sets can be formed at any given point by gathering together any number of objects already determined at that point. Any collection thus formed is an iterative set. At the most basic level, we have only individuals -- i.e. objects that are not sets or multitudes, and therefore have no

This view of Frege’s conception of set seems to be supported by the reconstruction of Fregean logicism in N. Cocchiarella (1986). On the other hand, it may indirectly conflict with the assessment of others, e.g., Bell and Demopoulos (1993), according to which Frege’s stress on the “extensions of concepts” does not prevent him firom having a viable notion of set (although not strictly an iterative one).

143 elements themselves — or perhaps just the empty set. As these are all the available objects at level 0, we can form at level 1 collections of them. Then sets of objects of level 0 are in turns ‘definite, separate objects’ at level 1, and then we can form sets of sets of level 1 and objects of level 0, and so iterate indefinitely the process. From the philosophical standpoint, the most attractive feature of this explanation of the notion of set is that it makes the existence of sets to depend only on what Parsons calls the priority requirement’: that is. the condition that the members of a set be determined at a preceding stage of formation. The purpose of the present section is to illustrate how far this sparingly furnished framework (including only the simplest basis of the iteration) can be extended on the way to a justification of set theory.

At the heart of the iterative conception is the idea the operation ‘set of can be applied to objects that are given in an intuitive sense (recall Cantor’s qualification of

definite, separate’ objects to be collected into a whole’). This is the restriction incorporated in the iterative conception which rules out the possibility that the extension of arbitrary predicates (of any predicate whatsoever) be considered a set.

It may be unclear how to develop this notion of‘givenness’. A prominent attempt at explanation^ suggests that we read this requirement through the lens of a constructive metaphor. On this view, the formation of a set is described essentially as a process of construction from available elements, although the construction methods may (in fact will) have to be more ‘idealized’ than, e.g., the methods of mental construction of mathematical objects allowed by a Brouwerian intuitionist. A set is formed, according to Wang, if the objects can be gathered into a

Wang (1974).

144 surveyable totality. The combination of objects into a whole’, as mentioned by

Cantor, must be such that it affords an ‘overview’ of the multiphcity at hand;

we can form a set from a multitude only in case the range of variabüity of this multitude is in some sense intuitive— An intuitive concept... enables us to overview (or look through or run through or collect together), in an idealized sense, aU the objects of the multitude which make up the extension of the concept.'

A question will emerge in due course about the tenabihty of this view in the face of the constraints required to “generate” the theory of the iterative notion of set to the extent that is necessary for it to be a plausible intuitive foundation for set theory simpliciter. This is the question taken up by Parsons in his paper on the subject - discussed below. From our standpoint in the present context, an especially interesting aspect of this issue is whether there is, impUcit in the iterative conception, a commitment to regard set formation as a constructive or quasi-constructive process. The constructive metaphor finds easily its way into the informal explanations commonly given of the iterative notion of set. But is the metaphor merely a descriptive or pedagogical device, or rather an essential ingredient? (Perhaps the question might be stated, more ambitiously, as: to what extent does the iterative conception involve an epistemological view about sets?).

For the time being, however, we set aside such considerations and follow G.

Boolos in laying out the “theory” (the minimal, informal theory of stages).^ Though we might accept individuals (non-sets) in the basic level as previously mentioned, we

" Wang, cit., p. 531.

“ G. Boolos (1971), and (1989). The main ideas underlying this kind of work on the iterative conception are also spelled out in D. Scott (1974). Another, more systematic presentation of set theory according to the iterative conception is in M. D. Potter (1993) and (1990).

145 may assume that there are no objects except sets. (We “start” then with the empty set and only talk about the hereditary sets.) The language L contains variables of two sorts, for sets - x, y, z... -- and for stages — r, s, t... There is a binary predicate symbol on stages <, which means ‘earlier than’, and a predicate letter F between sets and stages which means is formed at'.

The sets are divided into stages, at which they are formed. Stages are ordered by the ‘earher than’ relation. It is not assumed that this relation is a well-order or a linear order. As each set is formed at a certain stage, it wül be “available” at all later stages, so that the stages are cumulative. One could include this desideratum into a stipulation that stages are nothing more than the accumulation of the members and subsets of previous stages, as Scott does in his axiom of accumulation (the ‘earher than’ relation is membership in Scott’s presentation). This idea may seem attractive because it does away with the seemingly ungraceful introduction of other primitives called ‘stages’. Boolos leaves it open exactly how to treat stages, whether as primitives or as set-hke structures. The result arrived at, though, are essentially the same: this could be taken to show that there is nothing artificial about the introduction of stages. The fundamental restriction of the iterative notion of set is that every set is formed at some stage (this is called ‘axiom of restriction’ by Scott).

The following are the only “ordering” axioms about the ‘earher than’ relation:

TEA: VsVf [^ < s& s< 0 n r

NET: VfeVf 3r (s

The following axioms express the conditions on set formation in the iterative conception:

ALL: Vx3s xFs (the axiom of restriction)

146 WHEN: VxVs{xFs^Vy(yex^ yBs), where yBs expresses that y is formed before stage s, that is, it is an abbreviation for

3t (t

In particular, nothing implies the possibility of iterating the formation of sets beyond an arbitrary finite level To allow for that possibility is clearly part of

Wang's su^estion that the construction of new sets should abstract firom limitations such as those barring actual infinities from the domain of (orthodox) constructive mathematics. In keeping with that intuition, then, we might add a version of the axiom of infinity for stages, stipulating that there exists a stage that is not immediately later than any earher stage, even though it does have prior stages.

This allows to derive an axiom of infinity, but is not needed in the derivation of other axioms.

According to Wang, one can see that the axioms of Zermelo-Fraenkel set theory are true of the iterative notion once we have an intuitive picture of it. Thus, he justifies the axiom schema of subsets or comprehension as follows:

Since x is a given set, we can run through all members of x, and, therefore, we can do so with arbitrary omissions. In particular, we can in an idealized sense check against A [the property by which we select a subset of x] and delete only those members of x which are not in A. In this way, we obtain an overview of aH the objects in A and recognize A as a set."'®

To put it in terms of stages, the force of comprehension is that at any given stage we want to regard as formed ~ in effect, to regard as existing — any collection of sets that could have been defined into existence at a previous stage. (This is not

Wang, cit., p. 533.

147 to say, yet, that we regard all possible coUections of previously formed sets as available at the stage; in order to “fîU up” the process and obtain aU the sets that can be formed at a given stage, we must rely on the axiom of Powerset.) Thus, we bave the foUowing Comprehension axiom (actuaUy an axiom schema):

COMP: Vs'5xVy(yex = iA(y)& yBs)), one such instance for each formula A(y) of the language.

This is the extent of the theory S in Boolos’ papers. S is a “minimal” theory of the iterative conception. The aim is then to show how the more substantive principles of set theory can be derived from S. S does not imply a principle of extensionality, however. The justification of extensionality does not seem to depend on any specific feature of the iterative conception (Le., on any feature that is peculiar to the iterative notion). As for other important principles of set theory (notably the axiom of the choice, as discussed below), this should not be construed as casting any doubt on the truth of the axiom of extensionality. It appears that an assumption of extensionality may be justified on a more general level than any principle specific to the iterative conception. One inclines to view it as part of a metaphysical or logical background of “set talk”, as ‘a defining characteristic of sets (in contrast with properties)’ (Wang). An assumption of extensionality would then be a feature shared by any theory that is plausibly about sets or set-hke coUections of objects. (To paraphrase Kreisel’s observation about the ‘mixture’ of concepts, we might say that extensionahty is a presupposition common to aU notions of set included, in one form or another, in the mixture). At any rate, such considerations must be postponed for the time being. As Boolos notes, wtule it may be obvious that the iterative notion of set presupposes the truth of extensionahty, it looks as if that is “a principle for

148 whose evidence the iterative conception is not responsible.” The Appendix to this

chapter contains the semiformal arguments by which it is possible to derive the

axioms of ZF in stage theory.

With the axioms of Replacement and Choice we enter a different kind of

considerations. These axioms involve concepts that do not appear to follow directly

from the principles of the iterative conception as laid out in stage theory. The

arguments usually given in the heuristic style of Wang and Shoenfield purport to show that there is reason to believe these axioms true in a universe of sets shaped in

the mold of the iterative conception. But this, of course, is not the same as saying

that the iterative conception itself provides compelling reasons for the axioms.

Especially in the case of Choice, what emerges here is a limitation of the heuristic style of argument developed by Wang and others (it is in fact a merit of Boolos’s

approach over Wang’s that it involves basic “stage principles” and an intuitive notion of derivability, rather than working with just the intuitive notion of

“constructibility” Wang adopts).

It seems easy to give an intuitive justification of Replacement, or rather of the reasoning on the basis of which Replacement might be regarded as true by someone who accepted the iterative conception of set. Wang explains this by showing how an “intuitive range of variability” available for one set yields an intuitive range of variability for a set resulting firom the first by replacing its elements with other given sets. Here we must recall that a set is given, for Wang, once an “intuitive range of variability” provides us with an overview, a capacity to survey the totality of its elements. If the range of variability cannot be intuitively surveyed, the collection characterized by it is not a set. So the explanation of

149 Replacement relies on the fact that, once we have gained an intuition of the

multiplicity of elements as one whole, we maintain a capacity to survey that

multiplicity no matter what kind of things we put in place of each elements.

This is the intuition behind a modification suggested hy Boolos to the axiom

of Comprehension as presented above in the theory of stages. If susbtitution of the

elements of a set with the elements of another in a one-to-one fashion does not affect

the “surveyabUity” of the resulting set, then injection of a set into another might be

regarded as similar in kind to inclusion: the latter being the case where the correspondence is just the identity function. We have assumed, in Comprehension,

that any subset of a set is also a set; the suggestion is that any set that can be

mapped biunivocally onto a subset of a set is also a set. The modification described hy Boolos is perspicuous firom this point of view. The axiom of Comprehension is generalized to assert that we can “isolate” (by using a formula with one free variable, or by “running through with omissions”) not only those sets included among the sets formed before a certain stage s, but also the sets that are injectible into those. Let R he a two-place predicate letter standing for a one-one function.

The Comprehension schema previously stated then becomes:

\!s Vy (yex = (A(y) & 3z(Ryz & zBs)).

In the special case when R is replaced hy the identity function, we obtain the usual

Comprehension schema, in the version previously formulated. It should be noted, however, that this change is still somewhat spurious from the standpoint of the iterative conception: for the sets injectible in a set X need not have been spawned iteratively before X itself - unlike the subsets of X, which on the intuitive picture of set formation must have come, in a clear sense, “no later than” the X itself The

150 modified Comprehension axiom does help to derive one-one Replacement, from

which Replacement proper follows (cfr. the Appendix for details).

That Replacement is really linked to a different conception of sethood is most

evident from Wang’s explanation of it. The notion Wang is clarifying more closely

resembles Cantor’s idea that sets are those multiphcities that can be united into a whole, as opposed to those that can never he given together as a whole. In the iterative conception, the relevant metaphor is that a set is constructed in stages; from the Cantorian point of view that Wang seems to be adopting, the metaphor is that a set is something whose elements can be “surveyed”, or can aU be avaüahle for inspection at the same time, as it were. This suggests that, if the notion of

Replacement apphes to iterative sets, it does not apply to them in any special way.

Again, Replacement may be true of the sets of the iterative conception, hut it is not true of them because of the iterative conception. The independence of Replacement from the iterative conception is, however, not quite the same problem that we face with Choice. Replacement can be obtained by a modification of the theory S, while no such modification (or, anyway, no modification that can be motivated hy considerations in themselves independent from Choice) is readily available with

Choice. It remains to see how natural the modification needed for Replacement is.

How to be pro-Choice

We come then to the axiom of choice, and the issue is whether this axiom also finds justification in the iterative conception. Before tackling this issue, however, we should pause to notice that there are two types of considerations that can usually

151 be found in the literature with regard to the problem of supplying evidence in favor of a given set-theoretical axiom (the same types of consideration may apply to other

mathematical theory just as well). One is the type of reflections about the iterative sets that we have been sketching so far. As we noted, this type of argument rests

fundamentally upon the notion of a certain kind of object—in this case iterative sets- and appeals, more or less explicitly, to the possibility of learning its relevant features by “inspecting” it. Therefore, it is in a ilm e n ts of this kind that we would expect to see a more or less explicit appeal to mathematical intuition, as a faculty that purportedly enables us to inspect mathematical objects. It is well-known that

Godel believed not only that such a faculty is the source of much mathematical knowledge, but also that it is an indispensable source, one which is necessary to make sense of certain aspects of set theory. It should be noticed that we are not suggesting here that arguments of this type require implausible conjectures about our ability to have some sort of direct perception of set-theoretical objects, a Platonic faculty of actually observing universals, or such. We only need to assume, at this point, that the arguments proceed firom a predetermined understanding—however acquired-of the essential features of the type of obejcts in question: thus it is clear that arguments such as those examined so far belong to this genre, for they are based on the essential characteristics of iterative sets. (Incidentally, Grodel himself explicitly admits the possibility that intuition of set-theoretical objects may not be

“immediate,” as it would be if we were simply able to have “visions” of how things are in the upper reaches of the set-theoretical universe.) Following Maddy, we could call arguments in support of set-theoretical principles that are of this type

152 intrinsic.^

Another type of argument is usually favored by those who support an

“empiricist” approach to the philosophy of mathematics, and argue that

mathematics is much more like the empirical sciences (whatever this means) than

most have thought. Under this approach, axioms of set theory are introduced much

hke empirical hypotheses, to be “tested” by the consequences they hold for the more

settled or well established areas of mathematical study (including, typically, those

that are contiguous to applications in physics and elsewhere): the evidence mustered

in favor or against them, therefore, is Oust Uke for physical science, presumably) entirely a posteriori. A champion of this view was I. Lakatos, according to whom

mathematical knowledge is advanced by the introduction of new axioms as “bold

hypotheses” that are then evaluated against a backdrop of quasi-experimental “data” provided, e.g., by finitist claims of elementary number theory or basic plane

geometry. Set theory would be no special case, in this regard. This sort of empiricist justification may support extrinsic arguments in favor or against new axioms.

We shall consider more carefully the sort of intrinsic arguments that have

been advanced in favor of the axiom of choice on the basis of the iterative conception in the form that we have presented in the previous section; we shall then comment,

more briefly, on the extrinsic justification. I think neither type of argument is convincing in the case of choice. By way of anticipation, though, a further point

See Maddy (1990), Ch. 5, “Axioms.” Maddy’s notion of “intrinsic” evidence in mathematics incorporates several notions: “intuitive” knowledge, “a priori” evidence, and “self-evidence.” It is not clear that each one of these finds application to the instance of intrinsic argument I have suggested, namely the development of the iterative conception in stage theory, but for present purposes nothing of importance hinges on this.

153 deserves notice: appeals to either intrinsic or extrinsic justification have tended to

underwrite a realist or Platonist conception of set theory (and they have been, in

different occasions, used for this purpose). What is essential to a realist account of

in particular, set theory? I suppose something like this: such an account must

regard set-theoretical statements as answering to states of afiairs in principle

independent either firom the mathematical practice involving those concept or from

our cognitive undertakings in the relevant area, and set-theoretical concepts as

referring to real properties of objects. Now, on the face of it both intrinsic and

extrinsic arguments seem to support the presupposition that this aspect of realism is

essentially correct. Godefs position seems once again paradigmatic: intuitive

knowledge of the set-theoretical hierarchy can explain why, in the well-known

formulation, the axioms “force themselves upon us as true.” Intuition, in other

words, is what makes knowledge of sets “more like” perceptual knowledge of

everyday physical objects.®^ Extrinsic arguments, on the other hand, downplay to

some extent the analogy between mathematical intuition and perception, to the

advantage of that between mathematical knowledge and confirmation of hypotheses

in empirical science. This move may in fact yield a subtler, but more solid, realist

conception than intuition of objects; Maddy, for instance, makes extensive appeal to

Observe that, if - following Maddy - we read ‘intrinsic’ as meaning, among other things, a priori’, the realist thrust of Godel’s concept of intuition is somewhat diminished: if mathematical intuition were o priori, it would be harder to conceive of intuition as a faculty that gives access to a domain of objects existing in complete independence fi:om our cognitive practices, because experience of an external state of affairs does not seem required to have a priori knowledge. For Kant, who was convinced that mathematical knowledge iiad to be a priori, it became necessary to devise a “pure” form of intuition, the objects of which are given not in experience, but as preconditions of experience.

154 extrinsic arguments just for the purpose of amending (what she takes to he) the least

plausible aspects of Godelian Platonism; the resulting view is still solidly realist. I

think both varieties of rea lism unwarranted, but this will have to be discussed at

fuU length elsewhere.

Intrinsic arguments

Zermelo declares the principle of choice to be “self-evident” and avails himself

to considerations not too different in spirit ffom the preceding observation about

Foundation; the principle of choice is shown to he obvious by the naturalness with which it has been applied — unwittingly - by mathematicians who have hardly paused to take notice of it (some of those were among the most vocal opponents of

Zermelo’s axiom). But the axiom of Choice cannot be derived from the theory of stages. This seems to indicate that the iterative conception - or at any rate the iterative conception in its reincarnation as the theory S — is neutral about the axiom of Choice: it does not commit us to its truth. It is rather natural to observe, of course, that the axiom may very well be true of the kind of objects the iterative conception is about, namely iterative sets. This can best be shown by arguments such as Wang’s, which aim at giving an informal explanation of why anybody committed to the existence of sets in the iterative sense might accept the axiom of

Choice. In fact, that is the strategy that Wang adopts with regard to every axiom of

ZFC: every axiom is shown to be intuitively true of the objects of a structure that conforms to the iterative conception.

In general, along these lines no derivation is provided, at least not in the sense in which Boolos derives some axioms in the way described from the axioms of

155 s. But the explanation offered of the axiom of Choice is especially meager. It consists of exactly three lines (in the edition quoted so far): “since every member of x is got at an earlier stage than x, aU members of x are got earlier and any selection from these can be collected together to form a set.” But this explanation cannot be straightforwardly turned into a proof of the axiom of Choice from the theory S, as it was the case for several other axioms. In fact, on closer inspection it is questionable whether the explanation purports to give even an argument (however informal) for the belief that the axiom is true. We could draw, in this regard, a distinction between substantial and non-substantial explanations of a thesis or concept. A substantial explanation of a principle is such that it would provide sufficient justification of the principle to someone who reasoned on the basis of concepts that do not presuppose the truth of the principle itself. Thus, I think that one can argue that the previous description of the iterative conception has provided us with substantial explanations of some typical set theoretical axioms on the basis of a small stock of presuppositions about stages of formation. I shall argue, next, that a substantial explanation of the axiom of choice on the basis of the iterative conception is not forthcoming; there are, moreover, reasons to suspect that the problem would not necessarily be solved by adopting a different stock of conceptual primitives (i.e., a different notion of set).

Let X be a family of sets, pairwise disjoint, formed at stage s (we can assume that there is such an s, by ALL). Then, by WHEN, all members of x are formed at stages prior to s. What we want is a choice set for x, that is, a set containing exactly one element from each of the members of x. Consider an arbitrary y ex. All members of y are in turn formed before the stage at which y is formed, say t (where

156 t

“since all elements of y must have been formed before t, each one of them severally

has been formed before t, so we can take an arbitrary element of y, and put it into a set at t or any later stage with other similarly chosen elements of members of x.”

The problem is that, by so doing, we have just stated the axiom of choice: we cannot justify the formation of the required choice set simply by saying, at this point, “pick an element of y whatsoever,” for the possibility of “picking” under these conditions is exactly the statement of axiom of Choice. What we have done is more aptly regarded as an illustration of what stipulating the axiom of choice means-of what exactly one who accepts it is accepting, not of why the axiom itself is to be accepted by someone committed only to the concepts of stage theory. It is not just that the argument is circular: the circularity is such that it is not even clear that we have, in fact, an argument—an attempt at justification. Also, I do not think that the argument could be saved by claiming that the application of Choice just imported in the argument depends “just” on the assumption of Choice in the metalanguage; at any rate, this kind of move is destined to appear even weaker in the present context, in which we are trying to justify the axiom of Choice on the basis of a “thinner” notion of iterative set.

If Wang’s argument (and other like it) has initial plausibility, it is no doubt because it attempts to provide a rationale for adopting the axiom on the basis of other intrinsic features of the iterative universe, namely its articulation in the

“chronological” structure of stages. Again, what works (or seems to work) here on the intuitive level is the quasi-cons tructive metaphor of proceeding from a stage to the next. If we took the metaphor more seriously, we could consider the stages as

157 imposing a well-ordering of sorts over the sets. So, in the case at hand we could say: we need not make an arbitrary choice among the elements of y; we can choose the one formed at the earliest stage. But then, again, while supposing that such a choice can be made would not be in itself equivalent to the axiom of choice, the assumption that any arbitrary set is well-ordered by the ‘formed at an earlier stage than’ relation is at least as strong a presupposition as the axiom of choice—for it entails that any set can be well-ordered.

Perhaps we might eliminate the looming circularity hy parsing the damning move ‘pick an arbitrary...’ in terms of an application of Comprehension. This requires that we find a formula A(z) which is true of z just in case z is a member of y

(for a y such as above) and, moreover, z is the only member of y of which A(z) is true.

If we had such formula, we could use it to obtain the required choice set since the restriction to the elements available at the given stage (i.e. formed at previous stages) is, as noted, already satisfied. However, it can be immediately seen that the second part of the formula, the uniqueness clause, cannot he expressed without circularity. A(z), in other words, would have to look Uke the following (x is, once again, fixed):

3y [ye X & z ey & Vu; {{w ey & -‘z hu) o -'A(w))]

This is clearly illegitimate. It would he possible, of course, to guarantee fulfillment of the uniqueness condition if the element z of y were, for each ye x, selected by an effective procedure, or if we could assume that every such y is neatly ordered in some way that allows us to make the choice in every case. But this latter assumption would he, once again, the assumption of the axiom of Choice (the “neat

158 order” would have to be a well-ordering, after all). So the interesting claim-- the possibility to make choices in arbitrary cases—would remained unproved. On the other hand, to assume that the uniqueness condition can be fulfilled in the general case is exactly to assume the axiom of Choice. This shows that, as Boolos observes, according to the iterative conception we may easily form the concept of a choice set, as this does not violate any of the constraints relative to the formation of sets (this is what Wang’s explanation should convince us of). It is certainly possible that a choice set exists for a (or even any) given set of sets; but there is no compelling reason to assert that such a set exists, unless one assumes just the existence of such a set - but the assertion that that set exists is the axiom of Choice. We are then caught in the weh of Choice no matter how hard we try, it seems. It is just this realization that is at the basis of Zermelo’s justification of the axiom, as will be argued shortly.

Obviously, these considerations do not prove that there is no proof, nor were they intended to do so. The idea was not to provide a demonstration of the independence of Choice from the “axioms” of S, but rather to show that the heuristic arguments that S is after aH designed to capture invariably stumble upon an assumption which presupposes a principle of choice. The heuristic arguments themselves are not readily formalizable. Therefore, one cannot hope to provide a definitive proof that those arguments will never succeed in proving Choice without implicitly assuming Choice. However, the failure of the attempts described is evidence that there is no “leaner” intuitive conception firom which a principle of

Choice follows.

The upshot of this discussion can also be expressed in the following terms. In his exhaustive study of the history of the axiom of choice, G. Moore outlines four

159 distinct stations of increasing“eonimitment” through which mathematical practice has passed on the way toward explicit admission and then acceptance of choice as a new and independent principle.®" At each level, choosing an element from a collection of sets is permitted under certain conditions; acceptance, or implicit use, of the axiom itself corresponds to the level characterized by the most “permissive” conditions -- the frnal stage. The levels are distinguished thus:

I. Selection of an arbitrary element is permitted from a single set, or from each of a finite family of sets. n. Simultaneous selection of an arbitrary element is permitted from each of a (possibly infinite) family of sets, provided that the selection is effected according to a definite rule. in. Simultaneous selection of an arbitrary element is permitted for an infinite family of sets regardless of the explicit specification of a selection rule, even though such a rule could be provided.

IV. Simultaneous selection of an arbitrary element is permitted for an infinite family of sets, and no selection rule can be given, even in principle.

The idea of Moore’s analysis (in a section entitled “The origins of the assumption”) is to show the transition, which took place in mathematical practice, from methods of proof and reasoning involving only the lower levels of committment to the higher levels; the intent is to trace the genesis of the principle in its applications to actual mathematical problems. An argument at the first level does not require the axiom of choice: picking an arbitrary element as the “representative” of a given set is not an application of the axiom, even if the set is infinite. The difference between the third and second level is that the axiom may in fact have been applied in proving a certain proposition (third level), but its application may be

®- See Moore (1982), sect. 1.2.

160 unnecessary--the same proposition can be proved without applying the axiom. It is only at the fourth level that the axiom of choice is required, in the sense that a mathematical argument which rests upon the possibility of selecting an arbitrary element firom each member of an infinite family of sets cannot he carried out without postulating what amounts to the axiom of choice.

The intent of Moore’s reconstruction, of course, is at least in part to show that the axiom’s roots in mathematical practice run very deep: this corroborates

Zermelo’s central contention in favor of the adoption of Choice, that is, that the principle is widely used by mathematicians, though never acknowledged explicitly as an independent principle. But the scheme of historical development that Moore describes also illustrate, somewhat indirectly, what the heuristic arguments we have previously discussed -- the arguments hased on the iterative conception, such as

Wang’s " try to accomplish. To put it in terms of Moore’s scheme, the arguments trade on the suggestion that there is a more or less “natural” transition firom one level of commitment to the next one—and ultimately firom the earlier levels to the last one, which fully corresponds in effect to presupposing the truth of the axiom.

But the transition firom one level to the next in Moore’s scheme is not justified on an evidential basis: in other words, the reasoning permissible at the lower levels does not provide justification for allowing arguments that involves the axiom ineliminably. For reasoning at the lower levels does not involve the axiom: the existence of a choice set at those levels is unproblematic, and does not need to be postulated, because the choice set can be constructed. The gist of the matter is that mathematical arguments that take place at the lower stages do not represent uncontroversial cases of application of the axiom: a fortiori they do not represent

161 applications that can simply be “extended” to new, more controversial cases — thus introducing the axiom. Stepping to the fourth level is tantamount to the postulation of the axiom; and that is precisely what needs justification.

By comparison, consider the following: we can (and usually do, I suppose) conceive of infinite sets as an extension of a simpler and more familiar notion of finite collection. We know for a fact that there are potentially infinite sets, and we are easily led to reason about indefinitely large sets. It may seem arbitrary at this point not to apply the same rules, patterns of inference, etc. to completed, actually infinite collections. But notice that this case is explicitly formulated as one in which the same principles are applied: forming the union of a finite set is the same operation as forming the union of an infinite one. In the case of Choice, instead, the

“familiar cases” do not involve the principle at all: there are, in a sense, no familiar cases!

For much the same reason, it seems to me that a certain constructive interpretation of the axiom remains conceptually opaque.®^ In one respect, for instance, the statement of the axiom, when read constructively, seems to be trivially true — “trivially” because just in virtue of the meaning of the quantifiers under, say, an intuitionistic interpretation. “For all x, there is a y such that 0(x,y)” means, if one read the logical constants intuitionistically, “there is a particular method / such that, when given an n as input, it will yield an output fa for which 0(n, fa)": the existence of a choice function is thus ensured, it seems, by the very meaning of the quantifiers. On the other hand, the choice function so obtained is by definition an

See the explanation of the axiom in M. Dummett (1977), 52-54.

162 effective selection procedure; under such conditions (when the selection procedure is

guaranteed to be effectively calculable), can we still say that a “choice” principle is

necessary? The existential nature of the axiom is not expressed in this context -- nor

is it expressible, for the intuitionistic interpretation of quantifiers is meant precisely

to rule out the ontological “sleight of hand” which classical set theorists call “purely existential assertions.” A principle of choice might just be a feature of classical

mathematics for which there is simply no natural translation within a constructive, in particular intuitionistic, conception: with the intuitionistic interpretation of logical constants, the intended interpretation of choice is no longer available.^

This sort of observations also teUs against a specific feature of the analysis proposed in McCarty and Tennant (1987). These authors purport to show that one can repel the Skolemite challenge by “weakening” the logical foundations of set theory from a classical to an intuitionistic logic. Technically, the program has its attractions: on a plausible intepretation of intuitionistic set theory, one can formulate equivalents to many classical notions, including uncountability; one can then prove, in such a theory, that no equivalent of the Lowenheim-Skolem theorems is ever provable. But the philosophical analysis relies upon the assumption that one can neatly separate what the paper calls the “logic of sets” from the “ontology of

There are deeper issues in the constructive role of the axiom of choice than mentioned here; Dummett does not discuss them at aU. One is that, as argued in Beeson (1985), p. 163, the axiom of choice apparently implies the principle of excluded middle. This fact is known as Diaconescu’s Theorem; a proof is described in the appendix to ch. 5. (Dummett’s remarks about the provability of the axiom in intuitionistic mathematics still apply to weaker forms, the axiom of countable choices and that of dependent choices: these are indeed intuitionistic theorems.) The second was pointed out to me by Stewart Shapiro: the axiom of choice is incompatible, in general, with the key fact about functions in intuitionistic real analysis, i.e. that every total function over the reals is uniformly continuous.

163 sets.” One can then take the minimal ontological requirement of standard set theory, such as the null set and the Infinity axiom, and replace the traditional classical logic of the object language with an intuitionistic version.^ But it is not easy to see how this approach would work with the axiom of Choice. Classically, the axiom has a clear ontological significance, yet it would not belong on the side of unabashed ontological assumptions, on the McCarty-Tennant classification. On the other hand, intuitionistically the axiom is either unacceptable (as it implies, per

Diaconescu’s theorem, the principle of excluded middle), or else is not a postulate at all, but rather a trivial consequence of how quantifiers are interpreted. In neither case, it seems, the meaning of the classical principle can be represented in a constructivist framework.

Proving Choice as a theorem

The discussion so far was about intrinsic arguments that deploy the limited resources made available by the iterative conception; as before, the underlying assumption was that the iterative notion of set is of special interest in the context at hand, because it can be claimed, with some reason, that it is an “intuitive source of axioms.” It seems to me legitimate to conclude that an explanation of Choice based on an intrinsic argument of the kind formulated by Wang — the “intuitive” argument from iteration, we could label it — cannot be substantial, and therefore cannot in itself justify the introduction of the axiom on the basis of the iterative conception.

There is another avenue of intrinsic justification, which may offer ground to regard

See McCarty and Tennant (1987), pp. 171-176.

164 the axiom as “a priori,” insofar as it is, according to this argument, derivable as a consequence of other principles — though hardly “self-evident,” I would think, at least based on the principles used. The principles involved are those of the so-called limitation of size doctrine.

This view of sets harks back to Cantor’s original conception of infinite collections as belonging to two non-overlapping categories: for Cantor there are, on the one hand, collections that possess a certain “degree of infinity,” and can be therefore compared to each other. Le., “ordered,” in such a way as to mimic, in effect, the mathematical behavior of finite collections (such as those considered in arithmetic). On the other hand, there are collections that are “absolutely infinite,” and show no articulation or ordering in “degrees”; Cantor seems to believe (on several grounds, including religious ones) that collections of this second type exist, but are, given their structure (or lack thereof), mathematically intractable.^

Paradoxes arise when the distinction between the two types is muddled. Now, as 1 mentioned before, the iterative conception is usually represented as a sort of “non- response” response to the challenge posed by the paradoxes: ideally, the iterative conception should point us to a notion of set such that, provided we construct our set theory simply by unfolding this notion, we should be able to steer clear of the paradoxes; in other words, the “pure” iterative theory should require no special provisions to defuse the paradoxes (as the first part of the chapter should show,

Zermelo’s theory is not pure in this sense, although it is usually considered a good

See M. HaUett (1984), especially Introduction and Ch. 1.3-4. Chapters 4, 7 and 8 of this book provide a very complete reconstruction of the limitation of size doctrine.

165 approximation).

Cantor’s attitude more directly suggests another response, embodied in the

“limitation of size” conception: take as sets only those collections that are not “too

big.” J. Von Neumann proposed in the twenties a formalization of set theory based

upon just this principle. Von Neumann took Cantor’s talk of absolute totalities rather literally: “large” collections, in particular those involved in the paradoxes

(like the universal class and the class of all ordinals) exist and are sets; there is a limitation of size axiom internal to the theory, ensuring that in spite of the existence of the problematic entities the paradoxes cannot be derived. The axiom states that a set is “large” iff it is equivalent to the universal set (the set of all sets), i.e., there is a one-one mapping from that set onto the universal set. In Von Neumann’s theory, moreover, one can develop the theory of ordinals in a way that is now familiar (in fact, the contemporary treatment of ordinals in ZF set theory owes more to Von

Neumann than to Zermelo); this implies, among other things, that one can prove the extendabüity of ordinals - that every ordinal a has a successor, represented in canonical form as a u {a}. Thus one shows easily that the set ON of all ordinals is not a small s e t , otherwise the contradiction known as the Burali-Forti would immediately ensue. But if ON is a large set, then there is a one-one correspondence between ON and the universal set; moreover, ON is easily shown to be well-ordered by the membership relation. Hence, the universal set is well-ordered, which is equivalent to the axiom of choice; in fact, this produces the strong form known as the axiom of global choice.*"

As a further note, the axiom of replacement is almost analytic on this conception: if A is a (small) set and is no smaller than B, then B must be a small set

166 This shows that a sanguine interpretation of the concept of limitation of size will in effect entail a commitment to the axiom of choice, among other things.

Viewed as an “intrinsic argument” for the axiom, the justification provided by the

Von Neumann theory is obtained through strong assumptions: it yields the result firom above, so to speak, that is, as the consequence of assumptions stronger than the axiom itself The question before us, however, is not just that of finding a convincing argument for the truth of the axiom of choice; rather, we would like to find reasons for the acceptance of the axiom in set-theoretical practice even before the theory was explicitly formulated. The explanation sought, in other words, should also make clear why the axiom may appear so evident or “natural” that its application could take place, so to speak, seamlessly. This constraint might have been satisfied by a substantial explanation in terms of the iterative conception, if one accepts the claim (often advanced by supporters of the iterative conception, such as Zermelo) that iterative sets do represent the notion of collection of something else standardly employed in mathematical practice. But it is dubious that the

“explanation” in terms of limitation of size could achieve this goal. The problem is not so much that something even stronger than choice is assumed as rather that the assumption is a principle of “maximality.” Godel commented on the matter that

“this axiom [Von Neumann’s “definition” of large sets]... is a maximal principle, somewhat s imilar to Hilbert’s axiom of completeness... [I]t says that any set which does not, in a certain well-defined way, imply an inconsistency exist.”®® Like

as weU.

®® From a personal communication to Ulam quoted in HaUett (1984), p. 291.

167 Hilbert’s completeness axiom in the geometry of the real line, however, the limitation of size axiom is really a metatheoretical principle clothed as internal axiom: it seems to say something not directly about the ways of set-theoretical practice, but rather about the proper interpretation, the models into which that practice ought to be embedded. As such, it seems arduous to claim that the concept of limitation of size given to it by Von Neumann can be extracted directly from a preformal notion of set. To be sure, it is explicit in Cantor the notion that, as long as we can increase the size (Le. cardinality) of a set, we remain in the realm of the transfinite, which allows comparability of sizes, and the trappings of a transfinite arithmetic (this is the core of what HaUett calls Cantor’s “finitism”). The absolutely infinite, by contrast, is for Cantor beyond this realm, and cannot be increased. But even assuming that the Cantorian notion of absolute infinities is firmly rooted in pretheoretical intuition about sets, we still would not have the fuU extent of the limitation of size axiom. At the center of Von Neumann’s system is the idea that the very notion of size native to the finite and transfinite rungs — cardinality — can be applied in the same form to the Cantorian absolute: there is a supremum cardinal, and it is the power of aU large sets. Surely it is debatable whether the propriety of this solution, elegant as it may be, can be appreciated independently of highly theoretical concepts such as, for example, the analysis of the paradoxes.

Extrinsic arguments

So far, however, we have dealt only with intrinsic arguments - what about extrinsic ones, based on “empirical” corroboration? The advantage of this kind of justification is that it need only be indirect: roughly, one could point out at the

168 numerous applications the axiom has in core areas of “real” mathematics, such as analysis, as well as in the definition of central notions in set-theoretical terms, such as for instance the distinction between finite and infinite sets; Moore’s book gives ample display of the significance of the axiom in modem mathematics. Because of its significance, it becomes clear that one could not (or no longer) simply jettison the axiom in the name of philosophical or “a priori” concern; the axiom is “needed,” in other words, to do many things, some of which are considered even more “intuitive” than set theory itself One may well agree with Lakatos in attributing to some basic parts of mathematics the epistemological of empirical phenomena in the physical science; in this spirit, it may be then said that the axiom of choice is after all needed to “save the phenomena”—for example, to preserve the intuition that the boundaries between the finite and the infinite are not vague.

Such is the line of thought that an extrinsic argument spells out. Yet it seems to me that this indirect extrinsic argument does not do justice to the nature of the axiom either. The chief reason for this is that the axiom, by Moore’s account, slipped into the practice (through analysis, mostly) largely unnoticed', this is after aU exactly the point that Zermelo himself offered in corroboration of the axiom. This seems rather different firom the way it is with empirical confirmation, the preferred term of comparison for extrinsic arguments. Clearly, Lakatos’s talk of “bold hypotheses” seems grossly inappropriate: how can a bold conjecture seem obvious?

The inadequacy of this explanation is even more evident if one compares the status of choice in this respect with that of other axioms for which extrinsic considerations seem natural. Maddy (1990 and 1988) shows that most recent work in set theory involves the formulation of new axioms of large cardinals and determinacy, and the

169 analysis of the resulting extensions of ZFC is typically aimed at finding extrinsic justification for such axioms. Set theorists consider such axioms as by nature unsuitable to any but extrinsic confirmation. But these axioms are indeed akin to

Lakatosian “bold hypotheses" in the empirical sciences (assuming any such conjectures really exist): no intuition, whether a priori or not, can provide sufficient motivation for believing in their truth. Even though such axioms have consequences in the more “concrete” reaches of set theory, in particular consequences for the structure of real numbers (which provide the “empirical verification” for the truth of the axioms), it is not commonly believed that there is any such thing as an implicit commitment to axioms of this kind: no one, apparently, expects to find a theorem of analysis in the proof of which the author presupposes unknowingly the existence of a supercompact cardinal Just because of this, however, such axioms appear very different firom the axiom of choice, precisely in that the axiom of choice does have intuitive motivation. This fact must be explained if one is to understand its widespread application before it was explicitly formulated as an independent principle. Of course, to some extent any mathematical theory is to some extent confirmed a posteriori in the sense suggested by the simple fact of being in use; this is in no way a feature peculiar to mathematical theories (it seems to apply equally weU to any reasonably systematized body of belief), but there is no reason why mathematical theories should constitute an exception in this regard. Thus, a conjecture need not he “hold,” in whatever sense, to enjoy a degree of extrinsic confirmation. But the problem remains that extrinsic justification seems unable to account for the difference between intuitively evident principles and theoretical conjectures.

170 CHAPTER 5

CHOICE AS A LOGICAL PRINCIPLE

Hilbert and the Epsilon

Replying to those who had objected to the introduction of the axiom of choice in his proof of the well-ordering theorem, Zermelo writes in 1908:

That this axiom, even though it was never formulated in textbook style, has frequently been used in the most diverse fields of mathematics... is an indisputable fact, which is only corroborated by the opposition that, at one time or another, some logical purists directed against it. Such an extensive use of the principle can be explained by its self-evidence, which of course must not be confused with its provabüity. No matter if this self-evidence is to a certain degree subjective — it is surely a necessary source of mathematical principles...®®

He proceeds to fist a number of “elementary and fundamental theorems” that either require or are equivalent to the axiom of Choice. Clearly he intends to show that a theory of sets would be seriously disfigured by a refusal to adopt the axiom, if such a refusal was to involve a committment to rejecting any result that essentially rehed on principles entailing the truth of the axiom (a commitment which many early

®® Zermelo (1908a), p. 187.

171 opponents of Zennelo’s axiomatization would have found difficult to live up to).

Some critics of the original 1904 proof of the well-ordering theorem, where Zermelo first introduced the principle of choice, had attacked the principle because it was

“unprovable” (the most notable of these critics seems to have been Peano). Zermelo’s reply in 1908 is intended to defuse such objections by showing not the provability of the principle but its “validity.” But what does this mean? According to Zermelo:

Now even in mathematics unprovability, as is well known, is in no way equivalent to nonvalidity ... Thus, in order to reject such a fundamental principle, one would have had to ascertain that in some particular case it did not hold or to derive contradictory consequences firom it: but none of my opponents has made any attempt to do this. ”

Zermelo’s explanation amounts roughly to a sort of “transcendental argument” in favor of the axiom of Choice: even those who purport to deny the axiom must end up accepting some proposition provably equivalent to it. This shows, in his opinion, that the validity of the axiom cannot be impugned. Or, perhaps better: the anti-Choice camp cannot impugn the validity of the axiom and yet maintain a meaningful grasp of the concepts involved, such as set, collection and operations thereon. Note that this line of reasoning, however, makes Zermelo’s previous claim about the “self-evidence” of the axiom somewhat odd: why would so many object to something “self-evident”?

The strategy I propose is different firom Zermelo’s, but is aimed, in a sense, to the same target. I submit that Zermelo’s observations are best explained if we come to regard the axiom of Choice as having the status of a principle of logic.

Zermelo expresses the idea (in his point about “validity”) that, if the axiom is

Zermelo (1908a), ibidem. Italics are Zermelo’s, while underlining is mine.

172 an irreplaceable component of a “logic of sets,” then it is an essential component of

logic tout court. This of course takes us directly to the heart of the debate about logic

and the search for an “appropriate” logical language that occupied the first three

decades of the century. There we find, that Hilbert and others held a position about

logic that taUies up well with Zermelo’s suggestion on the axiom of Choice. ' ‘ The

conclusion I propose to draw is the following: although there is no specific element of

the iterative conception of set that forces upon us the axiom of Choice, the axiom

must be seen as part of the background logic. We are therefore advancing the

hypothesis that both elements are part of the intended notion of sets: the iterative

conception as the ontological underpinning of the intended interpretation, and the logic as the general laws governing any interpretation. Furthermore, I think that it is only as a principle of logic that the axiom can be said to be “self-evident” in the sense that Zermelo has in mind.

In his (1923) Hilbert exphcitly refers to the axiom of Choice as a logical

notion, stating that

in the logical analysis carried out in my proof theory ... the essential thought on which the principle of choice is based is a general logical principle, which is necessary and indispensable for even the most initial elements of mathematical inference. By securing these initial elements of inferring, we simultaneously prepare the ground for the principle of choice. "

He proceeds to show how the axiom of choice can in fact be applied to predicate logic

to produce definitions of the quantifiers, and derivations of all the logical rules directly concerning quantification (what we would call today introduction and

See Moore (1980), pp. 117, 120, 126.

■- Hilbert (1923).

173 elimination rules). The device by which he accomplishes this is the so called “epsilon constant”, introduced as a variable-binding term-forming operator. Where A is a formula containing x free, the standard semantic interpretation of the epsilon constant is a choice function over the universe of discourse: it selects, for a given nonempty set A, an arbitrary member of A. The intuitive meaning of exA is illustrated by Hilbert by an example very similar to the following: suppose A is the predicate ‘is honesf; then exA represents an individual of such unshakable rectitude that if he were found to be corrupt we would have to conclude that no honest person exists; ‘exA’ is called an epsilon term?^ In model-theoretic terms, then, we would say that a choice function for the universe of the model is the natural interpretation for the epsilon constant. This gives rise to a technical difficulty: while choice functions are typically defined over families of nonempty sets, we are allowed to form an epsilon term by binding the free variable of a predicate A even when there are no things that are A. Thus, the semantics for a language containing the epsilon constant will require some adjustments needed to specify the value of the choice function over the empty set. Various options have been formulated in this regard, including the possibility of regarding the logic based on languages with the epsilon constant as a free logic, along the lines suggested by N. Tennant for logics containing operators for definite descriptions and class formation; probably the simplest option

' ’ Actually, Hilbert’s example illustrates the use of an epsilon term by forming the converse of our sentence, thus introducing the term to define the universal quantifier: he takes the predicate of the example to be is dishonest’. In other words: Let E be the epsilon term for the one-place predicate A; then Hilbert’s own example is of the form “If A(E), then everything is A”. In the example I described, on the other hand, the relevant sentence is of the form “If anything at aU is A, then A(E).”

174 is to assign to the choice function the same value for the empty set and for the universe: this seems to correspond rather faithfidly to the conception of epsilon logic one finds in Hilbert's works (see Leisenring (1969) for details). At any rate, this problem will not concern us further.

The only axiom that needs to be introduced to govern the epsilon constant is the following:

A{t)^ A(exA)

We can refer to this as the epsilon axiom. Here A is a formula where the variable x occurs firee; A(t) is the result of replacing x with the term t in A With the help of this axiom, the existential quantifier can be defined contextually by setting

3r A s A(exA). A typical formulation of the rule governing the existential quantifier in axiomatic presentations of logic is: Let X be a set of formulas and t an individual parameter not occurring in A B or X; then if there is a deduction of A(f) 3 B from X, there is a deduction of 3x A firom X. It is easy to see that this rule can be derived with the aid of the epsilon axiom, thus justifying the previous stipulation.

(Exactly the same can be done in the style of natural deduction: in this style, to show that the existential rule is a derived rule amounts to showing that the elimination rule for 3 respects the above definition in terms of the epsilon constant. ') If we add the usual definition of the universal quantifier in terms of 3, we have then that every quantified statement can be proved equivalent to one without the quantifier and containing an appropriate epsilon term.

See Leisenring (1969), ch. II. For an approach in the style natural deduction, see Tennant (1980).

175 Hilbert’s proximate goal in introducing the epsilon axiom was to demonstrate

what he calls (1923) the principle of excluded middle (PEM), Le., the equivalence

-A/x

from the epsilon axiom goes through in classical logic only because an equivalent of

PEM, namely the law of double negation, is already available. BeU shows several other facts that I recapitulate here to illustrate the relations between Choice, the epsilon principle and PEM. The natural question to ask, from Hilbert’s point of view, would have been whether PEM can be derived from the epsilon axiom in intuitionistic logic, where neither PEM nor double negation are available. (We are now referring to PEM in the familiar form ‘A or not A,’ rather than to Hilbert’s statement.) The answer is that PEM does not follow from the epsilon axiom alone; one needs to add an extensionality principle for the epsilon constant, stating that the constant operates in fact as a function: if two predicates A and B have the same extension, then exA = exB. Here, again, it is hkely that Hilbert had simply assumed this further condition, because in light of the equivalence 3x A sA(exA) it seems natural for the epsilon constant to behave extensionady in the way required.

Under assumption of the extensionality principle for epsilon, then, one can derive PEM from the epsilon axiom. The proof works basically in the same way as that of Diaconescu’s theorem, which shows that in IZF (intuitionistic set theory) the

Axiom of Choice entails PEM. In the case of this theorem, there is no need for an additional condition mirroring the extensionality condition on the epsilon, since IZF includes the Axiom of Extensionality.

So it seems that the epsilon axiom is similar in logical strength to the axiom of choice where the arbitrary terms introduced are considered as having extensional

176 identity conditions. Note that by Hilbert's fundamental theorem (Leisenring, ch. 2, cit.) the epsilon calculus (predicate logic enriched with the epsilon constant and related axiom) is conservative over classical predicate calculus. Thus the axiom of choice does not become derivable from the axioms of standard ZF theory in the new language; it is easy to see, however, that if formulas containing the epsilon constant are allowed into the axiom schema of Replacement, then there is a "proof of the axiom of choice (of course, with the epsilon constant a formula defining a choice function is always available).

In the previous chapter I briefly mentioned attempts to give intuitive justification of the axiom of Choice from an intuitionistic standpoint (in fact, a quote from Skolem in ch. 2 already made a similar point). The proof by BeU of PEM from the epsilon axiom shows such attempts to be at best questionable. For not only does

Choice entail PEM in IZF; PEM can even be derived from a weaker, more “logical” thesis, the epsilon axiom, at least in any theory T satisfying a further extensionality condition. In fact, BeU concludes that it is extensionaUty, rather than bivalence (as often argued), that characterizes a commitment to classical logic. There remains, it seems, the possibUity of adding something Uke Choice to theories that do not satisfy the further extensionaUty condition (an intensional set theory?), though the intuitive meaning of an axiom of choice in the context of such a theory is somewhat difficult to grasp: on the basis of a proof of the antecedent of the axiom, the intuitionist could conclude to the existence of a choice procedure, but not of a choice function, it seems.

In any case, this substantiates the objection moved to the attempts at constructive justification. Choice in its fuU-fledged meaning is intuitionisticaUy unacceptable.

Less fuU-fledged versions, such as Dependent Choices, are acceptable, and in fact

177 can be proved as theorems in an appropriate intuitionistic theory.

But this suggests that in a constructivist conception Choice will appear to be an “obvious” principle simply because we have in feict constrained in certain ways the interpretation of quantifiers. Like the proponents of the “intuitive” intrinsic arguments discussed in the previous chapter, the constructivist cannot quite produce a case in which the postulation of Choice as an axiom is necessary. The claim that Choice is justified on an intuitionistic basis, therefore, rings hollow, just as the intuitive intrinsic arguments could not quite explain how, on the basis of such an argument, we should find ourselves convinced of the truth of a principle that has not even been applied in the course of that very argument.

Returning to the semantics, the natural interpretation of the epsilon constant is given by a choice function over the universe of a modeh the semantic value of the term exA under a given interpretation is the value of the choice function for the subset of the universe that represents the extension of the predicate A.

Hilbert beheved that the “thought on which the principle of choice is based” was necessary in logic in order to justify the adoption of the laws of quantification.

In Hühert’s conception, as is well-known, there is a more or less sharp dichotomy between those parts of mathematics that can be sohdly built upon a finitistic basis and those that cannot he so grounded. Much like the constructivists, Hilbert regarded the content of mathematics to be exhausted by finitary manipulations of the sort one encounters in elementary arithmetic or Euclidean geometry; Hilhert would of course have differed fi-om many constructivists — for instance, from

Brouwerian intuitionists — about the nature of those finitary manipulations:

178 operations on “concrete” symbols or symbol types, for Hilbert, mental constructs for the intuitionists. At any rate, Hilbert saw that the classical logic which prevailed in the usage of mathematicians could be explicitly grounded in terms agreeable to a finitist only with respect to finite domains; quantification over finite domains is essentially reducible to propositional logic, which is finitistically “secure.” But quantification in general was another matter -- although the classical laws are characterized precisely by their handling the general case of infinite domains as if there were no principled difference firom the finite case. Quantification in the general case requires a logic that makes sense only in the “transfinite mode of thought,” as Hilbert puts it (in the same article from which I last quoted). In this approach, the question then arises: what is the characterizing element of the transfinite mode of thought? Hilbert locates it in the principle of choice: the assumed possibility of choosing an arbitrary element, with no further specification, from any given element of a family of sets. The principle is represented in logic by the epsilon axiom, and the brief description above of the contextual definition of the usual quantifiers in terms of the epsilon axiom substantiate, in Hilbert’s view, his thesis that it is the assumption of choice that allows one to preserve a uniformity of logical behavior even in that part of mathematics that has no discernible mathematical content, and therefore is not amenable to finitistic justification.

Hilbert’s program, in its original intent and formulation, is no doubt part of the foundationalist currents of thought that were dominant in the philosophy of mathematics throughout the first decades of this century. As such, it has at the core a strong epistemological project: in particular, that of “securing” mathematical knowledge from the vagaries of the paradoxes. In the 1923 paper I have quoted

179 from, there are hints at the “dangers” of proceeding to the adoption of logical schemes (in this case, for instance, the classical interdefinabUity of the existential

and universal quantifier) the adequacy of which cannot be directly tested in the safety of one’s finitist home; paradoxes are exphcitly mentioned as the primary motive why the laws of quantification should be justified in the first place. To the modern student of foundations, the epistemological project is the least convincing aspect of those early foundational studies -- partly because at least two of those foundational programs (the logicists’ and Hilbert’s) appear to have failed on their own terms, and partly because the philosophical allure of foundationahsm has by and large vanished. It is now generally thought that mathematical notions are not particularly in need of securing, especially if the securing must rely on philosophical or logical notions that are in themsleves less understood than many mathematical concepts; further, many have argued that the prescription for curing mathematics from the paradoxes was, even in the heyday of foundationahsm, just a deeper mathematical analysis of the notions involved (such as that of set, for instance).

However, while the motives that drove Hilbert to his special concern with the axiom of choice may appear obsolete, there are aspects of his conception that seem to me fully deserving of attention - especially in Ught of our previous observations about the difficulty of producing non-question-begging arguments in support of the axiom of choice.

I am especially interested in two such aspects:

(1) the claim that the axiom of choice should in fact be regarded as a fundamental principle of a logical nature; on this, 1 think, Hilbert is substantially right and the introduction of the epsilon axiom even provides a way to find some support for this

180 claim (at least as much support, I shall argue, as one can expect to muster in favor of a thesis of this sort -- Le., an attribution of “logicality^; and

(2) the related suggestion that there is a difference in “logical quality” between, roughly, the domain of finitary mathematics and the higher, transfinite reaches of set theory.

With regard to the first point, I should remark that my proposal is not that we take “fundamental” quite in the sense that Hilbert seems to have understood it: for Hilbert, as mentioned, it was important that the principle of choice he appealed to in the form of the epsilon axiom (or indeed as a way to justify the stipulation of the epsilon axiom) be seen as both more “basic” and less questionable than the logical inferences it was meant to justify. This is a foundationalist requirement: it makes sense, and is indeed necessary, within the context of the foundationalist project of securing mathematics on a finitist basis. On the other hand, we need not worry whether a principle of choice, insofar as it is a logical principle, enjoys some form of justificatory primacy over the standard quantifiers (i.e., whether it is more basic firom the standpoint of justification). It is a key point, however, that the principle is at least as basic as those logical laws of quantification: purged of foundationalist motives, Hilbert’s suggestion could be interpreted as saying that the principle of choice is in any case part of our best understanding of how quantification works, and this suggestion seems to me still vital Of course, this line of thought runs into the usually difficult question of how to tell logical firom non- logical laws -- difficult, because there is as yet no universally agreed upon criterion by which the determination can be made. We shall see, however, that at least one plausible criterion of demarcation between the logical and the non-logical would

181 allow us to go some way toward justifying Hilbert’s suggestion.

The second point is, I am afiraid, even more in need of explanation — and even less amenable to conclusive demonstration. To begin with, I should note a well- known fact about Hilbert’s foundational program: an integral part of the philosophical underpinnings of Hilbert’s programme is the view that set theory and

“transfinite” mathematics are an entirely different sort of thing firom elementary arithmetic or basic Euclidean geometry. This thesis follows directly firom what we may call Hilbert’s “theory of meaning, ” that is, the view that mathematical discourse has content only insofar as it has finitistic import -- a view that also includes a conception of what the finitistic content of a statement can be, and so forth. Since

Cantorian set theory, for example, outstrips the limits of finitism, an Hilbertian will be compelled to construe the meaning of set-theoretical statements in a different way firom the rest, or else to propose revisions in several areas of pure mathematics, in particular set theory. This distinction is what gives the Hilbertian project its basic architecture: try to justify the addition of the transfinite element of mathematics not directly - finitistically impossible - but through consistency proofs about the resulting formal systems (although this would also prove impossible to achieve by purely fimitistic means). The rift between the finite and the transfinite with respect to content of statements implies prima facie that there will be a corresponding divergence with respect to the modes of reasoning sanctioned in the two distinct realms. The acknowledgement of this is explicit in Hilbert. He states, in the same 1923 paper:

In our proof theory, meanwhile, we want to go beyond the domain of finite logic and obtain provable formulae that are the representations [Abbüder] of the transfinite propositions of the usual mathematics.

182 And we shall see the real power and verification of our proof theory when we shall succeed in demonstrating consistency after the addition of certain further transfinite axioms. Where does the exit from the concretely intuitive and the finite first take place? Manifestly, already in the application of the concepts ‘all’ and there is’.

Thus, in summary, the realm of the “transfinite” in mathematics is heterogeneous in content with respect to the finite; the demarcation can be drawn at the point where quantification becomes necessary: in the finite, quantifiers can he construed as abbreviations of non-quantified, determinately contentful statements, while in the general case quantifiers cannot he so construed; to preserve uniformity of logical laws (that is, to “save” the validity of classical laws when applied to transfinite domains) across the two realms we need to appeal to the concept underlying

Zermelo’s principle of choice. This, I think, is a fairly accurate reconstruction of the meaning of Hilbert’s suggestion (2) above.

Logical notions

Thus Hilhert based his claim that the axiom of Choice is a “fundamental logical principle” just on the possibility of defining the existential quantifier with the epsilon constant. I think, however, that there is a stronger argument for the logical nature of Choice (and derivatively of the epsilon constant), one based on our best understanding of the logical vs non-logical distinction. I shall try here to articulate that argument.

It is somewhat difficult to say what makes a certain statement a logical principle. My argument that we should so qualify a principle of choice, whether in the form of the axiom of choice or an epsilon principle, is perhaps best seen as an

183 inductive attempt. That is, I believe that a number of “tests” of different natures are relevant, some more than others, but I still think that there could be more to be taken into account. The criteria that I see as relevant to the present case are basically the following:

4. Is a principle “fundamental” in the sense that its validity cannot be

justified by appeal to some other more basic principle?

5. Is a principle intimately connected to logical operators of a known

species, e.g., can it define or replace them in the theory of logic?

6. Is it a valid principle expressible in purely logical terms?

It seems to me that Choice satisfies these three criteria, modulo the considerations just developed about its connections to the epsilon principle.

Ad 1: 1 have already argued to some extent that the usual “intuitive” arguments for the justification of Choice either fail outright or are simply based on the prior assumption that Choice is true. I shall return to a more sophisticated argument based on an “intuitive” conception of set in the latter part of this chapter, but the discussion will show that that argument fails as weU.

Ad 2 : 1 have argued in Chapter 2 that Choice was needed, at a critical juncture, to settle the interpretation of dependent existential quantifiers. Furthermore, via the epsilon axiom we have seen that a principle of choice can be directly used in logic to represent quantifiers. We do not have interdefinability strictly: the quantifiers can be defined in a language containing the epsilon constant and a logic including the epsilon axiom and the extensionality condition on epsilon; but it is not possible to derive the epsilon axiom in a language containing only the usual quantifiers.

However, it seems reasonable to claim that semantically the classical quantifiers

184 behave as if the epsilon axiom was available (might this be considered “informal definability,” perhaps?).

A d 3: 1 shall presently argue that this is the case for choice, if we consider the content of a principle of choice to be fairly represented by the introduction of an epsilon constant. In this kind of arguments, we want to show that all the terms needed for the adequate expression of the principle denote logical notions. Criterion

(3) seems to me the more convincing of the three, because I believe that we already have a reasonably crisp account of what a logical notion is, in the work of TarskL

This account is much strengthened by Güa Sher’s claim that the demarcation between logical and non-logical notions one obtains by it is internally coherent with

Tarski’s account of logical consequence (a fact that Tarski himself apparently had doubted)."' Thus the argument I provide aims to show that a “choice term” can in fact be considered a logical notion by Tarski’s method. This strategy requires, then, that we shift the focus firom the logicality of a principle of choice to the logicality of the notion of choice. This is basically the approach adopted here.

When Hilbert claims in (1923) the need for introducing the principle of choice as a “general logical principle,” however, he is not making appeal to a firm criterion for discerning logical firom non-logical constants. His idea, rather, seems to be the following: since the concept of choice is needed in order to underwrite the full development of predicate logic in the general case (beyond the confines of finitistic mathematics), it ought to be counted as a “fundamental” logical principle. In other words, in this particular approach the attribution of logicality to the concept of

See G. Sher (1991), especially chs. 2 and 3.

185 choice flows from its indispensability to logic properly so called (viewed as the realm of rules of inference). A firmer version of this idea is offered in Tennant (1980)

(where its generalized form is attributed to T. Smiley): if it can be shown that rules governing the quantifiers can be derived with the help of the epsilon axiom, then one can in fact argue that quantifiers and the epsilon constant are semantically interchangeable, for the quantifiers can be defined in terms of the epsilon constant

(as shown previously); that suffices to conclude that the epsilon constant is a logical notion and the epsilon axiom a logical axiom. This line of thought is also made explicit in Hilbert and Ackermann (1938),’® where a form of the axiom of choice is simply introduced as an axiom formulated in the purely logical vocabulary of what

Hilbert calls “extended predicate calculus,” that is, a second-order language. Note that it is in the same vein that one can also introduce numbers - as predicates of predicates — as purely logical constructs: one shows, as Hilbert does, that in the language at hand number predicates can be expressed in the purely logical vocabulary, and that the basic facts about such predicates then follow as consequences solely of logical axioms. '

The problem with this style of argument is that it presupposes a criterion for discriminating logical firom non-logical apparatus, since the argument requires that we already have a “logical vocabulary” at hand. But if we are looking for a principled distinction between the logical and the non-logical components of a

’® Hilbert and Ackermann (1938), ch. IV.

’’ Hilbert’s observations concerning the expressive power of second-order languages with respect to the axiom of choice as well as number predicates are given fuller presentation in Shapiro (1991).

186 language, this style of argument is of no particular help: we would still need some way of individuating the basic logical vocabulary. (Note, in fact, that the “principled distinction” is not what many proponents of such arguments are after: in Hilbert

(1938), as in Shapiro (1991), the point is rather to show how powerful a certain logical apparatus, in particular second-order logic, could be. Le., how many notions can be give a purely logical definition i/w e agree to consider the apparatus as a part of logic.)

A criterion to distinguish logical firom non-logical notions was suggested by

Tarski (1986),'® and my proposal is simply that we apply this criterion to Hilbert’s epsilon constant: I argue that according to Tarski’s criterion choice functions are, in a reasonable sense, logical notions, so that an expression used in a logical language to designate a choice function - an expression Uke the epsilon constant - can be in fact considered a logical constant.

The idea is fairly simply described. Tarski considers it a generalization of

Fehx Klein’s characterization of various geometries in terms of notions invariant with respect to a certain set of transformations. As an example, we consider geometrical space as a set of points. A transformation over such a space is simply a one-one correspondence that associates to each space point another space point.

Physical motions, for example, are one type of transformation in three-dimensional space: under such a transformation, an object changes place, but does not change shape, as the distances between its “internal” points remain unchanged. Distances between points, therefore, are invariant with respect to physical motions (at least

Tarski (1986).

187 motions of the garden variety).

Given a generic space S, then, to each set G of transformations over S there corresponds a set of invariants, that is, properties defined on S and such that, whenever a point x has such a property, the point g(x) (the image of x under a transformation g in G) has the same property (of course, invariant relations, etc., are similarly described). By studying different classes of transformations, one will isolate different sets of invariants. For example, similarity transformations, which preserve the ratios of distances but not distances themselves, preserve such geometrical properties of figures as that of being an isosceles triangle, or a circle

(although not that of having a certain diameter, for example). Such properties are the subject of Euclidean geometry. Of course, motions are a special case of similarity transformations. In general, by extending the field of transformations, we reduce the corresponding field of invariant notions; that is, there are fewer properties that remain invariant. We can extend the set of permissible transformations further to those that do not respect similarity, but preserve only coUinearity and betweenness of points: if a plane figure is a right triangle, for example, its image under such a transformation wül still be a triangle, but not necessarily a right one. The laws of Euclidean geometry may no longer be satisfied in a space where such a transformation has taken place; the laws of projective geometry are, however: projective geometry has a smaller set of invariants.

Tarski’s idea is this: consider an arbitrary set of individuals; what notions are invariants under the set of all possible one-one transformations of this space?

Tarski’s conjecture is that the invariants will be the logical notions. The notion of

‘notion’ under scrutiny here is a thoroughly semantic or model-theoretic one: what

188 Tarski considers are properties, functions, operations, etc., that can be given a set-

theoretical construction. At the ground level, there are individuals; it is assumed that there will be infinitely many. One level up, there are classes of individuals, then classes of classes, and so on: among these, at one or the other level, are the operations designated by the logical constants in standard treatments of logic. It is assumed that any mathematically significant concept will be defined in terms of a construct appearing at some level of this hierarchy. As regards in particular logical notions, this means that they will appear in the hierarchy roughly in the form made familiar in the algebra of logic: disjunction, for example, wül appear as the logical sum,” or union of two classes; negation as complementation, and so on. On this basis Tarski proceeds to show that the application of his method yields the expected results - that is, that the notions that turn out to be invariant under the set of all transformations of the domain are indeed the notions that one would regard intuitively as logical. Thus, for example, at the ground level (individuals) there are no logical notions: no individual is invariant under arbitrary transformations. At the level of properties, i.e., classes of individuals, there are only two logical notions, the empty class and the universal class. Moving up the ladder of types, we find other notions: identity is counted as a logical relation; conjunction and disjunction, interpreted algebraically as described above, also are classified as logical notions, at the level of binary operations on classes, and so on. This heuristic procedure is summarized in the result that Tarski believes conclusive here: every prepositional function (semantically) definable “by purely logical means” is invariant with respect

189 to all one-one mappings of the universe of discourse onto itself'’’

Now, what happens if we apply what we may call Tarski's test’ to the notion

expressed by the epsilon constant — that is, choice functions? In the traditional style

of model-theoretic interpretation, the interpretation of a language in a model M is

defined inductively for each type of well-formed expression in the language. Epsilon

terms, for their part, are assigned an interpretation in M by a clause saying

something like the following: I^fzxA) =cp({y I , where 1^ is the given

interpretation in M and (p is the choice function over the universe of M. That is, the

semantic value of exA is the value selected by (p firom the set of individuals that

satisfy the property A (if any: as noted above, one needs to take care of the case in

which there is no such individual, but we are not concerned with that now).

Now in general the function (p itself is not a logical notion in the requisite sense. For suppose that g is a transformation of the universe of discourse: then in

general, whenever X is a subset of the universe, it is possible that

(p(g(X)) * g{^QC))?° On the other hand, the property that (p has of being a choice

function - the property by which

result of applying a transformation of the type under consideration to any choice

function is stiU a choice function. The notion choice function' itself is a logical

notion by the hghts of Tarski's test. But notice that the adequacy of the clause

This theorem was presented in an earher article (Lindenbaum and Tarski (1935)). The “purely logical means” referred to in the statement are those in the logic of Principia Mathematica.

“ The notation is to be read with the usual proviso: 'g(x)' is the individual mapped by g for x, while ‘g(X)' is the image of the set X under g.

190 specifiying the assignment of a semantic value to an epsilon term -- a clause such as the one written above -- depends exclusively on the fact that

(for the whole domain). Put in different terms, the semantic value of the epsilon axiom depends solely on the epsilon designating an object within the notion ‘Choice

Functions': any one of them wül do equally well, as long as it is defined for the domain at hand. So, in a sense, the internal properties of the individual choice functions within the class Choice Functions’ is never appealed to in the interpretation of Choice. And similarly, the fact that the epsüon constant can be governed by the deductive rules described at the beginning of this section (or by the natural deduction rules specified by Tennant (1980)) depends only on its “intended interpretation” as a choice function, that is, in particular, on the fact that, if we know nothing else of exA, we know that it is “an A.” Therefore, I think, it would be reasonable to regard not only the notion of choice function as a logical notion by

Tarski’s criterion, but also the epsüon constant as part of the logical apparatus of a language.

Of course, acceptance of this thesis wül be conditional, at the very least, upon one’s acceptance of Tarski’s test as embodying a plausible criterion to draw a boundary between logical and non-logical notions. Different criteria would presumably issue in different tests, and one might expect disagreement on the results. It is clear that to motivate and outline a comparative judgment on the merits of competing analyses of logical constanthood would exceed the scope of the present investigation, but a few comments are warranted, I believe, so that our choice of Tarski’s criterion may not appear as entirely arbitrary. Tarski’s method is clearly inspired by a semantic or model-theoretic conception of logic. It is

191 characteristic of this conception that it does not draw a sharp boundary between the proper domain of logic and that of the more abstract reaches of mathematics, in particular set theory. Thus, for instance, by probing further into Tarski’s criterion, one can see that it makes (set-theoretical) membership a logical notion, provided one regards it not as a relation between individuals in an ontology of sets, but as a

“higher level” relation between individuals and classes of individuals (and more generally between objects at a given level and classes of such objects). For the relation obtaining between an individual and a class it belongs to will he invariant under arbitrary transformations of the domain. In effect, the membership relation under this construal turns out to he indiscernible from the relation of predication: and indeed many set-theoretical principles can be stated in purely logical terms in the language of second order logic, as explained in Hübert-Ackermann (1938) and in greater detail in Shapiro (1991).

Does this mean that Tarski’s method involves arbitrary theoretical presuppositions? I do not think so. It is worth noticing that Tarski’s criterion seems theoretically consistent with his account of logical consequence and validity: that account, too, is formulated in model-theoretical terms, in such a way as to make our best understanding of logical properties of one piece with our understanding of set theory: for it requires that we have some grip upon the notion of‘all models’, or all possible worlds’, or some such. Therefore, even at a cursory glance, Tarski’s criterion cannot be dismissed as based upon ad hoc stipulations: it seems, on the contrary, an apt candidate to be considered in the present context. Of course, precisely on account of the theoretical baggage it carries, one cannot expect the criterion to be uncontested: the conception of logic embodied by Tarski’s method is at

192 variance with the (today) widespread assumption that logic should, by its own

nature, be independent of any ontological presupposition.®^ Tarksi himself is aware that even the same criterion can be applied in different ways, corresponding to different conceptions of the subject matter to be investigated: he notes in (1986) that one could start, in a manner of thought he attributes to Zermelo, with an ontology of sets, Le., with sets as the individuals of the initial domain; and that, under these conditions, one would of course end up with different results (we have already noted the obvious point about membership in this regard). We must rest, therefore, with the realization that a theoretically neutral way of resolving the distinction between logical and non-logical notions is not available.

The Self-Evidence of Choice, again

If it is true that Choice is fundamentally a logical principle, how does this bear upon the kind of arguments we examined in the previous chapter about extrinsic and intrinsic justifications of the axiom? To rehearse that discussion:

There seem to be two ways to go about justifying set-theoretical principles, and therefore Choice. The first, which issues in intrinsic arguments, is to draw on a pre­ existing conception of a type of object, a conception which is presumed as somehow given to people taking part in mathematical practice. To show that Choice is true

A criterion of logical constanthood designed to satisfy^ this assumption is offered by Peacocke (1976). Under Peacocke’s criterion, membership is not a logical notion (notice that the criterion is to be applied to expressions of a language, however, rather than to notions in Tarski’s sense). Nor is the epsüon constant, for reasons explained in a further debate: see Bjurlof (1978), as well as Tennant (1980), cit., together with Peacocke’s replies to these authors (immediately following the respective papers).

193 requires showing that the conception of such objects incorporates the principle of

Choice. The second way, which characterizes extrinsic arguments, conceives of principles like Choice as theoretical hypotheses that find corroboration indirectly through the confirmation of whatever consequences they hold in store for the more

“empirical” parts of mathematics.

It is also characteristic of the first way of justification that some appeal is made to a notion of self-evidence -- in this case, of set-theoretical principles. Choice, in this line of thought, must not only be true of sets (or whatever type of object is the relevant one), but it must also appear to be so upon inspection of these objects. The conception of these objects which we are alleged to have must therefore entail more or less straighforwardly the truth of Choice. This claim, however, portends at least two distinct interpretations. The first would take the self-evidence of a set-theoretic principle to be simply a matter of its “obviousness” for the cognoscenti: no explanation is really necessary. In this vein, several authors have claimed that the axiom of Choice, for instance, is self-evident on a certain conception of set

(Shoenfield and Wang say as much of the iterative conception; S. La vine thinks the same of what he calls the “combinatorial” or Cantorian conception).

A second interpretation would instead try to reflect the meaning of Godel’s famous remark according to which the axioms of set theory “force themselves upon us as being true.” A set-theoretical principle is, on this interpretation, self-evident in this sense much in the way certain perceptual beliefs can be self-evident: they need no further justification to be known, in addition to their being perceived. The idea here is that, even though an axiom may not be “obvious” in the first sense, nevertheless we may convince ourselves of its truth not by reasoning firom the other

194 principles, but rather by forming a more detailed intuitive model of the sort of universe under examination. That seems to be Godel’s point when he suggests that the continuum problem demands “not purely mathematical” ways of investigation.®"

In fact, the intrinsic approach naturally leads toward conceptions of mathematical objects that countenance the possibility of a cognitive faculty significantly hke perception in mathematics. Godel’s remark famously preludes to the suggestion that sets ought not to be considered as drastically difierent from physical objects and knowledge of sets ought not to be treated as different in kind from knowledge of physical objects. Several authors have developed the suggestion in diverging directions: Ch. Parsons’s Kantian notion of mathematical intuition and

Maddy’s perceptual justification of naive set theory, for example, are both developments of this Godelian idea. What is clear is that, on either view, certain basic truths of set theory or arithmetic will appear self-evident to the soundly functioning “observer,” for either in the case of intuition or in the case of perception we assume something hke the following: someone correctly positioned with respect to the subject matter (e.g, someone with the proper view or conception of the set theoretic universe) and with cognitive faculties unmarred by malfunctioning wül immediately know the truth of some proposition P concerning the mathematical subject matter in question; and therefore P will be self-evident for this person, as he wül be unable sincerely to deny P. This contention must be shared both by the intuition proponent and by the perception proponent, for in spite of significant

®" Godel (1964). At p. 259, Godel states that the undecidabihty of CH would be “only a precise formulation of the foregoing conjecture, that the difficulties of the problem are probably not purely mathematical”

195 differences between perception and (Kantian) intuition both faculties are typically construed (and they certainly are so construed in this context, since that was the whole point of Godel’s remark) as sources of immediate, non-inferential knowledge.

Neither of these characterizations of self-evidence, however, seems apt to account for another way in which something can be said to be self-evident in mathematics. The prime example of this third way is, I think, Zermelo’s observation that the “self-evidence” of the axiom of choice is appreciated when one reflects on the pervasive application of the axiom even prior to its explicit formulation and even by mathematicians who later would reject it (see page 171). To a certain extent, of course, Godel’s picture still applies: part of what Zermelo is there trying to demonstrate is that the axiom “forces itself upon us as true,” that we are in fact compelled to accept it — it is not up to us whether to postulate it or not, and Zermelo is not proposing that we simply stipulate it as part of axiomatic set theory. On the other hand, its way of thus “forcing itself’ is quite different from the sort of intuition or vision of set theory that Godel seems to have in mind. It has to be, because

Zermelo’s point otherwise would defeat the sort of criterion sketched above for self­ evidence: his point is precisely that people did not see the truth of the axiom, their tacit assumptions notwithstanding. Whatever the faculty presiding upon our epistemological interaction with the set theoretical universe, whether intuition or perception or something else, it was certainly issuing the wrong verdict in the case of

Choice if practising mathematicians were disposed to use it and deny it at the same time.

But why call the axiom self-evident, then? Here we must briefly reflect about

Zermelo’s reasoning, and ask: why is this an argument in favor of axiomatic set

196 theory as he has just formulated it, i.e., inclusive of the axiom of Choice? I guess the argument goes roughly hke this: Suppose a certain area of mathematics has had important apphcations, has uncovered deep and important notions, etc.; suppose, moreover, that it turns out that many significant results in this area rest on the presupposition of principle A; then, it seems, we already have justification in favor of

A, even though a "proof of A may not be available (Zermelo is exphcit in denying that provabüity is a necessary condition for “self-evidence” in mathematics). In the specific case of Choice, Zermelo thinks its apphcation is so pervasive in such basic areas that the axiom can be effectively considered scZ/-evident. Thus, imphcit in

Zermelo’s reasoning is quite a different style of cashing out a notion of self-evidence.

Let us take stock of the discussion up to this point. We can provisionally recapitulate at least three notions that fall under the umbrella of “self-evidence”: Let

F he an “area” of mathematics, a group of concepts all related to a unified subject matter, say.

(A) First account: a claim A concerning F is self-evident if it is immediately obvious to anyone who has a sufficient understanding of the concepts involved in F.

(B) Second account: A is self-evident if its truth wül be evident firom the proper intuitive understanding of the mathematical domain represented by the concepts of

F. “Intuitive” in the defining clause is to be taken as a somewhat technical term: a proponent of this notion of self-evidence must rely on the specification of a cognitive faculty that will do more or less the work that Godel appeared to reserve for intuition or “a kind of perception.”

(C) Third account: A is self-evident if it could not be rationally rejected by anyone with the proper understanding of the F concepts. In other words, the truth of A,

197 according to this view, is implicitly presupposed in F. Zermelo, of course, thought he

had the strongest possible evidence in support of the claim that Choice was imphcit in the current conception of set theory: it was already being apphed by practising

mathematicians.**

If allowance is made for the usual amount of inaccuracy in labelling, let us agree to call the second account of self-evidence ‘Godehan’ and the third ‘Zermehan’

(not much really hinges on these names: I do think that these two notions are closely matched with what their respective eponyms appear to have thought, but fidehty to the originals is not really the issue here). It seems to me that the first notion is in general of httle use, at least in mathematics: since what is obvious to Godel is bound to be quite different fi-om what is obvious to me, the obviousness of a proposition is an almost entirely subjective measure of its plausibility. Moreover, it is simply difficult to see what work could be done by a notion of self-evidence couched in those terms: some (or perhaps all) of Peano’s axioms for arithmetic, hke at least some of the usual Zermelo-Fraenkel axioms, are perhaps obviously true of numbers, or respectively, sets. Is this a determining or even just a significant factor in their acceptance? Even in these cases, what matters is not so much their “obviousness,” as rather the fact that such principles appear to flow naturally firom a coherent intuitive conception of a certain class of objects, which the principles may be seen as describing. If this is correct, then the claim that such principles are, on their face, simply obvious is best construed as a claim about their self-evidence under the

“ This classification of distinct notions of self-evidence is indebted to S. Shapiro and M.E. Smircich, who made several related observations in their review [forthcoming??] of La vine (1994).

198 second interpretation of the notion.

The central question in the present discussion is whether the remain in g two interpretations isolate genuinely distinct notions of self-evidence. The answer seems to me affirmative. The Zermehan account of self-evidence admits the possibihty that a practitioner of F (someone with the requisite understanding of F-concepts) finds A not at all obvious - or even downright counterintuitive - when stated in isolation, but is rationally compelled to accept it as an essential presupposition of what she understands of F. The Godehan notion would not seem to apply to A in such a predicament, or at least not without further elaboration: it is certainly fair to say, in this case, that one neither perceives nor intuits the truth of A, nor does it seem plausible to claim that the truth of A is “commanded” of us in anything hke the way certain certain behefe about physical objects seem to be commanded of us. If it stiU makes sense to maintain that A “forces itself upon us as being true” in the kind of situation envisioned, it must be through an indirect path, in ways incompatible with this notion of self-evidence.

In order further to underscore the distinction, it may be worthy of notice that the distinction thus sketched is not entirely of an exegetical nature, but seems grounded in actual explanatory demands arising in the epistemology of mathematics. Both types of self-evidence are “cahed upon,” so to speak, in different contexts. Once again, Godel’s reflections on the continuum problem are instructive in this regard. The “problem” is that not much is known about the power of the set of points on the real line, and Godel suspects that the CH, the conjecture advanced by Cantor, is undecidahle on the basis of the Zermelo-Fraenkel axioms (this turned out to be the case, of course). At the core of Godel’s considerations is his contention

199 that, in spite of this apparent indeterminacy within set theory, there is no reason to

believe that the question whether CH is true has “no meaning.” The question is

meaningful, Godel suggests, because the notion of set that Cantor first attempted to

describe is grounded in “intuition,” not just in the stipulation of formal principles.

Such intuition allows us to have insights into the notion of set, that is, into the

Cantorian universe, by which we can formulate more or less natural extensions of

the extant formal theories: thus, the proper assessment of the status of CH in set

theory and mathematics at large is that we have not yet uncovered the deeper

features of the Cantorian universe, and that we must keep looking -- not that CH (or

its negation) can be a matter of stipulation, in which case we should rather conclude

that there is no fact of matter as to whether CH is true. The upshot is that in order

to find an answer to the continuum problem, we must search for new principles, for

the extant ones are demonstrably insufficient (well, they would be demonstrably so just a short time after the pubhcation of Godel (1964)). But this search is actually to be conceived of as the analysis of a certain intuitive conception,®"' and therefore the

new principles will have to be selected among those that appear “self-evident” - in the second sense described above - with respect to that conception. In conclusion, self-evidence of the second sort is treated as a heuristic criterion for the analysis of set theory.

But the Zermehan notion could not play the same role, under the same

®"* Here, to repeat, ‘intuitive’ is not to be read in the ordinary sense of ‘informal’ or ‘preformal’, but rather as something like: given to the faculty of intuition. It remains to be seen, of course, what precisely such a faculty may turn out to be, whether Kantian intuition or something more hke perception, but in this context we need not worry about this.

200 circumstances. For the continuum problem consists partly in the fact that a question that is no doubt meaningful according to the current conception of set has no answer on the basis of the principles actually constituting the current conception: new principles are needed, if progress is to be made. But a principle is new, presumably, only if it is not already applied or actually presupposed by current practice. Thus, it is hard to see how it could he unreasonable, for someone thoroughly schooled in the current conception, to reject a principle that is neither presupposed by nor contained, however subtly, in the practice answering to that conception. In fact, part of Godel's point is precisely that it would not be unreasonable under the circumstances to reject any number of new principles, and it is because of this that intuition of sets is needed as a guide to deepening our understanding.

This somewhat lengthy recapitulation was necessary in order to understand how the considerations developed in the preceding subsection bear upon the general question of “how to be pro-Choice.” I have already argued that the Zermehan, but not the Godehan, notion of self-evidence is indeed appropriate to the case of the axiom of Choice, and furthermore that the source of self-evidence is not strictly hmited to the concepts of set theory, but includes concepts of logic. The latter thesis avoids, I think, the drawbacks that characterize, in turn, the intrinsic and the extrinsic approach to the justification of Choice. I shah come in more detail to the difficulties of the intrinsic approach in a moment; I shah here first comment on a problem concerning the extrinsic approach vis-a-vis Zermehan self-evidence. It ought to be clear that a principle justified “extrinsicahy” in the sense defined above

(subsection “Extrinsic Arguments”) in general whl not be a self-evident principle in

201 the Zermelian sense. Thus the notion that the Zermehan account is designed to

capture is not the notion of extrinsic justification that characterizes the so-called

“empiricist” approach to mathematics. Extrinsic justification, in fact, requires no

self-evidence: its best description is Godel’s, when he envisions the possibihty that a

new principle may be adopted as an extension of Zermelo-Fraenkel set theory simply on account of its “fiuitfidness” either for set theory or for other parts of

mathematics. Of course, a principle can be fruitful in many ways; certainly there is

no requirement that a fruitful principle be either (i) intuitively consistent with the previous conception of the theory to which it is added (that which concerns Godel, as we have seen), or (n) to any extent imphcitly presupposed in the current or previous practice of that theory.

Were Choice to be shown self-evident in the Godehan sense, it would thereby be shown to have intrinsic justification: for, plainly, to claim that the truth of a principle is evident from the mathematical intuition of sets amounts precisely to providing an intrinsic motivation for such a principle. I have argued already that the usual arguments of the “obviousness theorists,” such as the arguments based on the iterative conception, are not convincing in this regard; such arguments would, if successful, effectively establish the Godelian self-evidence of the axiom. But, as we have seen, those arguments only manage to articulate the conviction that the truth of the axiom of Choice is consistent with the iterative conception, not that it somehow follows from the iterative conception — in fact, in those arguments the truth of Choice is presupposed.

202 Indefinitism and Choice

This very point is well stated in Lavine (1994). A discussion of Lavine’s

account of these problems seems to me of great interest, because in spite of much

agreement between the present analysis and his we end up diverging — as I shall argue — at a critical juncture; I hope that a concise reconstruction of his thought may help the reader to see more clearly the point I am trying to get at. Thus Lavine says of the iterative conception:

Let S be a nonempty iterative set of pairwise disjoint iterative sets. If the Axiom of Choice is true, then there will be a set T that contains exactly one member of each member of S. Since each member of a member of an iterative set is itself an iterative set, and since aU of the members of S appeared at some stage before S, the set T will be an iterative set that appears at the stage at which S does. Thus, if the Axiom of Choice is true of sets, it will also be true of iterative sets.®°

This seems to me essentially correct. The existential claim of the Axiom -- the existence of the choice set - cannot be derived from the iterative conception.

Arguments to the contrary, which purport to exhibit the intuitive plausibility of

Choice, trade on a sort of generalization pattern from the case of simple finite sets

(for which the existence of a choice set can be demonstrated without recourse to an independent principle) to the sticky case of infinite sets, or infinite famihes of finite sets. The iterative conception itself is based on a similar “abstraction process,” in which one moves from a seemingly constructive environment, in which sets are

“built” in stages, to one where the “building” operations are not constrained by any constructive limitations. Our objection to that kind of reasoning was that the possibihty of generalizing, of simply abstracting from the limitations, is in itself that

See Lavine (1994), pp. 147-8.

203 which needs justification -- that is, it needs to be shown that in generalizing we are not in fact moving to a conception in which Choice is in fact presupposed. This is another point on which Lavine seems to be in agreement:

the “explanation” that Choice holds because the Omnipotent Mathematician has the power to pick members out of infinitely many nonempty sets is not so much an explanation of Choice as an exhortation to believe that Choice is true.®®

In fact, I would say, such explanations are at bottom clarifications of the meaning of the Axiom; they do not provide a reason to heheve it, rather they explain what someone who believed it would be committed to.

On the backdrop of this analysis of the problem, to which I subscribe, it is all the more surprising, however, to learn how Lavine in fact proposes to understand the introduction of the axiom of Choice in set theory. His own account seems to me open to at least two objections on the basis of the considerations I have so far developed (some of which are in fact analogous to Lavine’s), of which the third is probably the most instructive.

First off, it should be pointed out that Lavine does believe Choice to have intrinsic justification in set theory; on this 1 take no issue. In particular, he does heheve that there is intuitive evidence, of some sort, for Choice; on this, again, I am substantially in agreement. But, although quoting approvingly firom those remarks by Zermelo that I mentioned at the beginning of the previous subsection, he seems not to distinguish the sort of “self-evidence” to which Zermelo attests firom evidence deriving firom some kind of intuitive apprehension of objects; he reverts regularly to talk in which “intuition” and “self-evidence” are treated virtually as synonymous.

Ibid., p. 290.

204 Thus he says, for instance, that “Cantor, Zermelo and the rest needed to rely on their sense of self-evidence, on their sense of what was implicit in those... ideas” (my emphasis), whereas in the very next paragraph the “need for self-evidence” of the

“working set theorist” is explained as follows: “Most set theorists... have and rely on a feel for their subject matter to navigate their way through attempts at proofs” (p.

242).

Lavine claims that the evidential support in favor of Choice comes from the fact that the axiom is “self-evident” for the “combinatorial” conception of set, which he believes to be fundamental to Cantor’s work. In the words of Russell (as quoted by Lavine), a combinatorial collection is “defined by the enumeration of its terms,” that is, sets as combinatorial collections are simply “gathered by picking their elements in an arbitrary way, not necessarily in virtue of a rule” (Lavine, p. 104).

This notion stands in opposition to a notion attributed primarily to Russell and

Frege, that of “logical collection”: a logical set is, according to Lavine, “defined by a rule.” Now, the reference to a defining rule has clearly to be taken with a grain of salt: what I gather Lavine has in mind is that sets as logical are presumably the extensions of certain specified predicates, not that they are semi-constructive entities whose elements are selected according to a more or less effective recipe (an adherent to the logical conception of set would seem to be, by Lavine’s criteria, what is called a “definabihst”). The axiom of Choice, in its classical interpretation, asserts the existence of a set in complete independence from the availability of a way to define such a set or in any way to characterize it explicitly. So it seems correct to say, with Lavine, that Choice cannot be considered self-evident on a definabihst conception - as it is further confirmed by the negative reactions against the axiom,

205 most of them springing from definabihst misgivings.

But why would the axiom be self-evident for someone holding onto the combinatorial conception? Here Lavine’s justification is apparently twofold: in the first place. Cantor — in his view — simply considered it as a constitutive characteristic of sets that they are well-ordered, and of course if we assume that all sets can be well-ordered it is easy to prove the axiom of Choice. As regards Cantor’s ideas, this may or may not he true, but the main problem from our point of view is that, if such is the case, we obtain only a rather unsatisfactory justification of

Choice: we are simply left with the further question as to why exactly one should simply presuppose that all sets can be well-ordered, especially since the latter assertion is actually counterintuitive in the case of sets that have the power of the continuum. Here Lavine’s explanations seem less than forthcoming, even considering Cantor’s alleged “finitism” (as HaUett calls it), i.e., his conception of transfinite cardinals as a sort of well-behaved extension of the natural number system. When, for example, we are told, of combinatorial sets, that

they obey Well-Ordering because combinatorial collections are gathered by enumerating their elements, and they obey Choice because combinatorial collections are gathered by picking their elements in an arbitrary way, one sees Lavine falling into the same mistake he imputes to the proponents of the iterative conception: this can be a fine explanation of what commitments one undertakes by accepting Choice, but hardly a justification. For the obvious issue remains unaddressed: what is the justification for extending the notion of

‘enumerating the elements of a set’ from the “obvious” case in which the elements can actually be counted, to one in which such enumeration is completely arbitrary,

206 and of necessity disjointed from the possibility of counting? The fact that the elements of combinatorial sets are picked arbitrarily seems the key factor, for

Lavine, which leads one to find that combinatorial collections "obey" Choice and

Well-Ordering. But again, it was just Lavine’s own observation, in regard to the iterative conception, that the mere fact that sets of a certain type obey — that is, satisfy - the axiom of Choice is not conclusive evidence in favor of its self-evidence, or of its being implicit in the iterative conception. Why should the same objection not apply to the combinatorial conception, then?

The combinatorial conception, however, is mostly a piece of historical reconstruction, for Lavine. Its interest derives from the fact that it seems to have been the favored conception among the founding fathers, but Lavine does not think that it ultimately can provide the correct source of evidence for modern set theory

(for one thing, the axiom of Power Set is counterintuitive, from the combinatorial point of view). The second difficulty, on the other hand, concerns more directly

Lavine’s own intuitive conception of set theory, which is supposed to explain the intuitive basis for this theory, as well as to address longstanding issues about the epistemology of the transfinite.

We may convene to call his theory ‘indefinitism’ — really a more apt name that one might at first suppose, as it is actually related to Hilbert’s finitist program.

The idea seems to be this: While interesting mathematical theories — arithmetic, analysis, set theory, etc. — are ineUminably committed to infinite domains, in many contexts one can make do by presupposing not infinite domains as ranges of quantification, but only domains that are, in a sense to be made precise, indefinitely large. In a nutshell, every theory about an infinite domain has a counterpart theory

207 that only deals with indefinitely large, but (possibly) finite domains; such counterpart theories constitute the realm of “finite mathematics.” Lavine believes, furthermore, that he has found a way to make the notion of‘indefinitely large’ reasonably precise, by means of some previous logical work done on finite mathematics. As an example for arithmetic consider the fact that

(Vx)(3y)y =x + l (2)

With both quantifiers ranging over N, this sentence expresses that there is no finite termination of the natural number sequence. If the quantifiers were bound by any given finite number p, the sentence would no longer be true. But suppose that the first quantifier is bound to range within a finite, arbitrary collection of numbers Qq while the second quantifier is bound within another collection Q,, with the proviso that every number in has its successor (the number resulting from adding I) in

Q,. Then if Qq is finite, Q, need not be infinite, either: all that is required is that Q , be “indefinitely large” with respect to Qq. Then the statement

(VxeQo) (3yeQ^) x=y+l (3)

is true, and in fact any statement of this form will be true provided that the pair of subdomains Qg and Q, is replaced by another pair of subdomains whosemembers are related in exactly the same way as Qg and Q,: that is, the second contains aU elements that are obtained by applying a selected operation (in this case successor) to members of the first. In this way, Lavine suggests, the original statement, involving unbounded quantification, can be regarded as a schema each of whose instances involves only quantifiers relativized to indefinitely large subdomains. The

208 collection of bound instances constitutes the finite counterpart of the original theory

(in this example, of course, we have considered a “theory” containing a single axiom).

Lavine believes that there exists a “natural” relation that motivates the unbounded version of a theory on the basis of its finite counterpart, a relation he calls extrapolation: the unhounded theory can be reached firom the finite theory by

“extrapolating,” which apparently means abstracting firom any particular bound to the ranges of quantification - by simply dropping all relativizing clauses, Lavine seems to suggest. However, extrapolation is “evidence preserving,” according to

Lavine, that is, an unbounded theory is intuitively justified, by extrapolation, if the axioms of its finite counterpart are intuitively evident.

One point that needs to be underscored is that, if this approach is to serve its philosophical purpose, it will not do to have the omegas replaced by definite finite bounds on the quantifier. In other words, it is essential that the omegas be actually regarded as indefinitely large (this is what makes it indefinitism). The reason seems to be that extrapolation would not have the desired type of intuitive justification, in that case. If the omegas were identified with subdomains of determinate finite magnitude, then extrapolation would in effect require that we show the truth of every instance of the schema (2), i.e., of a statement of the form (3) for every finite subdomain. But this of course would mean that extrapolation involves as much of a commitment to an infinite domain as the original unbounded theory, and the presumed ontological and epistemological gains that the indefinitist is after would prove elusive. Perhaps it helps to think of extrapolation as supported by the kind of inferential transition exemplified by the rule of universal generalization, in which the premise is to be shown for an “arbitrary” individual Indefiniteness is a way to

209 ensure such arbitrariness: the bounds involved in the relativization of quantifiers

need to be as large as needed, while still finite. (As I understand it, this seems what

is expressed by Lavine's claim that indefinitely large magnitudes are

“indiscemible.’O

It has already emerged, then, that the property of being indefinitely large is contextually determined: to use the example once more, Q, is indefinitely large with

respect to Qq because it contains everything that one can obtain by means of a given set of operations (in this case, just the successor function) firom the elements in Qq.

We say that, given an initial collection of individuals and a collection of functions, the individuals that can be obtained by direct application of such functions to elements of the initial collection are available. Thus Lavine defines ‘indefinitely large with respect to X" as the property of containing aU elements that are available firom X. In conjunction with the observation made in the previous paragraph relative to the distinction between the finite and the indefinite large, this conception of availability leads to an important feature of indefinitism. Lavine’s account is committed to the following thesis: for each of the successively defined £\ if something is available, then it is in Qj.®' I shall call this the availability thesis.

However, the converse of the availability thesis is not in general true: if it were, it would in effect impose upper bounds on the omegas, thus making them not indefinitely large. In other words, one cannot deduce that a certain element is available firom the fact that it is in (one of) an indefinitely large domain: if one could, this would, once again, trivialize the conception of indefinitely large domains, and

See ibid., p. 262.

210 turn indefinitism into finitism.

The availability thesis implies, among other things, that, although the

omegas are generally intended to be indefinitely large but finite (they need not be

finite, but what is interesting is of course that they can in general be taken as finite

collections), one cannot simply argue on the basis of their finiteness that something

in them is “available,” that is, can be obtained or “picked out” by some function,

operation, definition, etc. This, I think, turns out to be a serious obstacle for

Lavine’s indefinitist justification of Choice (and possibly of other features of set theory).

Thus we come to the central part of the difficulty. At the core of Lavine’s approach is the concern that has been driving our discussion, that is, how to individuate the intuitive conception of sets, or the source of intuitive evidence for the principles of set theory. It should by now be clear what is the strategy that characterizes indefinitism with regard to this problem: in essence, the plan is to show that the axioms of set theory are indeed self-evident in the Godehan sense when motivated by a conception of set theory as a theory of indefinitely large domains. The project is, in other words: (i) show that any mathematical theory, including theories committed to infinite domains and in particular set theory, has a counterpart in finite mathematics which deals only with at most indefinitely large objects; (ü) show that the axioms of set theory are self-evident when interpreted within finite set theory. Le., with respect to indefinitely large domains; (hi) argue that set theory (the real thing) is simply the theory obtained by extrapolation firom the theory dealing only with indefinitely large domains. For the purpose of the present discussion, my objection is to (ü) as it regards the axiom of Choice; I have

211 only cursorily mentioned what Lavine seems to think of (iii), and (i) is based on several results in logic and model theory which we have not touched upon.

Why is the axiom of Choice intuitively evident when applied to “finite” set theory, that is, to the theory of indefinitely large domains? Here Lavine’s explanations are succinct: in his view the task consists simply in establishing that

“the principle of finite set theory can be motivated as self-evident principles concerning finite sets” (p. 288). ‘Finite’ here includes the indefinitely large; this is no stretch, since “[t]he ideal capacities required to manipulate indefinitely large sets require no great leap of faith, since the sets are, after all, finite” (289); “it is a trivial matter to see that the axioms of Fin(ZFC) [the finite counterpart of ZFC] are truths of finite set theory” (290). The only exception Lavine sees as intruding is the axiom of Infinity, to the indefinitist explanation of which he devotes considerable effort. As for Choice, “the explanation of Choice arises firom a simple fact about the finite”

(289).

The problem here is not just the brevity, however. By all appearances,

Lavine’s motivation for including the axiom of Choice in finite set theory seems to boil down to reminding us of its obviousness when considered in its apphcation to finite sets (more precisely, to finite families of sets, because for infinite famihes of finite sets the advantages of finitude are not there). NaturaUy, it is true that if all the sets of which we speak were finite, and we dealt exclusively with finite coUections of such sets, then the assertion that there exists a choice set for any such coUection would be trivial, because one could always perform, by an effective operation, the required selection firom the members of the collection. However, this is manifestly due to the fact that we conceive of the elements of any finite set as

212 being all in some sense “available” for inspection: a set is finite if it can be exhaustively ordered by counting its elements; once this is done, one can always pick out an element by a rule, without having to rely essentially on an arbitrary choice.

Of course, there are huge finite sets, that cannot be enumerated within the lifetime of the Sun; but the going idealization is that “in principle” the elements of a finite set are “all there” already simultaneously, and therefore one is always available to be selected.

This sort of explanation is no doubt trivial, as it rests entirely on commonsense intuitions about the finite. The trouble for Lavine is that, while these intuitions are indeed mundane, they cannot be easily adapted to serve as a justification for Choice under the conceptual constraints he has set upon finite set theory. For recall that the converse of the availability thesis is false, in general: one cannot argue firom the premise that something is in an indefinitely large collection to the conclusion that it is “available” as the result of an arbitrary operation. In the case of finite collection, this inference seems plausible: the converse of the availabihty thesis seems true of finite collections (so it would be true of the omegas

Lavine uses, if those omegas were in fact finite, not just indefinite large).

Let us look for a moment to the actual statement that according to Lavine expresses Choice in Fin(ZFC):®®

(t'x eQg) [ ^ y z e Q;) ‘if y and z are in x, then they are disjoint’ =» (3ye Qj)( Vze Q^) ‘If z is a nonempty member ofx, there is exactly one element that belongs both to z and y’];

“ Cfir. ibid., p. 296. In order to make the formulation more perspicuous, I have used sentences in “set-theoretical English” instead of spelling out everything in the language of ZFC proper, as done in the original Single quotes indicate that the sentence “encodes” the corresponding construct in the language of ZFC.

213 why is the existential assertion in the consequent of the embedded conditional justified? That is, how can we be sure that the choice set is available in Q,, given only what is available at the initial stage? Of course, that is the core of the problem.

The only suggestion I am able to make out is that, under the “natural” interpretation of the context suggested by the indefinitist, one can simply construct the choice set. And why can we assume that this is feasible? Because all sets involved, including the indefinitely large omegas, will be finite. But if this is the reasoning, it seems to me that the finitization of set theory brings us no closer to an answer — if the question to be answered is the one that Lavine himself thought was not solved by the traditional types of intrinsic argument. For we have a dilemma, either horn of which leads to failure. If we simply stipulate that all famüies of sets to be initially considered are finite, then the argument reduces to that of the constructivist we discussed in a previous section; we can construct the choice set, but what remains to be seen is whether the existential assertion can be correctly extended to cases in which no such construction is available. (Furthermore, constructivists would have initially a more plausible case than Lavine’s, because from their standpoint the meaning of an existential assertion just is that a certain construction is possible — a tack for which Lavine shows no affinity, given his avowed preference for the combinatorial conception of set.) On the other hand, if we allow indefinitely large families of sets -- constrained only by the size of the indefinitely large domains to which they belong, then we will not get the result that a choice set is always available “trivially” unless the converse of the availability thesis is appealed to — but in this case ‘indefinite large’ is treated really as longhand for ‘finite’.

214 To put things slightly differently, what Lavine appears to have done in

justifying Choice is electing to treat the indefinitely large collections as collections

that can be surveyed and enumerated exhaustively. Such collections can be counted

from their first to the last element, just like finite collections. Then, of course, the

axiom of Choice does appear trivial On the other hand, the availability constraint

on the indefinitely large is violated, and the force of Lavine’s appeal to extrapolation

as the rationale for introducing the fiill-fiedged, non-finite axiomatization is largely

lost. In other words, the explanation reduces to that kind of “exhortation” that the

author had previouly chided: it amounts to say, well Choice is obviously true for

finite sets, why shouldn’t it be for infinite ones?

There is also a broader difficulty with Lavine’s attempt to explain the axiom

of Choice in terms of an indefinitized relativization of set theory. In actual fact,

most readers of Lavine’s complex construction will be struck by an underlying

mismatch between his early arithmetic examples, such as the one I have used to outline indefinitism, and the later work to which the theoretical framework is put.

That sense that the analogy is faulty can be attributed, in my view, to the following problem. By adding I to any given natural number, the successor number is effectively produced; in any given case, the operation is as a matter of fact bounded

“from below,” from what was already available. The plausibility of the indefinitist relativization of the “schema” (2) seems due in large part to the fact that, in the end, the (possibly) finite size of the omegas plays no role in the determination of the successor operation itself: just as Lavine requires, in that case availability does not depend upon there being a finite bound on the size of the omegas involved. If we regard (2) as the schema and statements of the form (3) as its instances, it is easy to

215 see that the truth of each instance considered individually owes nothing to the fact

that the particular omegas occurring in it are finite: that shows that the restriction

introduced hy the omegas is irrelevant, in a sense. But when we move to finite set

theory, the truth of each instance of some key axioms (certainly Choice) has to be

guaranteed by explicit appeal to the finiteness of the omegas, in the explanation of

finite set theory, the finiteness of the omegas is prominent in ensuring that the

principles of set theory appear intuitively evident.

Conclusion

Lavine has made what is in my opinion the best and certainly the most sophisticated attempt to justify the axiom of Choice on an intuitive background.

This is what “intrinsic” arguments typically aim at, but except for Lavine’s these are

usually confined to a few cursory remarks that the authors (Wang, Shoenfield, etc.)

appear to regard as thoroughly platitudinous. If my analysis is correct, they are right: intrinsic justification does usually turn out to be platitudinous, because it is

usually impossible to picture a universe of iterative sets without already assuming the truth of th axiom in the description of the picture.

Lavine argues on the basis of a combinatorial conception, and in fact tries to provide justification both for it and Choice simultaneously. That the attempt fails seems to me an important indication of how deep the conceptual roots of the axiom of Choice really are. On the other hand, extrinsic arguments, especially when supported by the wealth of historical detail in Moore (1982), are fine as far as they go: they might suffice to convince an otherwise hostile mathematician that the

216 axiom of Choice is of such central significance that it cannot be discarded. But I think they remain inadequate to explain the intuitive roots of Choice, seen in the remarkable fact of its prior applications noted by Zermelo. And they do not tally up well with Hilbert’s and others’ early endorsement of the axiom on solidly intrinsic grounds. Extrinsic arguments tend to fit Choice into the same order of considerations that apply to genuinely unintuitive hypotheses like the large cardinal axioms. Thus we have seen the failure of most intrinsic arguments (in this and the previous chapter), the inadequacy of extrinsic justification (in the previous chapter), and an extended argument that Choice is justified (in some complicated way) by logic. I shall happily rest with the third option.

211 APPENDIX A

A BRIEF OVERVIEW OF PAPERS BY SKOLEM DISCUSSED IN THE CHAPTER

Several papers by Skolem figure prominently in the foregoing discussion, and the development of the ideas expressed in them is also of significance. Therefore, it was thought that having short descriptions of their respective contents in one place of reference might be helpful to the reader.

(1920) “Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by L. Lowenheim and generalizations of a theorem."

The first section of this paper, which is the only one referred to, deals with the theorem indicated in the title. It defines ‘first-order expression’ following the manner of Lowenheim, but while that author had used the technical device Skolem calls “Schroder’s development of products and sums” (which makes use of propositions with an infinite number of terms, represented in this notation by subscripts, subscripts of subscripts, etc.) the present paper attempts an original modification. The first result is that for every given first-order expression there is an expression in “normal form” which is equisatisfiable to the given one. This is done

218 in way similar to Lowenheim. The main result, following from this one, is that every

expression in normal form is either not satisfrable (Skolem says: “is a contradiction”)

or satisfrable within a denumerably infinite domain.. This is where Skolem modifres

Lowenheim's proof he does not use subscripts as symbols for individuals, but instead uses the axiom of Choice to introduce Skolem functions in place of the existential quantifrers. Consider a simple normal form expression like

(Vjc) (3y ) G(x,y) . By hypothesis this formula has a model D; thus assign to x an

arbitrary individual a in D: there must be some b such that G(a, b). Select one

(Choice is used here); then select another individual c such that G(b, c) - again, by the hypothesis there must be at least one, and so on. This is the so-called Skolem hull construction: the chain of individuals a, b, c ,... selected in such a way is a denumerable subset of D. In fact this proof establishes not only the truth of

Lowenheim's original proposition, but also that, if an expression is satisfrable in an infrnite domain D, it is satisfrable in denumerable subdomain D’ of D, and in fact D’ is by construction a submodel of D. In the remainder, Skolem extends the result to more general cases of frrt-order expressions, showing that the theorem holds even if the expression is the logical product or sum (conjunction or disjunction) of a denumerably infrnite set of formulas.

(1923a) “Some remarks on axiomatized set theory.”

This paper is the main source for the Skolem paradox. It is divided into several sections, each discussing a separate point about the Zermelo axioms. First,

Skolem states a general qualm about set theory: it purports to be a study of the

“general notion of domain,” but that notion has to be presupposed in the “logical

219 investigations” of the theory itself Second, an inductive definition of what Zermelo vaguely conveyed as “definite properties” (which define sets) is given: the notion defined by Skolem is in fact that of first order formula in the language of ZFC. Third,

the LST is again proved. This proof however, does not use the axiom of Choice.

Instead, it shows how, given a first order formula in normal form, directly to construct an interpretation over the integers; so this is not the submodel construction, rather what I have referred to as the numerical model construction.

The construction proceeds as described for the Skolem huU above, but it uses the intrinsic natural ordering of the integers to pick out the selected element, in stead of

Choice. The better part of the paper is then taken by reflections on the paradoxical consequences of this fact for set theory; relativism is considered a serious flaw of an axiomatic approach to set theory. Prominent among Skolem’s concerns is that basic notions hke finiteness, if founded set-theoretically, wiU turn out to be relative, while in his mind they are more solidly determinate than anything in set theory. He notes that the notion o f‘arbitrary finite’ is essential in investigations of deductive systems such as are central to formalism. Yet axiomatic set theory would provided only a relative notiont of finitude. Lastly, he discusses Poincare’s remark about induction: the axiomatic approach to arithmetic is vacuous, because it is impossible to prove the validity of induction without appealing to mathematical induction as a metatheoretical principle. Skolem registers his agreement with Poincaré, noting that the study of the properties of “finite expressions” in a formal system (formulas, proofe, etc.) already presupposed induction.

(1923b) “The foundations of elementary arithmetic established by means of the

220 recursive mode of thought, without the use of apparent variables ranging over infinite domains.”

The aim is to show that “a logical foundation can be provided for arithmetic without the use of logical variables.” The paper develops arithmetic in a language without unbounded quantifiers; defined predicates and functions are introduced by primitive recursive definitions; theorems can contain free-variables. All instances of a theorem, then, can be verified by direct computation and perhaps prepositional logic. This represents the “recursive mode of thought.” A concluding remark compares this mode of thought to Kronecker's view that aU definitions in mathematics should allow one to determine the defined object in a finite number of steps. Natural numbers (numerals) and the successor function are the primitives; the paper is devoted to show how much of the usual arithmetic relations and functions can be developed on this basis.

{1929a) ‘Tiber die Grundlagendiskussionen in der Mathematik.”

This is a conference on recent foundational debates. Much of it is occupied by a discursive presentation of the significance of set theory in the foundations of mathematics, starting with problems relative to the differential calculus. Skolem presents the three major approaches as, respectively, the logicist, the intuitionist and the axiomatic one. He mentions his (1923b) as an instance of the views one finds made expHcit in intuitionism. The “axiomatic” approach, i.e., the Hilbertian one, is presented in most detail, including a discussion of the “Vertreteraxiom,” which is

Hilbert’s epsilon axiom (analyzed below in Ch. III). Most of the comments in this regard refer to the more technical treatment of these matters in (1929b).

221 (1929b) “Über einige Gmndlagenfragen der Mathematik.”

This long work is dedicated to a detailed reconstruction of the reasons for set- theoretic relativism from (1923a), as well as of other foundational questions. The technical exposition of the LST is more detailed than in that paper, with a focus on clarifying why the result apphes to any axiomatic system. The impredicative nature of the reasoning used in set theory is also analyzed more closely, mainly to show, by examples, that impredicative modes of reasoning are not as secure (vis-a-vis the satisfiability of the theories obtained) as predicative ones. However, Skolem also notes that a philosophical antipathy toward impredicativity has no probative value in mathematics. The following is proved as a theorem: if a system of logic without

Hilbert’s Vertreter Axiom is not contradictory (i.e., is satisfiable), then so is the same system with the addition of the Vertreter Axiom. Since this axiom, Skolem notes, implies that the principle of excluded middle holds also for any quantified statement, the result shows that no contradiction can be produced by adding excluded middle. The piece concludes with a few observations about the impossibility of characterizing mathematical concepts by means of axiomatic systems; this point is stated in a more general spirit than in (1923a) - that is, it is not just presented as the problem of set-theoretical relativism.

(1941) “Sur la Portée du Théorème de Lowenheim-Skolem.”

Another formulation of the early results, as well as of the philosophical point of view of relativism, with special attention to set theory. In this work, however,

Skolem seems to have wholeheartedly embraced the skepticism he had criticized early. He says: “many mathematicians have found this relativism paradoxical; but in

9 9 9 reality there is in it nothing surprising.” In what sounds hke a one-sentence summary of Hilbert's program, he states that “the assertion of the existence of non- denumerable sets should be considered but a manner of speech, [for] such a non- denumerable absolute is no more than a fiction.” It appears, then, that the meaning of the Skolem paradox is just that this conception of “absolute” cardinahty properties is a mistake. The axiomatic method itself is no longer criticized; relativism is a characteristic, but not a defect of axiomatics.

223 APPENDIX B

THE AXIOMS OF ZF IN THE ITERATIVE CONCEPTION

What follows includes the derivations of the main axioms of Z-Inf (i.e. ZF

minus Infinity and Replacement) from the axioms of stages and set formation

presented in the first section of this chapter. The axioms are those of Boolos (1989),

and the “proofs” are either those given there or constructed from Boolos’s hints and sketches. The arguments are presented as informally as possible, focussing on the

intuitive reasons why they should be true in the present framework.

Pairing: Given sets z and lü, there will be stages s and t such that z is formed at s and w is formed at t, by ALL. By NET, there is a stage r later than both s and t.

So we have both zBr and ivBr (stages are cumulative). Then apply Comprehension to the formula fy =z^ y=w) to obtain the existence of the pair.

Union: Given z, for some r zFr (by ALL). Then if u; is a member of z, w must be formed before r. Le. wBr — by WHEN. Similarly, by WTIEN again, if y is a member of w, yBr. Then apply Comprehension to the formula 3ttJ (yçw &w ez) to obtain the existence of the set x which is the union of z.

Power Set: the axiom of power set can be justified, according to Wang, by

224 considering that we can ‘run through’ all elements of a given set with omissions’, thereby obtaining an ‘overview’ of every possible part of the original set. By this procedure we see that, if the original set was given to us by an ‘intuitive range of variability’ -- Wang’s criterion for the existence of a set —, then we obtain a novel

intuitive range of variability’ for the collection of aU. subsets of that set; thus, we can form its power set. Obviously, each subset corresponds to a single application of our principle of Comprehension to the original set. What is required now is that we form a concept of aU such applications.

In the framework of subsequent stages, the argument goes as follows: consider an arbitrary set z. All members of z are given earher than z is, and therefore any subset of z can be (and therefore is, given the assumptions implicit in the present context) formed at the same stage as z. But we assume that we can always move to a later stage. At that stage, the comprehension principle can be applied to yield the existence of the power set of z. In other words:

Given z, there is an s such that zFs (by ALL). By WHEN, zFs iff

\tw ( l u s ivBs). For any y, then, \fwiwGy 3 w€z)z>yFs. Let r be a stage later than s (by NET such a stage exists). We then get the power set r of 2 by an application of Comprehension taking the formula A(y) to be Vit; (weyz>w €z).

Emntv Set: the existence of the empty set follows by taking A(y) as ~'y=y and applying Comprehension. In effect, the existence of the empty set is proved by logic alone (if the identity sign is treated as a logical constant). This is customary in most presentation of set theory.

One of the most interesting features of stage theory is probably that it can dehver a principle of induction for stages from which in turn there follow easily the

225 Axiom of Foundation for sets and the proposition that stages are linearly ordered

(and therefore, hy the principle of induction for stages, well-ordered).

The argument showing this has independent philosophical significance relative to the question whether a principle of induction can be justified on independent grounds, but we do not dwell on this aspect. The (somewhat long) argument in support of “induction” is introduced only so that we can prove that the iterative sets obtained in stage theory are well founded.

The derivation makes “use” of the paradox of grounded classes, according to which there can be no set of all the y such that Va [yea 3 3xea(xfla=0)]. A member of a such as X is called a minimal element. What this formula says about y is; “every class to which y belongs has a minimal element.” If y has this property, y is said to be grounded. The key fact about grounded class is the following: any collection whose members are grounded classes is grounded. Here is the argument:

Let A be a class such that all elements of A are grounded. We show that any class B to which A may belong will have a minimal element. Suppose then that A eB . There are two possible cases: if BnA=0, then A is a minimal element of B and this terminates the proof of this case. Otherwise, there is some element of A, call it x, that also belongs to B. But x is grounded and xeB; therefore B must have a minimal element. This proves that any class A all of whose elements are grounded belongs to has a minimal element.

However, this implies that the class G = {y | y is grounded} whose members are aU and only the grounded classes cannot be a set. If G were a set, there would be a set {G} of which G is the only member. But since G is also grounded by the key fact argued above, G is a member of G. Thus Gn{G}=G*0, and therefore G belongs to at least a class that does not contain a minimal element - contradiction, since G is grounded. The key fact about grounded classes will be used in the following derivation of induction for stages.

226 We define a relation R between sets and stages by letting bRs iff all elements of b are grounded and are formed before s. Notice that, since all members of b are formed before s, a b such that bRs can be found by applying Comprehension at any stage s.

We now prove that, informally, that every property P is instantiated at some earhest stage: 3s Ps ^ 3 s [Ps & Vt(t< s 3 -

7. We assume as fixed an r such that Pr. First case: for all u such that u

-iPu. Then r is the minimal element we need in P and this terminates the

proof for this case.

8. Second case: for some u, suppose u

stage r we can form a set x such that x= {a 13s(s

since all such a s will have been formed before r. Note that by WHEN if aRs,

then a is formed at s, because all members of a will have been formed before

s. Then aBr is implied by aRs and x= {a| 3s(s< r &aR s & Ps)}.

As we noted above, for every s there is an a such that aRs. Moreover, ware

under the assumption that there is a u

nonempty. Now, if a is a member of x, every element of a is grounded. It

follows, by the key fact about grounded classes noted above, that any such a

is grounded. But then we have that every member of x itself is grounded; by

the same argument, then, x is grounded. Thus, x has a minimal element, call

it 6, and it follows that, for some s

Now we know that for every t

grounded (by definition of R plus the fact that all its members are grounded);

c is formed at t (since aU its members are formed before t, by definition of R);

227 so c is formed before s and therefore ceb, since bRs. Moreover, t

transitivity of earlier than’. Suppose for contradiction that Pt; then cex and

therefore b is not a minimal element of x, since c e (6 Ha) . But this is

impossible. Therefore iPt. This concludes the proof

The axiom of Foundation follows from the principle of induction for stages. We need to prove the schema 3xAx 3 3 x[Ax & Vy (y e x 3 -■ Ay)] for an arbitrary formula Ay.

Assume then Ax. Since every set is formed at some stage (ALL), there is a s such that 3x (Ax & xF). s

Let Ps stand for 3x(A r& xFs). By applying the previous result, we get

3spx(A x& xFs) &V£(f

Fix a stage s and a set x. Let y be an arbitrary member of x. By WHEN, there is a t

3£[£

The axiom of Foundation is not part of Zermelo’s original axiomatization of set theory. This is not because it is any less immediately evident than the axioms included by Zermelo. Some set theorists (for example, Kunen®®) observe that the axiom is mathematically “trivial,” in the sense (as I understand it) that no significant portion of mathematical practice assumes a universe where the axiom is be false. Normal presentations of set theory do not “do” anything substantive with the axiom, except noting its presence. So Zermelo’s neglect is probably a case of

^ K Kunen (1980), p. 101: “Foundation has no application in ordinary mathematics, since accepting it is equivalent to restricting our attention to WF, where aU mathematics takes place anyway.”

228 omitting what is obvious. The previous derivation of the axiom from the theory S,

then, does some justice to Kunen’s remarks: axiomatizations in which Foundation does not hold have been formulated (Aczel), but they would require some modification of the basic iterative conception.

We have thus a “strengthened” version of the theory of stages, which we may call S*. From S*, it is easy to derive the principle of one-one replacement: this is the version of Replacement in which the functional relation between two sets assumed to exist in the antecedent is required to be one-one. The informal argument goes as follows:

Suppose that for every member x of a set X there is exactly one y such that Rxy, where R is a two-place predicate such that if Rxy and Rxy’ then y=y’, and if Rxy and

Rx’y then x=x'. Of course, there is a stage s at which X is formed. Now let A(y) be the formula 3x(Rxy); this formula is true of all and only those y^s in R(X), which is the image of X under R. Then there is a one-one functional relation between the y’s such that A(y) and the x in X: namely, the inverse of R (which is a function whose domain is just the y such that A(y)). Let us call this function R’. Moreover, if A(y), the value of R’ for y (i.e., the x such that x=R’(y)) will have been formed before s, since it is a member of a set formed at s. Therefore, we can apply the modified

Comprehension schema to the formula A(y), thereby obtaining the existence of a set

Y={y:A(y)}. We have thus proved one-one Replacement.

The usual axiom of Replacement of course covers functional relations that are not one-one. One-one Replacement can be, however, generalized to this case. Let R be a many-one functional relation whose domain F is a set. Assuming one-one

Replacement we want to show that the domain of R is a set, too. Let R' be the

229 relation holding between x and y if and only if Rxz and Ryz. Since R is a functional relation, R’ is an equivalence relation and will partition F into mutually exclusive equivalence classes such as [x] (the equivalence class of x under R’). Let F(R) be the resulting partition. Now consider a two-place relation 8 defined by: Suv iff u e F(R) and, for all xeu, Rxv holds. It is easy to see that S is a one-one functional relation with domain F(R) and the same range as R. Therefore, we can apply one-one

Replacement and obtain that the range of S is a set - and so is the range of R.

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