Skolem's Promises and Paradoxes W. D. Hart The Journal of Philosophy, Vol. 67, No. 4. (Feb. 26, 1970), pp. 98-109. Stable URL: http://links.jstor.org/sici?sici=0022-362X%2819700226%2967%3A4%3C98%3ASPAP%3E2.0.CO%3B2-W The Journal of Philosophy is currently published by Journal of Philosophy, Inc..
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SKOLEM'S PROMISES AND PARADOXES*
HE LSwenheim-Skolem theorem is rather a philosophical tease; it has always excited philosophical imaginations, but without ever quite delivering on its suggestive intimations. In hopes of tempting it to reveal whatever philosophical charms it may have, let us first flatter it by describing its historical conception. Cantor thought he had domesticated the actual infinite and discovered whole new realms of actually existing infinities. Among these latter, Cantor seems to have been particularly proud of the endless hierarchy of transfinite cardinal numbers. His exposition of his set theory was no more axiomatic than that of elementary calculus; he followed his interests, assuming what seemed clear to him and arguing with varying degrees of success and rigor for the rest. But then the infamous paradoxes threatened Cantor's paradise, as Hilbert called it. Ernst Zermelo came to Cantor's aid, and as part of his program for salvaging Cantor's work, he axiomatized l as much of Cantor's set theory as precisely and carefully as he could.2 Zermelo's papers appeared just before 1910. I n 1915, Uwenheim argued for a general claim which we would nowadays express by saying that any quantificational schema that is satisfiable at all is already satisfiable in a denumerable domain. As is well known, this result generalized to denumerable collections of schemata, such as theories. Soon after Lowenheim's work came out, Thoralf Skolem got into the act. Lowenheim's argument had left a good deal to be desired. In 1920, Skolem published an improved proof of a version of Uwenheim's result, but without philosophical comment. Then, in 1922, Skolem gave his great address, "Some Remarks on Axiomatized Set Theory." 3 In it, he presented another improved proof of another version of Lowenheim's result. (The differences between these proofs and, more importantly, between these versions of Lowenheim's result have a certain philosophical interest. I shall at-
T
* In large part, this paper grows out of a seminar conducted by Burton S. Dreben at Harvard in the spring of 1968. Consequently, as in most philosophical matters, I am indebted to Professor Dreben-so much so in fact that he should perhaps share some responsibility for my errors. 1 Cantor was himself aware of paradoxes of set theory and sought remedies for them. See Cantor's 1899 letter to Dedekind in Jean van Heijenoort, ed., From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 (Cambridge, Mass.: Harvard, 1967), pp. 113-117; reviewed in this issue of this JOURNAL, 109-110. But it is not obvious that these remedies included axiomatization. 2 Even with Zermelo's work it remained for Skolem to make clear and precise Zermelo's notion of a "definite" property of sets and for Skolem (and Frankel) to add the axiom (schema) of replacement to Zermelo's set theory. 3 Translated and reprinted in Van Heijenoort, op. cit., pp. 290-301.
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tempt to describe the difference between the two versions of Lowenheim's result below; the two re-proofs of Lowenheim's result differ in that in the first but not the second, Skolem appeals to the axiom of choice.) In his 1922 address, Skolem attempted to draw some philosophical conclusions from his and Lowenheim's r e s ~ l tI.t~is at this point that the philosophical fur starts to fly. Skolem seems to have thought that Lowenheim's result was almost as devastating to Zermelo's attempt to salvage Cantor's set theory in an axiomatic form as the paradoxes were to Cantor's original, informal set theory. But not quite. For the paradoxes of set theory were inconsistencies within set theories which showed by reductio ad absurdum that those theories were false, pure and simple (whether they could be patched up is another story, and not one we are telling just now). But no one, least of all Skolem, ever claimed that Skolem's paradox, as it has come to be known, is an inconsistency deducible in any set theory, axiomatic or naive. But then what is Skolem's paradox? What, if anything, does it show? And where--on the basis of what assumptions-is this shown? A crucial point here is this: Skolem was perhaps resisting not so much set theory per se or even axiomatic set theory per se as axiomatic set theory as a foundation for mathematics. He says at the close of his 1922 address:
.
. . I believed that it was so clear that axiomatization i n terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. B u t i n recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore, it seems to me that the time had come to publish a critique (301). (Incidentally, it is of some historical interest to note that Cantor himself does not seem to have been motivated, at least primarily, by a desire to use his set theory as a foundation for any of extant mathematics; this desire seems rather to have sprung from the other 4 An interesting statement of a Skolem-type position occurs on pp. 426-427 of Stephen Cole Kleene's Introduction to Metamathematics (Princeton, N.J.: Van Nostrand, 1952). 3 This of course does not hold for all set theories. Godel remarks that set theories whose natural models are the iterative sets arranged in transfinite ranks have "never led to any antimony whatsoever"; see Kurt Godel, "What Is Cantor's Continuum Problem?" reprinted in H. Putnam and P. Benacerraf, eds., Philosophy of Mathematics, (Englewood Cliffs, N.J.: Prentice Hall, 1964), p. 263. However, such natural models do not seem to have been described in the literature until von Neumann used them as a device to prove the relative consistency of the axiom of regularity.
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two creators of set theory, Dedekind and Frege. Frege of course axiomatized nearly in accord with contemporary standards of formal rigor, since he was originating those standards. Thus, Frege should perhaps be credited with originating the program to use axiomatized set theory to found mathematics, even though in a more scrupulous history, one would carefully distinguish Cantor's Mengen and Frege's Wertverlaufe.) Skolem, then, objected to axiomatized set theory as a foundation for mathematics. The philosophical notion of a foundation is, or at any rate should be, notorious for its obscurity. Unfortunately for our present exegetical purposes, Skolem does not seem to have found it particularly so. He was perfectly happy with the view that mathematics has a foundation, even an "ultimate" one, whatever that might be. It was just that axiomatized set theory wouldn't fill the bill; inductive inferences and recursive definitions would.6 Whatever the rights and wrongs of this dispute, Skolem's general attitude toward foundations leaves us with the problem that he never seems to think he needs to say what he means when he denies that axiomatic set theory is a satisfactory ultimate foundation for mathematics. Instead, he contents himself with casting the conclusion of his philosophical discussion to Lowenheim's result in such exciting but frustratingly opaque terms as "a relativity of set-theoretic notions" and a relativity of the existence of the higher infinities in which Cantor took such pride. T o be sure, there is something natural and intuitive about associating the philosophical idea of a foundation and a denial of relativity; but surely contemporary philosophical conscience demands that this association be worked out and described clearly. A first approximation to Skolem's line of thought might go like this: Assume for reductio that set theory is the foundation of mathematics. The paradoxes have shown that, at best, we cannot trust naive set theory A la Cantor but, at most, only rigorously axiomatized set theory, such as was begun by Zermelo. Hence, what is in question is whether axiomatized set theory is the foundation for mathematics. And whatever else may be required of it, axiomatized set 6 More precisely, Skolem claimed that: "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 i n 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." See Thoralf Skolem, "The Foundations of Elementary Arithmetic Established by Means of the Recursive Mode of Thought, without the Use of Apparent Variables Ranging over Infinite Domains," in van Heijenoort, op. cit., p. 304. The science so founded is primitive recursive arithmetic, not classical mathematics.
theory should at least be fixed and invariable, in some sense absolute, if it is to provide a firm foundation for mathematics. Now of the subject matter of set theory, the transfinite cardinals are prominent and characteristic. So if axiomatized set theory is to provide mathematics with a firm foundation, it should have a fixed and invariable subject matter, including the transfinite cardinals. But if the set theory is really thoroughly axiomatized, then, by the Lowenheim-Skolem theorem, it will have a countable model which (granting certain straightforward technicalities about transitive models) cannot contain anything uncountable. So, axiomatized set theory cannot have the transfinite cardinals in its invariable subject matter, and so will not do as a foundation for mathematics. Of course, the argument rehearsed just now raises more questions than it answers; the rest of our time will be devoted to questions of both these varieties. Notice first that the argument requires that we lump together set theory as a theory of the transfinite cardinals, or more generally, of infinite structures or sets, with set theory as a foundation for mathematics. But must set theory wear both these hats at once? T h e transfinite cardinal numbers do not seem to have played much of a role in classical analysis and algebra; and though the continuum was discussed throughout the second half of the nineteenth century, its cardinal number was not subject to the same notoriety. So why not jettison that part of set theory which is explicitly a theory of transfinite cardinals, retain what we need for a foundation of standard mathematics, and perhaps thereby avoid the upsettingness of the contrast between the intended, nondenumerable interpretation of set theory (if such there be) and the unintended, denumerable interpretation? Do we really need to be able to distinguish denumerable and nondenumerable collections in order to do standard mathematics? It seems to me that, strangely enough, time has answered this objection. Whatever may have been the case in 1900, in 1970 some form or other of Cantor's theory of infinite sets and even perhaps of infinite cardinals has become a received part of mathematics. So even if N, is not mentioned in elementary calculus, still no foundation for present-day mathematics would be adequate unless it also founded some version of Cantor's theory of the infinite. (This reply does not require, as it might seem, that we must always slavishly accept whatever is currently received doctrine among mathematicians; rather, the point is that one who wants to found mathematics should found as m u c h of it as he can. T h e object of the game is to give a firm foundation to as much as possible of all the mathematics that there is.
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Historical conscience requires a remark to the effect that many people seem to have included Cantorian set theory in the classical mathematics for which the type theory of Principia was to provide the foundation. Type theory, which can be construed as a weak set theory, was treated as higher-order "logic," and higher-order logic was not recognized as a quite considerable strengthening of quantification theory. Quine explains this circumstance more fully on pages 257-258 of Set Theory and Its L ~ g i c . ~ ) Let us next consider the sort of objection to Skolem's argument exemplified by Quine's remarks at the end of the appendix to Methods of Logic.8 There Quine writes of the universe of all real numbers:
. . .we are told that the truths about real numbers can by a reinterpretation be carried over into truths about positive integers. This wnsequence has been viewed as paradoxical, in the light of Cantor's proof that the real numbers cannot be exhaustively correlated with integers. But the air of paradox may be dispelled by this reflection: Whatever disparities between real numbers and integers may be guaranteed in those original truths about real numbers, the guarantees are themselves revised in the reinterpretation. I n a word and in general, the force of the Lowenheim-Skolem theorem is that the narrowly logical structure of a theory-the structure reflected in quantification and truth functions, in abstraction from any special predicates-is insufficient to distinguish its objects from the positive integers (259/60).
These remarks are rather brief, but Quine's idea, applied to our case, would seem to be that, although the formalism of set theory can be given a denumerable interpretation, this is true only because the sole primitive predicate of that formalism, namely "E," has been given an interpretation quite divorced from its intended interpretation. As it were, if you change the input to the formalism, then its narrowly logical structure is insufficient to prevent the output from changing as well. This is where the difference between the two versions of the Lowenheim-Skolem theorem become important for us. If we are thinking of the version of the theorem which Skolem proved in 1922 and which Quine proves in his appendix using the Law of Infinite Conjunction (a corollary of Konig's Infinity Lemma), then Quine's comment is quite apt. This result says only that a standardly formalI Cambridge, Mass.: Harvard, 1963; reviewed in this issue of this 111-114. 8 New York: Holt, Rinehart & Winston, rev. ed., 1959.
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ized theory with a model has a countable model; but the countable model need bear no natural relation to the original model, and so the intended interpretation of the enshrined in the original model, will no doubt be lost in the new model. But if we assume the axiom of choice, then we may prove, as Skolem did in 1920, that a theory with a model M has a countable model that is an elementary submodel of M. T o explain this terminology, consider a set theory with "6" as its sole primitive predicate. A structure of the appropriate sort to be a model for this theory will consist of a set S and a two-place relation R on S. Call another structure ( S l , R l ) a substructure of (S,R) if and only if S1 C S and R, = R ? S1. Then ( S l , R l ) will be an elementary submodel of ( S , R ) if both are models of the theory, if (S,,Rl) is a substructure of ( S , R ) and if, for every open sentence 4 (x,, . . . , x,) of the theory and every n-tuple (a,, . . . , a,) over S,, (S, R, a,, . . . ,a,) satisfies 4 (xi,. . , x,) iff (S1, R1,a,, . . . , a,) does too. Having salved my technical conscience, I shall say that intuitively this means for our case that in the countable model of the set theory, the relation that interprets "E)' is the restriction of the interpretation of "a" in the original, intended model to the countable model; in this sense we have not changed the interpretation of "E)' at all, but only restricted it to a countable part of the original model.9 Accordingly, Quine's comment does not seem germane if we have this stronger version of the theorem in mind. (It might be asked how Skolem, a critic of set theory, can be allowed to use the axiom of choice, a characteristically set theoretic assumption. One answer might be that as a critic of set theory he may assume any part of it for purposes of reductio; but it is not at all clear whether Skolem thought that a successful defeat of set theory by his 1922 paper would require him to abandon the main result of his 1920 paper.lO)
.
9 Indeed, consider the somewhat unfashionable doctrine of implicit definition. This doctrine seems intuitively of a piece with Skolem's basic philosophical presupposition (see A and 3 below); indeed, when Skolem says on page 295 of his . sets are nothing but objects that are connected with one 1922 address that, another through certain relations expressed by the axioms," he seems to be opting for a version of the doctrine of implicit definition consonant with his basic presupposition. But then, since the elementary submodel and the original model satisfy exactly the same sentences of the set-theoretic language, it would seem that on the doctrine of implicit definition, the predicate "e" as confined to the set theoretic language has exactly the same meaning in both models. l o T o be sure, Skolem explicitly says in 1922 that "here, where we are concerned with an investigation into the foundations of set theory, it will be desirable to avoid the principle of choice" (''Some Remarks on Axiomatized Set Theory," p. 293). But this does not solve the basic dilemma: for the 1922 proof, as a critique, does not seem subject to Quine's reply; and the 1920 proof, though not subject to Quine's reply, requires choice, a characteristically set-theoretic assumption.
". .
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Let us now try to spell out Skolem's argument in a bit more detail. Here we might say: A. If it is an absolute mathematical truth that the Cantorian, settheoretic hierarchy of infinities exists (as it should be if an axiomatized version of set theory is to be the foundation of mathematics), then it ought to be possible to give a mathematically precise and unambiguous statement and justification of this truth. B. T o give a precise statement and justification of this putative absolute truth, it will be necessary (mere sufficiency will not suffice for Skolem's argument) to specify a formalized language in which this truth is proved. This formalized language will not be satisfactorily precise unless the formal system corresponding to it is countable and first-order. T o give an unambiguous statement and justification of this putative absolute truth, it is necessary that the aforementioned formalism be categorical. (Being derived in a categorical, countable, first-order formalism may be viewed as a proposed explication of being univocally and precisely stated and proved, as was demanded under A.) C. Points A and B and the claim that it is an absolute mathematical truth that Cantorian hierarchy of infinities exists (which should involve the more sedate claim that the formal set theory has an uncountable model) are jointly incompatible with the LijwenheimSkolem theorem. So, assuming the theorem, A, and B, it is not an absolute truth that the Cantorian hierarchy exists. (Of course, to deny that it exists absolutely may not yet be to affirm its existence relative to something or other.) There are at least three places at which this reconstruction seems to generate interesting philosophical dialectic. (1) Why should precision be explicated in terms of countable, first-order formalisms? That it should be explicated in terms of formalisms is not too distressing, but why must these formalisms be first-order and countable? In particular, anyone intimate with the Lijwenheim-Skolem theorem knows that the countability of the countable model results in good measure from the countability of the formalism. So why shouldn't we take the theorem as showing at most limitations on the powers of formalization by countable first-order formalisms rather than some sort of failure of absolute existence? Here it might be replied that if it is to be possible that people use a system of items, that system must be at most countable; that if a system of items is to be worthy of being called a language at all, it must at least be possible for people to use it; and that, conse-
quently, any formalized language worthy of the name should be at most countable. (A rationale for our first premise might be that the system should be eflectively enumerable.) It might also be replied that a formal system for set theory is introduced in order to specify such notions as uncountability in the first place and that, consequently, it would be a foolish attempt to pull oneself u p by one's bootstraps to think of using an uncountable collection to establish set theory.ll Whereas the first reply is not implausible, the second is; there does not seem to be anything worse about using phenomena of a sort described by a set theory in establishing that set theory than there is about using ink to write a description of the manufacture of ink. (2) Why should lack of ambiguity be explicated in terms of categoricity? This question has at least two parts. First, what candidates Aternative to categoricity are there for the office of explication of univocality? Second, what claim does categoricity have to that office on its own merits? Taking up the second question first, recall that what Quine called the narrowly logical structure of a categorical theory determines its interpretation up to isomorphism. Thus in a mathematically acceptable sense of "unique," categoricity means that the quantificational structure of a theory suffices to determine the interpretation of its predicates uniquely. This seems sufficient for univocality all right, but is it necessary? We might hold more than just the quantificational structure of the theory fixed; this is the thrust of Quine's remark in the appendix to Methods. It also seems to be the point of an illustration Myhill gave in 1953. He points out that in pure quantification theory, the following conditions on the cardinality of the field of a relation are expressible:
1. It is void. 2. For each n, it has at least n numbers. 3. I t is infinite. But we cannot impose any upper bounds or any nondenumerable lower bound. However, if we assume a predicate with the identity relation as its fixed interpretation, then we can impose finite upper llSkolem suggests such an objection on page 296 of "Some Remarks on Axiomatized Set Theory." 1 2 John R. Myhill, paper 2 of "Symposium: On the Ontological Significance of the Lowenheim-Skolem Theorem" with George D. W. Berry in Academic Freedom, Logic and Religion, APA, Eastern Division, Proceedings, vol. r ~ ,M orton White, ed. (Philadelphia: University of Pennsylvania Press, 1953), pp. 57-70.
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This dispute seems to dissolve in one's hands. Consider an analogous case (which foreshadows the answer to our second question). As is well known, any two dyadic relations that are reflexive and satisfy Leibniz's law are, as a matter of pure quantification theory, coextensive. Intuitively at least, this would seem to show that in some respectable sense of 'unique', the interpretation of "=" can be fixed uniquely by just a first-order, countable axiomatization. In contrast, "E" has yet, if ever, to be so axiomatized. Here we have a difference between identity and membership. But it is hard to see what philosophical significance, if any, should be attached to this difference; how could one characterize this difference perspicuously and in a philosophically informative fashion? (Hence, perhaps, the obscurity of Skolem's talk of relativity.) Categoricity is probably not the only possible explication of univocality. It is no doubt sufficient for some sort of univocality, but it is surely not necessary for any sort; and it is necessity that Skolem needs. A categorical axiomatic theory is complete, and a complete axiomatic theory is decidable; the converses of both these propositions fail. It is evident that both properties, completeness and decidability, have some prima facie claim to explicate univocality (as do others less well known). Both mean that all questions in which a notion figures and which are posed in a pre-set vocabulary can be definitely settled; this would seem to provide an impressively determinate grasp of that notion, and determinateness seems to be a key to univocality. Nevertheless, the three properties-categoricity, completeness, decidability--differ in extension; so not all of them can explicate univocality. I shall conclude only that greater exploration of this area seems indicated. (3) Lastly, one might ask after the rationale of A. We do not require the expression of any part of physics to pick out its subject matter uniquely; we are confident of forces without worrying about whether we have descriptions with them as their sole interpretations. What right, then, does Skolem have to raise doubts about the existence of uncountable collections simply because they cannot be uniquely specified by a formalism? In other words, a weak version of something like nominalism seems to be required for Skolem's argument to go through; unless a mathematical object can be specified uniquely by linguistic means, its existence can at best be only relative. This of course is not grounds for a straightforward accusation of circularity, since Skolem is not out to prove nominalism. But it does seem as though Skolem requires an anti-realist bias in order to establish a conclusion congenial only to an anti-realist; as such, his argument seems unlikely to convert anyone.
Hilary Putnam once stated in conversation a case which illustrates this point. Suppose we had a countable, first-order formalization of physics. Presumably it would have a nondenumerable model LM, since M would include all the points in space. If we then formed the Skolem hull H of M (making sure that H contains all physical objects and so forth), then H could contain at most countably many points of space. But then space as discussed in our theory and interpreted in H would be recognizable from outside H as highly discontinuous. Yet surely this is highly implausible; and if one is not willing to accept "relativity of continuity of space," then, by parity of reasoning, one ought not to accept relativity of the existence of nondenumerable infinite sets. Of course one might be willing just to accept a relativity of continuity of space and the cardinality of its points. Nor need this move be quite so arbitrary as it might seem. I have been told that there is a not quite respectable movement afoot in physics to quantize space and time, and, once quantized, space and time would be regarded as composed of countably many units. Unfortunately my ignorance prevents me from pursuing this line of thought profitably, so I must regretfully drop it. As noted above, the argument under A-C is at best a refutation of the absolute existence of the higher transfinite cardinals. But this need not automatically make it a proof of their relative existence. For instance, are we really clear yet about what is supposed to be relative to what? In this connection, it may perhaps be noted that an eminent logician has accused Skolem of confusing relativity to an axiom system with relativity to a model of it. Some idea of his point may be got from the following considerations. Suppose we show of Zermelo-Frankel set theory (ZF) that
I-eF
z
( 3 f)(f:P(w) -+ w & f is bijective)
Here we prove that P(w) is uncountable "relative" to the axioms of ZF. Of course, any theorem of ZF is "relative" to its axioms in this sense; so this sort of "relativity" reflects no special ill on uncountable infinities. Outside ZF we now assume a domain B such that B is a model of ZF (B need not be a set, that is, an object whose existence is demonstrable in ZF; the sentence displayed above will not be provable in ZF if it is consistent). Since B I= ZF, there is an object
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in B answering to the terms
in ZF (here I have made certain innocuous syntactical simplifications). Applying the Lowenheim-Skolem theorem, we now infer a countable elementary submodel B' of B (and we may assume B' transitive). Then there is an object
in B' which also answers to
in ZF; but now we know that {P(w)},, is countable. Of course, there can be no one-to-one map f of {P(w)),, onto w in B', but still there is such a map. Here we might imagine Skolem glossing these considerations by saying that if there is an absolute notion of uncountability, then there ought to be n o way of giving an argument for the countability of P(w). And the proper response to Skolem's gloss would be: What do you mean by "P(w)"? If you mean {P(w)},,, then this set is indeed countable. But you have no right to assume that {P(W))B' = {P(w))B We have said nothing to suggest that {P(w)}, is countable. I n B' there is no counting of {P(w)),,; but from outside B', we can count it; and no one has offered a vantage point from which to count {P(w)1B. Let us trade in this metaphor of vantage points for a more literal statement; the philosophical value of the latter may be easier to assess. Read as intended, ZF asserts the existence of uncountable infinities of objects. So to show that ZF has a countable model, we have to prove the existence of a model for ZF and of a counting function not in that model. Of course, neither result can be proved in ZF if ZF is consistent. And to show in the manner we have considered that ZF is not categorical, we would need not only that ZF has a countable model but also that it has an uncountable model; this too is not demonstrable in ZF if ZF is consistent. Hence, we can show that ZF is not categorical only in a theory stronger than ZF. Of course, there is (to date at least) no theory in which we can show all models of ZF to be countable. Putting the pieces together, some but not all models of ZF can be proved countable in some but not all theories. And thereby may hang a tale of relativity.
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For whether a model can be proved countable in a theory could be said to be relative to the model and to the theory. But now the question is: What is the significance of this sort of relativity for Skolem's philosophical claims? T h e answer, so far as I can see, seems to be: Precious little, except perhaps for what we said before to the effect that if Skolem needs assumptions stronger than ZF to infer results intended to cast doubts on ZF, then the force of his criticism is obscure. A great deal more could be said on these topics. But it seems like wasted effort. At the inception of the investigation, there is always an intimation of ontological ecstasies, and prompted by this promise, one plunges eagerly into the inquiry. But disappointment always ensues, and so eventually one is left with only a slightly irritated, dulled boredom. W. D. HART
University of Michigan
BOOK REVIEWS From Frege to Godel: A Source Book i n Mathematical Logic, 18791931. JEAN VAN HEIJENOORT. Cambridge, Mass.: Harvard University Press, 1967. x, 655 p. $18.50. This is a very instructive book. It presents a pregnant selection of papers, lectures, and letters on mathematical logic and foundations of mathematics from the time of 1879-1931 (beginning with Frege's "Regriffschrift"), each of them introduced by an expository article. All the texts are in English. Thus most of them had to be translated. T h e translations are performed with great care. Therein the editor had the cooperation of Stefan Bauer-Mengelberg (who translated most of the German papers, which constitute the majority), as of Dagfinn F@llesdaland Beverly Woodward. Especially valuable are the comments, given partly in the introductory articles and partly in additional footnotes (which are distinguished by brackets from the original footnotes). T h e comments afford helpful explanations, discussions, and plenty of historical information, whose concern goes far beyond the limit of 1931, nearly to the present. At the end of the book the papers to which reference is made in the comments are alphabetically registered. Almost all relevant authors are mentioned here. One might miss Leon Chwistek and Paul Hertz. Most of the introductory articles are by Jean van Heijenoort himself. W. V. Quine contributed the introduction to the paper by
