Mathematics without Foundations. by Hilary PutnamReview by: Georg KreiselThe Journal of Symbolic Logic, Vol. 37, No. 2 (Jun., 1972), pp. 402-404Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272992 .

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402 REVIEWS

closed under substructures and unions of chains. A class K is U-closed if for every set A and

every element a of LA the set {inf{a, 21}: 21 e K} is unclosed. If a class K is U-closed, then the

class HK (i.e., the closure of K under homomorphisms) is U-closed and the class of all subdirect

products of elements of K is U-closed. In ?6 characterizations of universal classes of relational

structures in the terms of the closure operations are given. A class of relational structures is

universal (i.e., it can be axiomatized by a set of universal sentences) if and only if it is U-closed.

If the similarity type of class K is countable, then K is universal if and only if it is u-closed.

Using these characterizations and the results of ?5 the author obtains a new proof of a result of

C. C. Chang which says that if K is universal, then HK is universal. Moreover, if K is universal,

then the class of all subdirect products of elements of K is universal. In ?7 the author considers

classes which are universal and closed under direct products. The class K is unclosed if and only

if it is universal and closed under direct products. If a class K is closed under direct products

then it is universal if and only if it is sup-closed and closed under substructures. In ?8 the author

applies the previous results to classes of abstract algebras. LESZEK PACHOLSKI

S. R. KOGALOVSKIJ. Univirsal'nyi klassy modilkj (Universal classes of models). Doklady

Akadimii Nauk SSSR, vol. 124 (1959), Pp. 260-263.

TADASHI OHKUMA. Ultrapowers in categories. The Yokohama mathematical journal, vol.

14 nos. 1-2 (1966), pp. 17-37. The author defines the notions of reduced product, ultraproduct, and ultrapower in terms of

category theory. Let A be a set and r an ultrafilter on A. A well-known result of model theory

shows that there is a canonical map which embeds the structure 21 into its ultrapower 2A/1r. The author remarks that in the category of topological spaces the canonical map d from A to

AAIr is not necessarily an injection. This leads to the study of the relationship between the

injectiveness of d and the property of A of being finitary. The following result is obtained: Let

W be a concrete, set-theoretical perfect category, A an object of W. If for every set A and every

ultrafilter r of A the canonical map d from A to AA/r is an injection, then A is finitary.

The paper is self-contained and hence even a reader not familiar with category theory will

find it understandable. It would be interesting to investigate the relationship between the cate-

gorical and the classical, i.e., model-theoretic, reduced product even further (especially as the

reviewer does not agree with a remark of the author that these two notions coincide for alge-

braic structures, consider for example the category of Abelian torsion groups). A change in the

definition of a concrete category seems indicated, i.e., the condition ","(A) = (B) implies

A = B" makes it impossible to take !L(A) equal to the set of elements of the group A. ANNE PRELLER

FREDERIC B. FITcH. Quasi-constructive foundations for mathematics. Constructivity in

mathematics, Proceedings of the cofloquium held at Amsterdam, 1957, edited by A. Heyting,

Studies in logic and the foundations of mathematics, North-Holland Publishing Company,

Amsterdam 1959, pp. 26-36. The author presents three formal systems of combinatory logic, of increasing strength, with

operators for equality, negation, disjunction, and quantification (but apparently lacking the can-

cellation combinator K). The systems are "quasi-constructive" in the sense that some of the

rules have infinitely many premisses. In the strongest system, the combinators give the effect of a

weak set theory, so that various mathematical notions can be represented in the system, as in

the author's earlier papers XIV 68 and XV 137. BRUCE LERCHER

HILAAY PUTNAM. Mathematics without foundations. The journal of philosophy, vol. 64

(1967), pp. 5-22. The paper starts with such unpromising declarations as (p. 5) ". . . I do not believe that

mathematics either has or needs 'foundations.' . . . the various systems of mathematical

philosophy, without exception, need not be taken seriously." Even those interested in founda-

tions still readily agree that (repositioning the author's inverted commas) the bulk of mathe-

matics does not 'need' foundations, for example for reliability or intelligibility. And the various

systems need not, in fact must not, be taken 'seriously' in the sense of 'to be valid in detail'

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REVIEWS 403

since their defects are well known. However, as will be seen below, the author would have done well to take seriously classical work on two familiar foundational schemes, namely Hilbert's programme and intuitionism. In any case, as the paper proceeds, he not only sketches a foun- dational scheme, but makes such impeccable observations as (p. 16): ". . . the mere fact that we may never know whether the continuum hypothesis is true or false is by itself just no reason to think that it doesn't have a truth value!" or (p. 22): "Such a weak set theory may well give us all the sets we need for physics.... But the fact that we do have an intuitive conviction that standard models of Zermelo [stronger] set theory ... are mathematically possible structures is a perfectly good reason for asking ... whether the continuum hypothesis necessarily holds in such structures."

Under the title A sketch of my view, the author stresses, on page 7, line 11 from below, that (many) mathematical propositions have equivalents formulated in terms of quite different primitive notions, for example in set-theoretic and in modal terms. In other words, those propositions can be (re)interpreted within different foundational schemes. In support of his view he refers on pages 9-11 to two properties of first-order logic: (i) validity (in all realiza- tions of the relevant language) is equivalent to necessity, in fact to derivability by means of the (necessary) principles formalized in predicate logic, and (ii) number-theoretic truth of any S0 sentence E, such as the negation of Fermat's conjecture mentioned on pages 9-10, is equivalent to validity of an (effectively) associated first-order sentence E'; cf. 4183, page 194. He neither mentions that (ii) fails to extend to HO1 sentences, and, more generally, to validity in models of Peano's (own second-order) axioms; nor, more importantly, that such results as (i) and (ii) are the basis for familiar formalist foundations. In fact, Hilbert's programme, while much more precise and explicit about adequacy conditions to be satisfied by a modal reduction, is quite similar to the author's general proposal on page 21, lines 3 and 2 from below. Hilbert wanted to replace assumptions concerning abstract objects by (finitist) principles concerning our (formalized) reasoning about them; the author finds (p. 19, line 11 from below) that the general modal "notions of mathematical possibility and necessity are clear" and he does not restrict himself to (necessity established by) finitist methods. He believes (p. 11, lines 23-24) the "one special example [(ii) above] . , to represent, in some sense, the general situation," as no doubt Hilbert did at the turn of the century (that is, 'represent' with respect to his programme). The reader is asked (p. 20, line 12 from below) to "accept it on faith" that the author's very sketchy modal equivalent for the notion of being a standard model for Zermelo set theory contains no " non-nominalistic" notion except O, though the formulation looks very much like simply putting the symbol 0 in front of an ordinary set-theoretic expression, which we under- stand only because we understand set-theoretic notions. To bolster up his faith, at least this reader would need a hint about the kind of properties of Q which imply the author's modal equivalents (and, presumably, the consistency) of Zermelo's axioms. Without denying the clarity of modal notions, it seems difficult to make strong assertions about them; specifically, stronger assertions than those formulated in the intuitionistic systems which are used in current work on extensions of Hilbert's programme. (Reviewer's note: On page 21, line 16 from below, the author asserts in passing that 6nonconcrete' models of Zermelo's axioms can be extended. This is not plausible if the domain of a non-concrete model is a Vielheit and not an Einheit, in the sense of Cantor's famous phrase, which is quite clear enough for the particular use just made of it.)

On page 20, lines 5-4 from below, the author objects on pragmatic grounds to foundation- alism, by which he means the selection of a privileged or fundamental system of concepts. This overlooks the fact of experience that at least occasionally there exists such an order among systems, for example in the sense that the concepts of one can be defined in terms of the other (more fundamental) system, but not conversely. Proper respect for this fact has been (p. 20, line 5 from below) "of real heuristic value" to scientists and logicians, however unreasonable this may appear to some philosophers of science. In connection with usefulness it should be remarked that the paper does not contain the remotest hint how the modal concepts, of neces- sity and possibilities in standard models of set theory, could be used to find new axioms or at least improve existing arguments for axioms which are at all problematic, for example the replacement axiom (in some model of rank > co).

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404 REVIEWS

Though somewhat outside the scope of this JOURNAL, the author's blithe references on page 8 to various "equivalent descriptions" in quantum theory-which he connects with the "equiva- lent formulations" mentioned above in the statement of his own view-should not pass unchallenged. Dirac (Lectures on quantum field theory, New York 1966, pp. 3-5) conjectures that one familiar "equivalence," between Schrbdinger's and Heisenberg's pictures, does not extend from elementary quantum mechanics to quantum field theory. If he is right, this could be compared to the failure of extending to III sentences the equivalence between number- theoretic truth of ZD sentences and validity in first-order logic. GEORG KREISEL

MARIKO YASUGI. Intuitionistic analysis and Godel's interpretation. Journal of the Mathe- matical Society of Japan, vol. 15 (1963), PP. 101-112.

The present paper studies in some detail the Dialectica interpretation, introduced by Godel (XXV 351), for an extension of intuitionistic arithmetic to all finite types.

Notation in this review: a. r denote types; (a)r denotes the type of mappings of objects of type a to objects of type ar. xO, yO, zU denote variables for objects of type a, ,d p, a denote sequences of variables, and t, I denote sequences of terms. Type superscripts will often be omit- ted; types will then be assumed to be appropriate. Conventions for application are as in Spector's XXXII 128: xl-... *x abbreviates (a .((xlx2)x3)... *), and when T = xl, , * *, x", then yp = Xip, ... , Xn.

Let T denote Gbdel's quantifier-free theory of functionals of finite type, and let HAW (intro- duced by Kreisel in this JouRNAL, vol. 24 (1959), pp. 284-285) be obtained by adding quantifica- tion over all finite types. Let AC denote the choice-schema VxO 3yt A(x, y) -3 3zB() VxO A(x,zx) for all finite types a, 'r; let IPO denote the schema (Vy A -) 32 B) -> 3p(V7 A -> B), A quantifier- free, T not free in B, and p not free in A; let M' denote the schema -.Vy A -* 3By-A, for A quantifier-free (a version of Markov's principle). Let H abbreviate HA + IPO + M' + AC, The principal result of the author may be paraphrased as follows. THEOREM. Let AG _=

3,x Vp A(o, V, 3) denote the transformation of A into 38-form as described by Godel in XXV 351 (or Spector in XXXII 128). Then H F AG *. A, and H F A implies T F A(ta, I, 3) for a suitable sequence of closed terms of T. Hence H characterizes exactly the Dialectica-interpretable formulas.

It should be noted that M' and IPO may be replaced by the slightly more general schemata M and 1PO, where the condition "A quantifier-free" is replaced by adding to the premiss "IV(A v -A) &". This is seen by noting that V7(A v -A) implies in H: 3y(A 4-.yy = 0). Then, relative to H, M and IPO are trivially obtained from M' and IPO . Kreisel (XXXVI 169(4)) already noted the role of M' and LPO in the proof of A *_ AG (2.11), and has given a Dialectica-interpretation for M (footnote 1 to 3.52).

There is an error in the author's detailed description of HA@; equations t = s between terms of any type are regarded as prime formulas. But then Gddel's interpretation cannot be carried through for HAO without further axioms. There are two possibilities for correction: either one uses an "extensional" version of the theory where only equality between terms of type 0 is treated as a primitive; then we must read "t = s" for terms t and s of type a in T as an abbre- viation of t7 = sc, where X is such that tT and sc are of type 0. The author's axiom (5) then also has to be weakened to a rule of extensionality: if P -t = s, then P -tt] = t'[s], where P is quantifier-free. This is Spector's solution (XXXII 128). The other solution, closer to Godel's intentions, is to use an "intensional" version of HA0, where equality between objects of type a

is a primitive notion (which may be conceived as " definitional," decidable quality), and to add a functional E, for each type a such that Eaxy = 04-. x = y, E~xy = 0 v E~xy = 1. An example, due to W. A. Howard, shows that the theorem does not hold for the system as de- scribed by the author: The interpretation of Vy?) 0-'Vu 0-(u = 0*-. y = Ax 0 .), when y = An 0 . is regarded as a prime formula, requires a functional z of type ((0)0)0 such that (*): Vy(?) '_ -y = 0 <-. y = Ax- . 0). It can be shown that all functionals of type ((0)0)0 in Tare continuous (= computable from initial segments), and no continuous z satisfies (*). The over- sight is in the treatment of right-& in the proof of the main result; the obvious interpretation of A --. A & A requires prime formulas to be decidable. A. S. TROELSTRA

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