The Class of Recursive Functions by J. R. ShoenfieldReview by: Paul AxtThe Journal of Symbolic Logic, Vol. 24, No. 3 (Sep., 1959), pp. 238-239Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2963819 .

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238 REVIEWS

these wider senses often stress provability alone, instead of proof also, and practically ignore the all-important though vague requirement that the relationship between 2t and A,, be simple. It appears therefore that the flexible spirit of (a) cannot be caught by a general mathematical concept, even if the attempt to accomodate (b) also is abandoned. Mathematically precise analogues will have to be tailored to particular situations.

Attention to (a) may also have detracted attention from the following aspect of (b): If possible, association of Al, A2, ... with 9t should be significant not only for theorems 9t expressing assertions but also for other sentences 2t expressing conjectures or hypotheses (in science) and also for formulas 9t with free variables expressing conditions or concepts (in science or also in number theory). Whether or not (b) is attempted for this larger class of formulas, this significance is largely measured by the extent to which logical (or number-theoretical) relationships between 2t and 21' are reflected by relationships between the associated A 1 , A 2, . . . and A '1 , A '2 . In this respect Herbrand's Theorem gives a rather satisfactory association for the predicate calculus. The reason is that the "infinite disjunction" of formulas (xi)... (xi,)A n (xl . . ., n) associated with 9t is not only a sufficient condition for W expressed in terms of formulas (Xi)... (xk)B, An and B quantifier-free, but a weakest condition of this kind. Not only is each (Xi).. . (Xn)A n 2 W valid, but also when (xi) ... (xk)B 2 W is valid, then (xl) ... (xk)B j (xl)... (x'n)A n is valid for some n. This fact is not implied by the author's conditions on disjunctive interpreta- tions but implies them, when combined with the simplicity of the relationship between 9t and An and the fact that if 9t is provable, then e.g., (x)(R(x) D R(x)) D 9t and hence some (xi) . .. (x'n)A n is provable. Dually, conjunctive interpretations serving (b) should, if possible, yield simply related strongest necessary conditions for 2t. This would often be useful in science. (For (a) it seems best to make the sufficient condition for 9t as strong as is compatible with the requirement that provability of 2t implies that of some An and the requirement of simplicity.)

As regards number theory, Kreisel's extracting finitist or intuitionistic "sense" from classical theorems and proofs is an achievement. It should be noted, however, that his applications of Ackermann's consistency proof (just as his algebraic applica- tions of Herbrand's theorem) are at least partly independent of his no-counter- example interpretation in terms of functionals. As regards number-theoretic concepts, the above sharpening of the concept of co-consistency seems an isolated instance, and it is not clear to what extent (b) has been or can be accomplished.

WILLIAM CRAIG

P. C. ROSENBLOOM. Konstruktive Aquivalente fur Sdtze aus der klassischen Analysis. Actes du Deuxieme Congres International de l' Union Internationale de Philosophie des Sciences, Zurich 1954, II Physique, mathematiques, pp. 135- 137.

Some theorems of classical analysis cannot be accepted from a constructive point of view. However sometimes it is possible to find a constructive version of a non- constructive theorem of classical analysis such that the equivalence between it and its constructive version classically (mostly by the axiom of choice) holds.

The author sketches two examples for this program, both concerning compactness theorems for function spaces. To get a constructive version, using recursive functions, the topology of these spaces is based (as usual) on the concept of a metric, and compact- ness is defined with the aid of convergence. Fuller details and proofs will be published in a forthcoming joint paper with P. Erdds. GERT H. MWLLER

J. R. SHOENFIELD. The class of recursive functions. Proceedings of the Amer- ican Mathematical Society, vol. 9 (1958), pp. 690-692.

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REVIEWS 239

The predicate "a is recursive" is expressible in the form (Ex) (y) (Ez)R(a, x, y, Z) with R recursive. With the aid of the Baire category theorem, the author shows it is not expressible in the form (x) (Ey) (z)R(ce, x, y, z) with R recursive. PAUL AXT

WILLIAM W. BOONE. An analysis of Turing's "The word problem in semi-groups with cancellation." Annals of mathematics, ser. 2 vol. 67 (1958), pp. 195-202.

The proofs in Turing's XVII 74 are frequently deficient and in error. The reader who has grasped the general outline can, laboriously but without difficulty, correct the proofs for himself. Alternatively he may consult the paper under review, where he will find the necessary alterations and amplifications clearly set out in the form of a line by line commentary. Perhaps the least obvious of Boone's corrections is to the proof of lemma 11: here Turing overlooked the fact that the length of a proof P may be less than the sum of the lengths of its component parts; for a step which is common to several of the parts need only be proved once in P.

One further correction to those listed by Boone is needed; namely, lemma 12 must be proved for all normal words. Boone's version of Turing's proof must therefore be replaced by a direct appeal to the forms of the fundamental relations and to the restrictions to which machine tables are subject. We then have a complete and correct proof of Turing's result. But, although Boone makes explicit two vital and obvious facts which Turing uses without mention (lemmas 16.1 and 18.1), the proof is still not transparent; and one wishes that Boone had taken this opportunity to simplify it.

In fact, Turing's rather involved definition of the 'length' of a proof is not necessary; only lemmas 11 and 19 depend essentially on its use, and there it can be dispensed with if the following modifications are adopted:.

LEMMA 11'. If (AG1G2F, BG3G4H) can be proved without cancellation ("w.c."), where G1G2 and G3G4 are barriers and either AG1 and BG3, or G2F and G4H contain no barriers, then (AG1, BG3) and (G2F, G4H) can also be proved w.c.

LEMMA 19.1. For any words A and B, if (GA, GB) or (AG, BG) can be proved w.c., then (A, B) can be proved w.c.

The proofs of these lemmas follow the outlines of Turing's proofs, but are con- siderably simpler. From 19.1, Turing's lemmas 5 and 6, and Boone's lemma 18.1 (which states that a normal chain yields a normal w.c. proof), it is easy to prove:

LEMMA 19.2. If (A, B) is a relation and A is normal, then there is a normal w.c. proof of (A, B).

And this is just what is required for the proof of Turing's final lemma. These alterations make sections D, F, and J of Boone's paper unnecessary. R. 0. GANDY

JEAN PORTE. Systemes de Post, algorithmes de Markov. Cybernetica, vol. 1 no. 2 (1958), offprint pp. 1-36.

For the most part, this article can be regarded as a sketch of well known salient facts about recursive functions, Turing machines, Post systems, and Markov algorithms. There is an underlying purpose, commendably achieved, of showing precisely in what sense these four notions are equivalent.

The exposition has some novel features, the most important of which is a generali- zation of Markov's normal algorithms (XXII 77). A normal algorithm can be regarded as derived from a "bilateral", axiom-free Post system with rules of the form XAY produces XBY, by designating certain rules as "conclusive" and prescribing a choice among possible alternative applications of any rule. Porte's algorithms are analogously derived from all axiom-free Post systems with rules of one antecedent. Porte studies, for example, the interesting algorithms derived from "unilateral" Post systems (with rules of the form AX produces XB); algorithms of this type, in which a part of a word may be transposed as well as transformed, were abandoned at an early stage by Markov.

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