Konstruktiver Aufbau Eines Abschnitts der Zweiten Cantorschen Zahlenklasse by WilhelmAckermannReview by: Alonzo ChurchThe Journal of Symbolic Logic, Vol. 17, No. 2 (Jun., 1952), pp. 152-153Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266282 .

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152 REVIEWS

the form P Q in which P and Q are words in the letters a, b and which holds in SG,H holds already in S3. It follows that if G *-+ H does not hold in S3, then SG,gv does not have the property $3.

This shows that the existence of an algorithm which would allow one to decide whether an arbitrarily given system does or does not possess the property v would imply the positive solution of the word problem for S3 and hence for S1 .

If $3 is an hereditary property, i.e., if every subsystem of a system with the property v

always has the property A3, then condition (1) can be simplified: it is sufficient to assume that there exists a system not having the property A. ANDRZEJ MOSTOWSKI

A. MARKOV. Ob odnoj nrrazrdimnoj problgmg, kasats'4jsd metric (On an unsolvable prob- lem concerning matrices). Ibid., vol. 78 (1951), pp. 1089-1092.

By matrices we mean in the following exclusively matrices with positive integral ele- ments. A matrix X is said to be expressible by means of matrices Y1 , Y2, * , Y. if there exist integers X, i ,i2 , * ix such that 1_ i r for j 1, 2, ** ,Xand

x = II Ye, v-1

The author considers the two following decision problems, which he calls respectively the general and the particular problem of expressibility of a matrix:

1. Does there exist for a given integer n an algorithm permitting one to decide for ar- bitrarily given matrices U, U1 , U2 , * , Uq of rank n whether or not U is expressible by means of U1 , U2 , *-* * , Uq ?

2. Does there exist for a given integer n and for q given matrices U1 , U2, *--, UQ of rank n an algorithm permitting one to decide for an arbitrarily given matrix U of rank n whether or not U is expressible by means of U1 , U2 , - - - , Uq ?

In the paper under review the author constructs one hundred two matrices U1 , U2, * U102 of rank 6, for which problem 2 is unsolvable. This result is obtained by an ingenious reduction of problem 2 (for suitably constructed matrices U1 , U2 , - - - , Uq) to the following problem, which was shown unsolvable by the author in XIII 53(2): Given matrices C, Z1, Z2, *-, ZP, Z1 , , Z, and integers il , *-- , ix satisfying the conditions 1 S is S p for j = 1, 2, , X, does there exist a matrix Q such that

A A C11 z, H= Z, *Q?

1-1 P-i

Because of many technicalities, no detailed explanation of the author's proof is possible in a short review. ANDRZEJ MOSTOWSKI

WILHELM ACKERMANN. Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse. Mathematische Zeitschrift, vol. 53 (1950-1), pp. 403-413.

A system of notation is set up for a certain initial segment A of the second number class, which provides a unique notation for each ordinal in A, and which is constructive in the sense that there is given an effective decision procedure (in terms of the notation) for each of the following: (1) to determine of two ordinals in A which is the lesser; (2) to determine of an ordinal in A whether it is of the first or second kind; (3) to determine the predecessor of an ordinal of the first kind in A; (4) to determine a fundamental sequence for an ordinal of the second kind in A. (Compare II 87(1), III 168, IV 93(2).) It is moreover demonstrated, in a way which is in a certain sense constructive, that transfinite induction is valid up to any ordinal a of A.

The system of notation is based on the usual notations for the ordinal number 1 and for addition of ordinal numbers, together with a notation "( , , )" for a certain ternary function (a, ,9, -y) of ordinals a, ,9, Sy of A. The author remarks that the notation can readily be extended farther into the second number class, without loss of constructiveness, by em- ploying a quaternary function (a, ,9, -y, 5) analogous to (a, ,9, -y); still farther by employing an analogous quinary function; and so on.

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REVIEWS 153

The purpose is to devise a system of notation which extends far into the second number class, assigning a unique notation to each ordinal, while remaining comparatively simple in character, and retaining the properties of constructiveness listed above.

This is done in a different way by Veblen in a paper in the Transactions of the American Mathematical Society, vol. 9 (1908) (see Example 6, pp. 290-292). Veblen's method differs from Ackermann's in being based on continuous increasing functions of ordinal numbers, and their properties, whereas Ackermann's function (a,,g, 'y) is discontinuous in all three arguments. Nevertheless there is evidently a close relationship between the two methods, and a detailed comparison would be of interest. ALONZO CHURCH

MOH SHAW-KWEI. The deduction theorems and two new logical systems. Methodos, vol. 2 no. 5 (1950), pp. 56-75.

This paper treats of a variety of topics centering about several versions of the deduction theorem, each holding for systems whose implication relation satisfies some appropriate set of conditions. A consideration of these conditions leads the author to suggest several im- plication-systems, and he studies their relation to well-known systems. Only a rough indi- cation of the (somewhat uneven) contents of this paper can be given.

A property of rules of inference which the author terms "normality" plays a pivotal r6le in the discussion. A rule of inference of a system states that under certain syntactical con- ditions on the Ai and B, B may be inferred from Al, A2, X * -, An ; such a rule is weak normal (in S) if, under the same conditions on the Ai and B, and with a certain syntactic condition on the variables of C, rC D B1 may be inferred in S from rC D A1l , rC D A21, * *, rC D

A,-. A system is weak normal if all of its rules of inference are. (If S is weak normal, it follows by induction in the meta-language that it is also strong normal in the author's sense. Hence one can speak simply of normal systems.) One of the author's versions of the deduction theorem is (essentially) this: If a system S (i) is normal, (ii) has the rule of modus ponens, and (iii) has as assertion schema rFA D . A2 D D - - - D . An D AiI, where i = 1, 2, . .. , n, and n < an; then S will have a deduction theorem stating that if there is a proof of B from the hypotheses B1, B2, X * , B,,m (with suitable conditions regarding variables), then rB, D . B2 D. .- - * D. Bm D B1 is asserted in S. And this result also obtains when only those rules required for the proof of B from the hypotheses Bi satisfy a normality condition.

The author proceeds to study several systems (U-systems, V-systems) in which some rules of natural deduction hold, i.e., the rules of Gentzen's system NK, as far as they do not involve quantifiers, and except for the "paradoxical" rule: From A to infer rB D A-. Further, these systems satisfy conditions requisite for the applicability of some of his formu- lations of a deduction theorem. Here, his question (p. 68) as to the independence of the rules 19 and 20 of his system U3 is readily settled-rule 20 is a consequence of the remaining rules. Also, the claim that the U-systems satisfy the eight-valued truth-tables given on page 71 is false in the case of U2 and U3. In these systems one has the theorem schema, rFA D B D . P-B D CAR1 and this is assigned a non-designated truth-value by these truth-tables for the assignment of 1 to A and 6 to B.

By way of illustrating system-types in which various versions of the deduction theorem hold, the author offers a system of strict implication S1.2, claimed to be independent of S2 and weaker than S3, obtained by adding to S1 the assertion 'p q D . Op Oq' (here strict connectives are intended). Further he studies the systems obtained by taking the implica- tion relation of his U and V-systems as strict implication, and defining r0Al as r,(A D -A)1. Also, a system of strict implication claimed to be intermediate between S1 and S2 is proposed. This is obtained from the axioms B6, B7, p D p, p D q D . pr D q, p D qr D . p D r, and (p D q)(p D r) D .p D qr; and "the definitions and rules as in SiX" (p. 75). It is to be assumed that the phrase quoted assures the inclusion of B4 and B5.

Finally, the author studies the rule of inference, "G6del's rule": From F A to infer F rOAl. (His wording of this rule is misleading; his formulations fail throughout to dis- tinguish between rules of the forms: From A to infer B, From F A to infer P B.) It is re- marked that "any strict system in which we could omit 'T' in all the T-principles must con- tain Gddel's rule" (p. 74). Hence there are T-principles even in S2: the author cites 'Tq D.

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