On Hierarchies and Systems of Notations by Hilary PutnamReview by: Wayne RichterThe Journal of Symbolic Logic, Vol. 31, No. 1 (Mar., 1966), pp. 136-137Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2270672 .

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136 REVIEWS

are notations for the same ordinal, then Ha and Hy- (the predicates in the ordinary jump hierarchy indexed by x and y, respectively) are Turing equivalent. Thus a unique degree of unsolvability is associated with each ordinal for which there is a notation in C. The author observes that the same proof of uniqueness will work for many extensions of C. He asks whether there is a "natural" definition of a class of systems of notations, each possessing the uniqueness property, and such that for every ordinal a in the classical second number class there is some member of the class which contains a notation for a.

The uniqueness result for the system C is obtained by proving a stronger theorem. A partial recursive function h is defined by the recursion theorem and a proof by induction (to be supplied by the reader) shows that if x, y E C and 1x1 IJyj, then Hx is recursive in Hy with G6del number h(x, y). The definition of h is quite complex and involves some seventy special functions and G6del numbers. The author has supplied some helpful remarks which make it possible for the less ambitious reader to follow the main lines of the argument without struggling through the details.

WAYNE RICHTER

HILARY PUTNAM. On hierarchies and systems of notations. Proceedings of the American Mathematical Society, vol. 15 (1964), pp. 44-50.

The author provides simpler proofs in a more general setting of the principal results of Kreider and Rogers (reviewed above). A schema of "inductively defined" systems D of notations for ordinal numbers is introduced as follows. Let Na be the set of notations for the ordinal a. (t) If a = 0, then Na = {1}. (2) If N# has already been defined for all fi < ?, then Nx+l = {2X I xe Na}. (3) If a is a limit number and N0 has already been defined for all fi < a, then Na = q{9(x _?a y)], where q' is an arithmetic operation and x ?a y is an abbreviation of: (Bfl) (3y) [x E N# & y E NY & fi ? y < xc]. It is required that the sets Na be disjoint and that the smallest ordinal for which there is no notation be a limit ordinal. The author remarks that all so far proposed inductively defined systems are of the form D.

Let CD= UN,. A short and elegant argument is used to show that CD E LA41

Thus D systems do not provide an example of a complete two-function-quantifier system sought by Kreider and Rogers. Since the systems C and C of Kreider and Rogers are of the form D, we have C, C E Al . The author's proof for D systems is much simpler than the original proofs of Kreider and Rogers for the particular systems C and C.

If certain minor additional restrictions are placed on q' it is possible to extend the hyperjump hierarchy through D systems. The author shows that all sets in this hier- archy (for any D system) belong to AI.

The same methods can be used to show that even if the notion of D system is generalized to permit q' E Al (instead of being merely arithmetic), then still CD E Al? And if, for these generalized systems, a more powerful jump operation j e Al is used, the corresponding hierarchy still consists only of Al sets. These results demonstrate the very strong closure properties of the class Al

It is of interest to obtain more precise information about the forms of the sets CD of notations of D systems (with, say, arithmetic Ap). In correspondence the author has shown that there is a fixed Al set in which all such sets CD are recursive. G. Kreisel (Mathematical reviews, vol. 28 (1964), p. 224) in his review of this article makes the much stronger claim that all the sets CD are recursive in the well-ordering functional of type 2, but this appears to be incorrect. For by modifying the definition of the system C, it is possible to obtain a D system whose set of notations is not recursive in this functional. And by extending this modified system to include notations for "constructively inaccessible" ordinals it is possible to obtain D systems whose sets of

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REVIEWS 137

notations are not recursive in functionals obtained by "diagonalization" of the well- ordering functional. (By the "well-ordering functional" we mean the functional W such that W(oc) is 0 or 1, depending on whether or not c is a well-ordering.) It thus appears that the sets CD are an extensive (but proper) subclass of A4.

WAYNE RICHTER

C. SPECTOR. Hyperarithmetical quantifiers. Fundamenta mathematicae, vol. 48 no. 3 (1960), pp. 313-320.

SHIH-CHAO Liu. Recursive linear orderings and hyperarithmetical functions. Notre Dame journal of formal logic, vol. 3 (1962), pp. 129-132.

The main theorem of both papers is the same, namely that for any recursive pre- dicate R(o, a, x) there exists a recursive predicate S(o, a, x) such that (3cc)HA(X)S(cc, a, x) - (a) (3x)R (o, a, x). This is theorem 3 of Liu's paper and the theorem mentioned in section 4 and proved in section 6 of Spector's paper. The two proofs of this im- portant and useful theorem are, both of them, interesting, illuminating, and difficult.

Spector makes a rather detailed analysis of Kleene's predicate C(b), using ideas and methods in Kleene's XI 127. Kleene proved that, for b E 0, C(b) is well-ordered by X e C(y). Spector considers basically those b's for which C(b) is linearly-ordered by X e C(y). He forms a first-order theory L(a, <a) which describes these linear orderings. He then shows that if a E 0 then there is exactly one binary predicate " <a" which is a model for the theory (lemma 9). This unique existence gives immediately a E 0 => (3f)HAL(a, fi). Also he shows (lemma 10) that if a < 0 then any "<a" satisfying L(a, <a) must be, in degree, higher than any HA set. Hence the result that a E 0 = (3fl)HAL(a, fi). From this, using the completeness of 0 for 1 FQV forms, we get the theorem mentioned in the first paragraph.

Liu uses a different approach. His treatment is based on Kleene's result (XXVII 82(1)) which gives a basis (a certain segment of the hyperarithmetic hierarchy) for the 1 FQ3 expression of a given HA set. In proving the main theorem, Liu proves two theorems which are interesting on their own account. He proves in theorem 1 that a recursive linear ordering / has a well-ordered segment of type IJy (where y E 0) iff / is well-ordered with respect to all functions recursive in Hy** . And in theorem 2 he shows that, for any recursive R(a, a, x), there is a recursive S(s, a), such that (oc)(3x)R (o, a, x) --(a)(3x)S(& (x), a) and (G)HA(3x)S(& (x), a).

Liu's paper is in some ways more elegant and interesting. But Spector's was the first paper and gives a better understanding of the historically important system, 0, of notations for constructive ordinals.

The only typographical error in Spector's paper is the omission of a negation sign over the symbol Qo on page 315, line 8.

The following are typographical errors in Liu's paper: (1) page 129, line 6, replace the Roman numeral L by the Roman numeral I; (2) page 129, line 12, supply "-<" directly under "/" just before the right curly bracket; (3) page 129, line 19, place a negation bar over the universal quantifier (immediately after H-**) and change the type of the quantified variable from roman to italic; also insert "c" immediately after the "less than" sign; (4) page 130, line 8, change the last letter of the line from a "c" to a "u"; (5) page 130, line 9, the first two letters of the line, "c" and "u", should be transposed; (6) page 130, line 7 from the bottom, supply the function variable "cc" between the parentheses after "hand"; (7) page 131, line 19, the last symbol of the line, "y", which appears as a superscript on "t" should be brought down to the line.

GUSTAV HENSEL

J. W. ADDISON. Hierarchies and the axiom of contstructibility. Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell Univer-

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