Recursive Equivalence Types and Combinatorial Functions by John MyhillReview by: J. C. E. DekkerThe Journal of Symbolic Logic, Vol. 31, No. 3 (Sep., 1966), pp. 510-511Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2270491 .

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510 REVIEWS

Brouwer) a maps the unsecured (with respect to F) sequences c of increasing lengths ,nto a chain in ap and p applied to the least element of the chain provides the value F(c).

For course-of-values recursion a schema is considered in which, again very roughly, a functional X is defined by course-of-values recursion over f from a functional G, where G arises from Rl(a). That is, X(x) is computed from a-recursive G using X(y) for various y -<0 x. It is shown that this schema is derivable in SI if Rl(X'O) is. This re- duction of course-of-values recursion to recursion on a,0 is a generalization of the author's earlier result (XXVIII 103) reducing nested recursion on f to ordinary recursion on co:. The proof turns out to be very neat, but the reviewer had difficulty because of a number of misprints and one small slip, for which the author supplied a correction when it was pointed out to him. Misprints: page 168, line 9, read "X" for "1V"; lines 13ff., in the displayed computation, read "Xx(O)", ... ,"X,(k)" for "XO(x)", ... "Xk(x)"; in the bottom display, on the left, move the superior arrow outside the brackets, on the right, put a horizontal bar over "p(<uo, . r. n, >, z) - 1", do not include the brackets under the horizontal bar in "{m + 1}", read " Ka" for " -<a" in (i), read "-<#" for "-<a" in (ii); page 169, line 5, read "S2" for "s2"; line 8, read "zi" for "t; line 12, read " )m" for t2mt"; in the definition of X, read "X" for "X". In addition, the definition of 0 given fails to satisfy (6.1), spoiling the argument, but the author has corrected this as follows. We can suppose that there is a primitive recursive successor function x* for e<,a, with x h<a y -+x* y provable in PRA. E.g. a in the proof can be replaced by co 0 a. Then set

0(s) -7171 u(Kiio . ..,jms>, zi p (m(<roXirrn>,> 2r)) * i-s Pi ZjZr

Then (6.1) holds. The author defines S1 to be a conservative extension of S if every formula of S

which is a theorem of SI is also a theorem of S. He then shows that if S1 is PRA' extended by adding Rl(a) for certain a and if SO is PRAO with RO(a) for those same a,

then S1 is a conservative extension of SO. Letting Ra = PRAO + RO(a) and I, = PRAO + I(a), he derives the consistency

of Ra in I(C'expto).co, and (for a _ O)) the consistency of I, in R,. R. E. VESLEY

JOHN MYHILL. Recursive equivalence types and combinatorial functions. Logic, methodology and philosophy of science, Proceedings of the 1960 International Congress, edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski, Stanford University Press, Stanford, Calif., 1962, pp. 46-55.

This is an expository paper which is an elaboration and continuation of an earlier paper by the same author and with the same title XXV 356(3). The subsets a and # of the set E of all non-negative integers are recursively equivalent (a f), if there is a partial recursive one-to-one function p of one variable such that a is included in the domain of p and P(a) = f. The equivalence classes into which the class of all subsets of E is partitioned by the relation are the recursive equivalence types (RETs). An RET is an isol, if it consists of sets which have no infinite recursively enumerable subset. Q denotes the collection of all RETs and A the collection of all isols. The RET which contains {x E I x < n} is identified with the non-negative integer n. Thus E c A C Q, where A and Q have the cardinality of the continuum.

The function / from E into E is combinatorial, if the function c from E into the set of all integers related to / by I(x) = X='ctCx i assumes no negative values. Combinatorial functions of n variables are similarly defined; in particular x + y and x -y are com- binatorial. While combinatorial functions were introduced by Myhill in XXV 356(3), almost combinatorial (a.c.) functions were later introduced by Nerode in his funda-

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REVIEWS 511

mental XXV 359. Every a.c. function / from en into e can be extended in a natural manner to a function IA from An into A, its so-called canonical extension.

In the present paper the author considers sentences which are truth-functions of equations between recursive a.c. functions. He summarizes the results obtained up to 1960 by himself and Nerode concerning the characterization of sentences which when true in e, are also true in A. We mention one such result. Theorem. Let H[x, y, z] be a Horn sentence. If (Vx)(Vy)(3z)H[x, y, z] holds in e and there exists a recursive a.c. function / such that (Vx)(Vy)H[x, y, /(x, y)] holds in e, then (Yx) (Vy) (3z)H[x, y, z] holds in A. The paper concludes with some remarks about f2-A and the difference between the arithmetic of isols and that of Dedekind finite cardinals without the axiom of choice (studied by E. Ellentuck).

Errata. Page 48, line 18, delete "XZ -A"; page 48, line 30, for "+", read -

page 49, line 12, in the binomial coefficient, for "A", read "n"; page 49, line 14, for

"(x/k)", read page 51, line 1, for "number," read "numbers"; page 52, line 18,

for "c", read "x"; page 52, last line, for "here," read "have"; page 55, line 7, for "Frankel," read "Fraenkel." J. C. E. DEKKER

G. KREISEL, J. SHOENFIELD, and HAO WANG. Number theoretic concepts and re- cursive well-orderings. Archiv fur mathematische Logik und Grundlagenfor- schung, vol. 5 (1959), pp. 42-64.

This paper discusses some relationships between recursive ordinals and arithmetic predicates and propositions. In the first part of the paper degrees of unsolvability are obtained for arithmetic sets of integers associated with initial segments of recursive ordinals. In the last two sections relationships between true sentences of formal arithmetic and well-orderings are discussed.

Both the Markwald-Spector set W of G6del numbers for recursive well-ordering relations (XX 283, XXI 412) and Kleene's set 0 of notations for constructive ordinals (IV 93) are considered in the first part of the paper. For x c W (x c 0), Ix (IxI) is used to denote the ordinal associated with x. The aim of this part of the paper is to determine degrees of unsolvability with respect to many-one reducibility (many-one degrees) of arithmetic sets W(a) and 0(a) where, if a is a recursive ordinal, W(a) =

{x I x c W & IxI < a} and similarly 0(a) ={x i x c 0 & IxI < a}. For a > coo) it is found that 0(a) and W(a) are not arithmetic. For a < coo), the authors' methods are not sufficient to determine many-one degrees in all cases, but they do obtain at least Turing degrees for these cases. The results have since been completed by Liu in two papers which are reviewed immediately below.

First, a relationship between the many-one degrees of sets W(o) and sets 0(a) is established, so that only the degrees for W(a) need be obtained directly. It is shown that for all recursive ordinals a,

degMW(a) degMO(co -(1 + a)) degMO(a) degMW(1r)

where degMW(a) denotes the many-one degree of W(a) and, for a given a, r is the least ordinal such that a < col (1 + r). An error in Theorem 2 (p. 47) should be noted, though it does not affect the validity of these results. The order type lIv(e) i of the ordering (e) defined in the theorem is not IeI* as stated in (1 1), but rather IeI** + 1 where, for lei 2 a), lei** is defined to be the unique ordinal such that lei = cl (1 + lei**) + A with k finite. (For lei < cl, lIv(e)II = 0.) The third line of the statement of the theorem must also be changed to read "e c 0 -? Ifl(e) I = lei** + 1JJ.

Next, degrees for the sets W(a) are studied. To summarize the main results, let en (an) be the highest many-one degree represented by a predicate form consisting of a

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