On Analytic Well-orderings. by Hisao TanakaReview by: Wayne RichterThe Journal of Symbolic Logic, Vol. 38, No. 1 (Mar., 1973), p. 155Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271753 .

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REVIEWS 155

theorem (p. 152) is as follows: If the recursive series EF(x, v) converges in a given interval I to a function G(x), then G(x) is recursive. From this it follows that log, exp, sin, and all other elementary functions are recursive. BRIAN H. MAYOH

HISAO TANAKA. On analytic well-orderings. The journal of symbolic logic, vol. 35 (1970), pp. 198-204.

The author presents a useful summary of facts about the order relations between certain countable ordinals.

Let K be a set of number-theoretic predicates, and w(K) be the least ordinal not the order type of a well-ordering belonging to K. For A a set of non-negative integers, aol is the least ordinal not the order type of a well-ordering recursive in A, and wl = wo . 0 is the complete l1 set of Kleene XI 127.

The author proves the following principal results. (1) w(IIH) = co4. (2) CO(Al)=

w(HI) < Cu(Z1) < w(Al). Assuming V = L he further shows: (3) W(Al) = W(H3) <

CO(Z') < - --, etc. The equality in (2) is obtained by a simple application of the Novikoff-Kond6-Addison uni-

formization theorem. The inequality (and proof that) w(Al) < W(Z2) is credited to Rogers (XXXVI 141). The equality (1) is actually a special case of a more general result (Theorem 7.3 together with 4.8) of the reviewer (XXXVI 341(1)). WAYNE RICHTER

MARIAN BOYKAN POUR-EL and HILARY PUTNAM. Recursively enumerable classes and their application to recursive sequences of formal theories. Archive fur mathematische Logik und Grundlagenforschung, vol. 8 no. 3-4 (1965), pp. 104-121.

MARIAN BOYKAN POUR-EL and WILLIAM A. HOWARD. A structural criterion for recursive enumeration without repetition. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, vol. 10 (1964), pp. 105-114.

A. H. LACHLAN. On recursive enumeration without repetition. Ibid., vol. 11 (1965), pp. 209-220.

A. H. LACHLAN. On recursive enumeration without repetition: a correction. Ibid., vol. 13 (1967), pp. 99-100.

In these four papers a study is made of recursively enumerable (r.e.) classes which can or cannot be enumerated without repetition. Their common inspiration is the now-classical theorem of Friedberg (XXV 165(3)) that the class of all r.e. sets can be so enumerated. For a time it was thought possible that any r.e. class had this property, but as the first paper shows, this is false even for r.e. classes of finite sets. However, the second and third papers demon- strate that Friedberg's methods are capable of considerable generalization, so that a wide variety of r.e. classes do possess one-to-one enumerations.

Call a r.e. class k-r.e. if it has an enumeration in which each member appears at most k times. Pour-El and Putnam construct, for each k 2 2, a k-r.e. class of finite sets which is not (k - 1)- r.e. They also produce a r.e. class of finite sets which is not k-r.e. for any k yet has an enumera- tion in which each set appears finitely often, and a r.e. class of finite sets such that in any enumeration of it infinitely many sets are repeated infinitely often. Further strengthening is im- possible, since if Q4 is a r.e. class and F is finite, then Q4 - {F} is r.e. However, the authors do construct a r.e. class of infinite sets such that every member of the class is duplicated infinitely often in every enumeration of the class. They do this by partitioning w into infinitely many dis- joint complete sets. The authors translate their theorems into analogous results about r.e. classes of axiomatic theories.

In the second paper, the following theorem is proved. Let 2t be a r.e. class and suppose h (called a height function for Zt) is a partial recursive function with domain all finite subsets of members of 2t such that A c B -h(A) < h(B), AO c A1 c A2 C ... C: W & W C 21 -

(3m)(Vn)(n > m -+ h(A.) = h(Am)), and (VA)(3B)(A c B & h(A) < h(B)). Then 2t can be enumerated without repetition. This theorem has several corollaries, of which we give one. Suppose 21 is a r.e. class, that for every integer n there exists a member R of Q4 such that {m I m < n} c R, but that w f 21. Then Ql is r.e. without repetition. Let V53 be the class of all r.e. sets. Then 3 - {c} is r.e. and satisfies the above conditions, hence is r.e. without repetition. Friedberg's theorem follows immediately. Other r.e. classes to which the theorem or its corollary

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