Degrees of Unsolvability of Constructible Sets of Integers. by George Boolos; Hilary PutnamReview by: A. S. KechrisThe Journal of Symbolic Logic, Vol. 38, No. 3 (Dec., 1973), pp. 527-528Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2273076 .

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REVIEWS 527

By examples analogous to examples in Feferman's XXXI 269, the author shows (where S is Peano arithmetic) that (i) M can be taken so that no recursive function is provable, and (ii) for any recursive f, an M can be found making f provable.

The author asks: Can more invariant significance be given to the concept of provable function either by restricting the choice of M or by modifying the definition in some other way? One possibility is to restrict M to be a primitive recursive (PR) formula. (See XXXI 269.) Another is to say that a function f is "provable" if a formula A(x, y) exists defining (semanti- cally) the graph of f, where " Vx3yA(x, y) " is provable in S and where A satisfies some further specified condition Ci . The author shows that the condition C1 that A define a recursive rela- tion is too weak and makes all recursive functions "provable," while the condition C2 that A define a primitive recursive relation is too strong and forces some recursive functions to be "absolutely unprovable" (in the sense that no axiom added to S can make such a function "provable"). The author conjectures that use of the condition C3 that A be a recursively enumerable formula (in the sense of XXXI 269) is equivalent to restricting M to be a PR formula. HARTLEY ROGERS, Jr.

B. A. TRAHTtNBROT. Ob avtosvodimosti. Doklady Akadjmii Nauk SSSR, vol. 192 (1970), pp. 1224-1227.

B. A. TRAHTENBROT. On autoreducibility. English translation of the preceding by M. Machover. Soviet mathematics, vol. 11 no. 3 (1970), pp. 814-817.

The author defines the notion of autoreducibility. A set of numbers A is autoreducible if it can be computed from itself via an oracle algorithm such that on input n the algorithm does not ask the question "n e A ?" of the oracle, that is, via an autoreduction. Several theorems and interesting questions about the notion are given. However, the reader should be warned that no proofs of the theorems are provided in the text. Of particular interest are the following two facts. There are recursively enumerable sets which are not autoreducible. There are arbitrarily complex recursive sets A such that no autoreduction of A to itself is essentially better in computing A than a direct computation of A (better in terms of Turing machine space).

RICHARD E. LADNER

GEORGE BoOLos and HILARY PUTNAM. Degrees of unsolvability of constructible sets of integers. The journal of symbolic logic, vol. 33 no. 4 (for 1968, pub. 1969), pp. 497-513.

This very important paper gives a detailed account of the fashion in which the sets of integers (for simplicity reals) appear in the levels of Godel's constructible hierarchy. It is, in fact, the first systematic study of what is nowadays called the fine structure of the constructible hierarchy. As a result of this study, a hierarchy of degrees of unsolvability is constructed which extends the usual arithmetic and hyperarithmetic hierarchies and covers all the constructible reals.

Let La, for a an ordinal, be defined as follows: Lo = z, La+i = {x C La: x is first- order definable over <La, E>}, LA = Ua<ALa, if A is limit. Of course La is the ath cumulative level of the hierarchy of Godel's constructible sets. Put L = UaL, = the constructible uni- verse. If x E L, the order of x is the ordinal a such that x e La+1 - La . An ordinal a < =

constructible c1 is called an index if there is a real x such that order(x) = a. The authors' main technical lemma (whose method of proof has now become classical) shows that if a is an index there is a real Ea coding the structure <La , E> such that Ea e La +1 . Whenever every real of order _ a is arithmetic in x, x is called (arithmetically) complete of order a. Thus Ea is the unique up to arithmetic equivalence complete real of order a. In fact Ea can be chosen so that if x is a real and order(x) < a then x is one-to-one reducible to Ea .

For A e w, let 0A be the relativized Kleene 0 and HA (where a < WA = the first ordinal non-recursive in A) be the relativized H-sets. Of course HA is defined only up to Turing-degree equivalence. If A = z we omit the superscript. To state the next results it is convenient to start the constructible hierarchy by putting Lo = hereditarily finite sets. The authors prove now that for 0 < a < o, , H,, a (where w a denotes ordinal multiplication) has order a and the reals of order a are exactly those arithmetic in H.,, , but not in any H3 with : < co a. Thus HJ) -a is complete of order a, 0 is complete of order c1, every a-< c1 is an index and the reals in LK,.1 are exactly the hyperarithmetic ones. Relativizing they show that if Ey is a com- plete set of order y then for any 0 < a < w o', HEY has order y + a while the reals of order

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528 REVIEWS

y + a are exactly those arithmetic in HEY- but not in any HB Y. where a < w-a. Thus HEYg is complete of order y + a, 0EY is complete of order y + wlv = Y every y _ a _ wly is an index and the reals in

are exactly the reals hyperarithmetic in E, . The hierarchy of Turing degrees that was mentioned in the beginning is now defined by d(O) = degree of 0, d(a + 1) = jump of d(a), d(w-f/) =

degree of EV, where y is the /3th index >w.

In the last part of their paper the authors compare the hierarchy LBl r) 2W with the hierarchy of ramified analytical sets A4, where AO = finite reals, A0+1 = reals definable by analytical predicates with constants from Afi and quantifiers restricted in A0, AA = Ufl < AAI, if A is limit. Putnam and Gandy independently confirmed a conjecture of Cohen by proving the existence of a smallest /-model of analysis. Moreover, they identified it to be A,60, where go is the first place where the hierarchy Af stops, i.e., the least A such that Af = A+1 . The authors prove here that for all /3 < Po, A,3 = LB r 20 and thus Po = the least non-index. A. S. KECHRIS

SAUNDERS MAC LANE. Categorical algebra and set-theoretic foundations. Axiomatic set theory, Proceedings of symposia in pure mathematics, vol. 13 part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 231-240.

This article poses to logicians a "technical question" concerning the use of proper classes in category theory (the "general philosophical question" also stated is a vague and, in the reviewer's opinion, misleading suggestion for answering the first question). In most mathe- matics such problems do not arise because only sets of small rank are considered, and in set theory no interesting problems have arisen that cannot be dealt with in Godel-Bernays set theory with a few simple circumlocutions. In category theory, however, it is desirable to consider large categories such as the category of all groups and to use constructions such as functor categories which lead to categories whose objects and morphisms are themselves proper classes. The author gives some examples of the use of such categories. These categories cannot be dealt with by the usual methods; the question is: What system will effectively handle them?

This question does suggest foundational problems of some interest but is quite vague as stated. One reason for this may be that the author is less interested in the formal foundation he asks for than a system which (as he states as a "practical requirement") "could be used 'naively' by mathematicians not sophisticated in foundational research." In otherwords, the system should serve as a guide to mathematical thought in this area as naive set theory and category theory itself frequently do in other areas of mathematics. In the reviewer's opinion this insight will depend more on developments in category theory than on logical considera- tions. WILLIAM MITCHELL

SERGE GRIGORIEFF. Combinatorics on ideals and forcing. Annals of mathematical logic, vol. 3 no. 4 (1971), pp. 363-394.

A uniform filter F on K is selective if any partition of K into pieces less than K of which are never in F must have a choice function whose range still generates a uniform filter when added to F. Grigorieff considers selective ideals on NO; one calls an ideal selective if its dual filter is selective. It is as yet unknown whether or not one can find selective ultrafilters using only the axiom of choice.

For an ideal I in S(c) one can form Cohen-like forcing conditions which provide information about only a set of numbers in I. In this way one obtains "I-Cohen" reals.

Grigorieff proves: A real I-Cohen over M is minimal if and only if I is the dual of a selec- tive ultrafilter. By " minimal " one means that it is constructed by any set which does not already lie in M.

To show the minimality of selective Cohen reals, he introduces a new characterization of selective filters in terms of trees of finite sequences. The converse is easier: Here one uses a partition having no large choice function to generate projections of I-Cohen reals which are themselves generic but do not construct the I-Cohen reals.

Finally, one calls a real F-Silver, where F is a filter, if it is generic for the conditions which give information about only a set of numbers not in F; I-Silver reals are just the F-Silver reals

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