Review: Eugene M. Kleinberg, Infinitary Combinatorics and the Axiom of Determinateness

monotone operator mapping /* into 7 has closure in / \ Other theorems presented include the following : If C has the prewellordering property then | C | < | -iC | (Aanderaa), IND^S) s en(8) for Spector theories 9 (Grilliot), and IND(C) = IND(C-mon) under certain general conditions (Harrington and Kechris). Chapters 5 and 6 deal primarily with admissible structures and the corresponding degree theory. A domain 8( with a prewellordering is called admissible if IND^-positive) = 2, on %. The characterization theorem states that for any admissible S(there is an "infinite" computation theory 6 such en(8) = Jx on 2t. A "next admissible set" theorem is given for Spector theories. Several results are given for Spector theories 6 over a, including (i) the plus-one theorem that the family sc(6) of computable relations equals sc(F) for some normal type-2 functional F and (ii) the fact that the family of hereditarily countable sets with 0-computable codes is an abstract 1-section. In Chapter 6, the partial ordering "B is ©-computable in C" is introduced for infinite theories. B is said to be regular if B f~) K is 0-finite whenever K is. If 8 is "adequate," then every semicomputable B has the same degree as some regular semicomputable set. The splitting theorem states, roughly, that any regular B in en(8) sc(0) can be partitioned into two sets of strictly smaller degrees. The question of the density of the degree ordering on en(0) is discussed. This material is based on the degree theory of a-recursion theory, where a is an admissible ordinal. Extension of the theory to non-admissible ordinals is discussed at the end. Chapters 4 and 7 deal with recursion on a domain of two types, S and <u. A list L of functionals and relations is called normal if it contains equality on S and the weak /4-quantifier. If L is normal then en(L) is closed under (3s e S) but not under (3x e <o). A characterization is given for prime computation in a normal list. The plus-two theorem states that for any "normal" theory 6, there is a normal list L such that en(0) D P(S) = en(L) fl P(S). Chapter 8 introduces functions on the universe Vof sets that are set recursive in some relation R; these include Jensen's rudimentary functions. The characterization theorem gives the equivalence of set recursion and Kleene recursion in a normal functional of type k + 2. A priority argument is given for set recursion relative to the quantifier functional E. An epilogue (8.5) points out some open problems, some omissions, and some areas of future research. As the above outline shows, this book contains a tremendous amount of material packed into only 205 pages. Ample references are provided where details are omitted or where supplemental reading is appropriate. The scholarship is impressive; theorems and definitions are credited to their originators and the history of each topic is given. There are 231 citations in the 205 pages of text; these can be located with the help of an author index as well as a list of 171 references. Also very helpful are the subject index and a list of over 120 notations. There features should make the book an excellent reference for researchers working in general recursion theory or some related area. While it develops axiomatic recursion theory in a systematic and convincing fashion, the book does not devote sufficient space to the standard examples to be fully self-contained. In particular, though most of the generalizations are based on hyperarithmetic and a-recursion theory, the treatment of these topics is, as the author states, highly selective and rather brief. Together with the absence of any exercises, this seems to indicate that the book is not intended as a textbook. As one rather expects in a book on recursion theory, the notation becomes involved at times; this is one particular drawback to the index approach to recursion theory. The situation gets out of hand on page 101, in the midst of a complicated six-page proof, when the letter a is used to represent two quite different objects simultaneously. The contradiction is pointed out in a note and no serious problem results. DOUGLAS CENZER