Gerald E. Sacks

POST'S PROBLEM, ADMISSIBLE ORDINALS, AND REGULARITY

1. Introduction. The basic notions of metarecursion theory were introduced in [7]. Metarecursioni theory is an attempt to generalize the ideas and arguments of recursion theory from the natural numbers to the recursive ordinals. Ordinary recursion theory concerns itself with finite sets of natural numbers. Metarecursion theory deals in an analogous fashion with metafinite sets of recursive ordinals. In [7] it was seen that two of the deepest results of ordinary recursion theory, the solution of Posts problem [3] and the maximal set construction [4], generalize to the metarecursive case. Theorem 4 of [7] states that there exist two metarecursively enumerable sets of recursive ordinals such that neither is metarecursive in the other. Kreisel [6] casts doubt on the contention that Theorem 4 of [7] is the correct generalization of Posts problem. He wonders how to correctly formulate the notion of Turing reducibility for recursive ordinals and he discusses four possible choices. It is not our place to decide the correct notion of Turing reducibility for metarecursion theory; however, in ?4 we show that there exist two metarecursively enumerable sets such that neither is reducible to the other by any of the four methods discussed by Kreisel. In [7] two sets of recursive ordinals are said to have the same metadegree if each is metarecursive in the other. It is not known if the metadegrees of metarecursively enumerable sets are order-isomorphic to the Turing degrees of recursively enumerable sets, but all the existing evidence seems to favor an affirmative answer. Driscoll [2] has shown the metadegrees of metarecursively

Some minimal pairs of α-recursively enumerable degrees

Exploitation of the model theoretic properties of G6dels constructible sets led in [61 to a generalization of the Friedberg-Muchnik finite injury (or priority) method from ¢o to every E l admissible ~. In order to generalize, it was necessary to sacrifice the standard indexing of arecursively enumerable sets, and hence of the requirements associated with finite injury arguments. For some as tile indexing was demonstrably not a-recursive. 131 gave an alternative view of [6] that centered on the nature of the indexing. This paper continues the study of indexing of requirements, and applies it to construct minimal pairs of a-recursively enumerable sets for some, but not all, ~x. The Friedberg-Muchnik solution of Posts problem generalizes in a trivial fashion to every 2; 2 admissible ordinal. All the complications of [31 and [61 resulted from forcing a 2;1 admissible ordinal a to do the work of a ~2 admissible ordinal. In this paper a is forced to do a much larger share of that work, and even

The 1-Section of a Type n Object

Publisher Summary This chapter provides an introduction to plus-one theorem. The proof of the plus-one theorem for the case k > 1 is presented. It is largely a consequence of a stability lemma described in the chapter. k sc U is the set of all objects of type k recursive in U, and is called the k -section of U. The plus-one theorem states that all k-sections generated by finite type objects (in which the appropriate equality predicates are recursive) are generated by type k + 1 objects. The first result on k -sections is that 1 sc 2 E is the set of hyperarithmetic reals. The chapter presents some of the elements of recursion in objects of finite type. The chapter introduces the notion of abstract 1-section and shows that many familiar collections of reals. It is proved in the chapter that every countable abstract 1-section is the 1-section of some type 2 object in which 2 E is recursive by means of a forcing argument of the sort associated with generic classes rather than sets. The nature of abstract k -sections when k > 1 is discussed.

On a theorem of Lachlan and Martin

In [3] we raised the following question: does there exist a recursively enumerable degree d such that 0(n)< <(n) < O(n+1) for all n ?0? This question was answered affirmatively by Lachlan [I] and by Martin [21. Lachlans proof combines a familiar priority argument with the fixed point theorem of Kleene. Martins proof is a new form of priority argument based on Theorem 3 of section 6 of [3]. In this paper we give a vanishingly short proof of the Lachlan-Martin result without any use of priority. Our argument is an exercise in the fixed point theorem. We exploit, possibly for the first time, a uniformity concealed in most proofs of the fixed point theorem. Let A be an arbitrary set of natural numbers, and let WA, WA, W2, be a standard simultaneous enumeration of all sets recursively enumerable in A. For each e ? 0, let

On the Number of Countable Models

Publisher Summary Scott discovered that each countable structure M can be characterized up to isomorphism by a single sentence F of L ω˥, ω —that is, M≠F and [N≠F]→N≃M. A canonical choice for F, denoted by F M , is developed from M by an inductive procedure that terminates at a countable ordinal r(M), the Scott rank of M. Scotts proof that r(M) exists and is countable is in essence a counting argument. The enumeration of countable models is briefly discussed in the chapter.

Metarecursively enumerable sets and admissible ordinals

Here we describe some results which will be proved in detail in [13] and [l4]. The notion of metarecursive set was introduced in [8]. Kreisel [7] reported on some of the model-theoretic deliberations which preceded the definitions of [8]. Metarecursion theory is a generalization of ordinary recursion theory from the natural numbers to the recursive ordinals. Theorems about finite sets of natural numbers are replaced by theorems about metafinite sets of recursive ordinals, some of which are infinite. Initially, metarecursive sets were defined in [8] in terms of hyperarithmetic sets, II} sets, and notations for recursive ordinals [6], [16]; however, it later proved convenient to utilize an equation calculus devised by Kripke [9]. The purpose of Kripkes theory is to generalize recursion theory from the natural numbers to certain initial segments of the ordinals [9], [lO], [ l l ] . He calls an ordinal a admissible if the ordinals less than a have certain closure properties definable in terms of an equation calculus modeled on Kleenes. Kripkes equation calculus has numerals denoting ordinals, finitary substitution rules, and one infinitary deduction rule. If an ordinal a is admissible, then an «-recursive function ƒ is defined by a finite system of equations: each value of ƒ is computable using Kripkes rules, and only correct values can be so computed. It turned out that the first admissible ordinal after co was Kleenes coi, the least nonrecursive ordinal, and that the metarecursive functions were the same as the cot-recursive functions [8], [9]. In this paper we concentrate on our first love, metarecursion theory, but we cannot resist noting, whenever appropriate, which of our results generalize to arbitrary admissible ordinals. A set of recursive ordinals is called regular if its intersection with every metafinite set of recursive ordinals is metafinite. (The metafinite sets coincide with the bounded, metarecursive sets.) It was observed in [8] that there exist bounded, metarecursively enumerable sets which are not metarecursive; each such set is a constructive example of a nonregular set. It would not be unfair to say that the interesting arguments of metarecursion theory, if it is granted that

The differential closure of a differential field

Good afternoon ladies and gentlemen. The subject of mathematical logic splits fourfold into: recursive functions, the heart of the subject; proof theory which includes the best theorem in the subject; sets and classes, whose romantic appeal far outweigh their mathematical substance; and model theory, whose value is its applicability to, and roots in, algebra. This afternoon I hope to sketch some theorems about differential fields first derived by model theoretic methods. In particular, I will indicate why every differential field si of characteristic 0 has a unique prime differentially closed extension called the differential closure of si. Model theory has proved useful in the study of differential fields because the notion of differential closure is surprisingly more complex than the analogous notions of algebraic closure, real closure, or Henselization. The virtue of model theory is its ability to organize succinctly the sort of tiresome algebraic details associated with elimination theory. The first concepts of model theory are structure and theory. Typic structures are groups, rings and fields. A theory is a set of sentences. A sentence is about the elements of some structure. The language of fields includes plus ( + ), times (•), equals (=) and variables that stand for elements of fields. A typic sentence in the language of fields says: every polynomial of degree 7 has a root. A typic theory is the theory of algebraically closed fields of characteristic 0 (ACF0). A structure si is said to be a model of a theory T if every sentence of T is true in si. Thus the models of ACF0 are what men call algebraically closed fields of characteristic 0. Pure model theory, at first thought, appears to be too general to have any mathematical substance. But that hasty thought is given the lie by several theorems, one of which is due to Vaught [1]: Let T be any coun-

Atomic models higher up

Abstract There exists a countable structure M of Scott rank ω 1 C K where ω 1 M = ω 1 C K and where the L ω 1 C K , ω -theory of M is not ω -categorical. The Scott rank of a model is the least ordinal β where the model is prime in its L ω β , ω -theory. Most well-known models with unbounded atoms below ω 1 C K also realize a non-principal L ω 1 C K , ω -type; such a model that preserves the Σ 1 -admissibility of ω 1 C K will have Scott rank ω 1 C K + 1 . Makkai [M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 (1981) 301–318. [4] ] produces a hyperarithmetical model of Scott rank ω 1 C K whose L ω 1 C K , ω -theory is ω -categorical. A computable variant of Makkai’s example is produced in [W. Calvert, S.S. Goncharov, J.F. Knight, J. Millar, Categoricity of computable infinitary theories, Arch. Math. Logic (submitted for publication). [1] ; J. Knight, J. Millar, Computable structures of rank ω 1 C K J. Math. Logic (2004). [2] ].

Measure-theoretic uniformity

Here we present the principal ideas and results of [5] with some indications of proof. The general notion of uniformity is difficult to harvest; nonetheless, various offshoots of it have borne fruit in all fields of mathematical logic. In this paper we introduce the notion of measure-theoretic uniformity, and we describe its use in recursion theory, hyperarithmetic analysis, and set theory. In recursion theory we show that the set of all sets T such that the ordinals recursive in T are the recursive ordinals has measure 1. In set theory we obtain all of Cohen’s independence results [1,2]. Solovay [8,9] has extended Cohen’s method by forcing statements with closed, measurable sets of conditions rather than finite sets of conditions; in this manner he exploits the concepts of forcing and genericity to prove: if ZF is consistent, then ZF + “there exists a translation-invariant, countably additive extension of Lebesgue measure defined on all sets of reals” + “the dependent axiom of choice” is consistent. Solovay’s result is also a consequence of the notion of measure-theoretic uniformity.

Post's Problem, Absoluteness and Recursion in Finite Types

The unresolved character of the power set operation stymies the solution of elementary problems arising in Kleenes theory of recursion in objects of finite type. E.g. Posts problem for 3 E has a positive solution if V=L (NORMANN, 1975), and a negative if AD holds. Let δ be the class of all sets R ⊆ 2ω such that R is recursive in 3 E, b for some real b . A forcing construction shows δ is not recursively enumerable in 3 E when there is a recursively regular well-ordering of 2″ recursive in 3 E. It follows that the concepts of Σ * , and weak Σ * , definability differ. With the aid of the notions of indexicality and ordinal recursiveness, an absolute version of Posts problem for 3 E is devised, and studied when certain recursively enumerable projecta are equal.