We prove that there exists a noncappable enumeration degree strictly below 0' e.

When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss bothcodings of models of arithmetic into the (...) recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by cofinality) then such constructions maybe performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natu. (shrink)

We show that every nonzero Δ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{2}}$$\end{document} enumeration degree bounds the enumeration degree of a 1-generic set. We also point out that the enumeration degrees of 1-generic sets, below the first jump, are not downwards closed, thus answering a question of Cooper.

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

Using properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree (...) is high; every nonzero low enumeration degree can be capped by degrees of any possible local jump ; every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degreeacan be capped belowa. (shrink)

We prove that every countable relation on the enumeration degrees, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}, is uniformly definable from parameters in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}. Consequently, the first order theory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document} is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the (...) first order theory of the enumeration degrees of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Sigma^0_2$\end{document} sets is not decidable. (shrink)

We show that every nonzero $\Sigma _{2}^{0}$ enumeration degree bounds a nonsplitting nonzero enumeration degree.

We prove that every non-computable incomplete computably enumerable degree is locally non-cappable, and use this result to show that there is no maximal non-bounding computably enumerable degree.

A set A is symmetric enumeration (se-) reducible to a set B (A ≤\sb se B) if A is enumeration reducible to B and \barA is enumeration reducible to \barB. This reducibility gives rise to a degree structure (D\sb se) whose least element is the class of computable sets. We give a classification of ≤\sb se in terms of other standard reducibilities and we show that the natural embedding of the Turing degrees (D\sb T) into the (...) enumeration degrees (D\sb e) translates to an embedding (ι\sb se) into D\sb se that preserves least element, suprema, and infima. We define a weak and a strong jump and we observe that ι\sb se preserves the jump operator relative to the latter definition. We prove various (global) results concerning branching, exact pairs, minimal covers, and diamond embeddings in D\sb se. We show that certain classes of se-degrees are first-order definable, in particular, the classes of semirecursive, Σ\sb n ⋃ Π\sb n, Δ\sb n (for any n \in ω), and embedded Turing degrees. This last result allows us to conclude that the theory of D\sb se has the same 1-degree as the theory of Second-Order Arithmetic. (shrink)

We characterize the class of c.e. degrees that bound a critical triple as those degrees that compute a function that has no ω-c.e. approximation.

We prove that each Σ 0 2 set which is hypersimple relative to $\emptyset$ ' is noncuppable in the structure of the Σ 0 2 enumeration degrees. This gives a connection between properties of Σ 0 2 sets under inclusion and and the Σ 0 2 enumeration degrees. We also prove that some low non-computably enumerable enumeration degree contains no set which is simple relative to $\emptyset$ '.

§1. Introduction. Natural sets that can be enumerated by a computable function always seem to be either actually computable or of the same complexity as the Halting Problem, the complete r.e. set K. The obvious question, first posed in Post [1944] and since then called Post's Problem is then just whether there are r.e. sets which are neither computable nor complete, i.e., neither recursive nor of the same Turing degree as K?Let be the r.e. degrees, i.e., the r.e. sets (...) modulo the equivalence relation of equicomputable with the partial order induced by Turing computability. This structure is a partial order with least element 0, the degree of the computable sets, and greatest element 1 or 0′, the degree of K. Post's problem then asks if there are any other elements of.The solution of Post's problem by Friedberg [1957] and Muchnik [1956] was followed by various algebraic or order theoretic results that were interpreted as saying that the structure was in some way well behaved:Theorem 1.1. Every countable partial ordering or even uppersemilattice can be embedded into.Theorem 1.2. For every nonrecursive r.e. degreeathere are r.e. degreesb, c < asuch thatb ∨ c = a.Theorem 1.3. For every pair of nonrecursive r.e. degreesa < bthere is an r.e. degreecsuch thata < c < b. (shrink)

We give a simple structural property which characterizes the r.e. sets whose (Turing) degrees are cappable. Since cappable degrees are incomplete, this may be viewed as a solution of Post's program, which asks for a simple structural property of nonrecursive r.e. sets which ensures incompleteness.

We show that the Π 0 2 enumeration degrees are not dense. This answers a question posed by Cooper.

We examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b' = 0'). This extends Csima's result that every CAD theory has a low prime model. (...) We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d < c and c not low₂ (c" > 0"), then for any CAD theory T, there exists a c.e. degree b with d < b < c such that T has a prime model of degree b, where b can be chosen so that b' is any degree c.e. in c with d' ≤ b'. As a corollary, we show that for any degree c with 0 < c < 0', every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models. (shrink)

Lachlan observed that any nonzero d.c.e. degree bounds a nonzero c.e. degree. In this paper, we study the c.e. predecessors of d.c.e. degrees, and prove that given a nonzero d.c.e. degree , there is a c.e. degree below and a high d.c.e. degree such that bounds all the c.e. degrees below . This result gives a unified approach to some seemingly unrelated results. In particular, it has the following two known theorems as corollaries: there is a low c.e. (...) degree isolating a high d.c.e. degree [S. Ishmukhametov, G. Wu, Isolation and the high/low hierarchy, Arch. Math. Logic 41 259–266]; there is a high d.c.e. degree bounding no minimal pairs [C.T. Chong, A. Li, Y. Yang, The existence of high nonbounding degrees in the difference hierarchy, Ann. Pure Appl. Logic 138 31–51]. (shrink)

This paper is dedicated to the study of properties of the operations ∪ and ∩ in the upper semilattice of the e-degrees as well as in the interval (c,c') e for any e-degree c.

We prove that a sequence of sets containing representatives of cupping partners for every nonzero ${\Delta^0_2}$ enumeration degree cannot have a ${\Delta^0_2}$ enumeration. We also prove that no subclass of the ${\Sigma^0_2}$ enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to ${\mathbf{0}_e'}$ by a single incomplete ${\Sigma^0_2}$ enumeration degree.

We show that there exist downwards properly $\Sigma _{2}^{0}$ (in fact noncuppable) e-degrees that are not high. We also show that every high e-degree bounds a noncuppable e-degree.

This paper continues the investigation into the relationship between good approximations and jump inversion initiated by Griffith. Firstly it is shown that there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^{0}_{2}}$$\end{document} set A whose enumeration degree a is bad—i.e. such that no set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X \in a}$$\end{document} is good approximable—and whose complement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{A}}$$\end{document} has lowest possible jump, in other (...) words is low2. This also ensures that the degrees y ≤ a only contain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{3}}$$\end{document} sets and thus yields a tight lower bound for the complexity of both a set of bad enumeration degree, and of its complement, in terms of the high/low jump hierarchy. Extending the author’s previous characterisation of the double jump of good approximable sets, the triple jump of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{0}_{2}}$$\end{document} set A is characterised in terms of the index set of coinfinite sets enumeration reducible to A. The paper concludes by using Griffith’s jump interpolation technique to show that there exists a high quasiminimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{2}}$$\end{document} enumeration degree. (shrink)

Ambos-Spies, K. and R.A. Shore, Undecidability and 1-types in the recursively enumerable degrees, Annals of Pure and Applied Logic 63 3–37. We show that the theory of the partial ordering of recursively enumerable Turing degrees is undecidable and has uncountably many 1-types. In contrast to the original proof of the former which used a very complicated O''' argument our proof proceeds by a much simpler infinite injury argument. Moreover, it combines with the permitting technique to get similar results (...) for any ideal of the r.e. degrees. (shrink)

We prove that there exists a nonzero recursively enumerable Turing degree possessing a strong minimal cover.

We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.

Lachlan [9] proved that there exists a non-recursive recursively enumerable degree such that every non-recursive r. e. degree below it bounds a minimal pair. In this paper we first prove the dual of this fact. Second, we answer a question of Jockusch by showing that there exists a pair of incomplete r. e. degrees a0, a1 such that for every non-recursive r. e. degree w, there is a pair of incomparable r. e. degrees b0, b2 such that w (...) = b0 V b1 and bi for i = 0, 1. (shrink)

We consider the set of jumps below a Turing degree, given by JB={x':x≤a}, with a focus on the problem: Which recursively enumerable degrees a are uniquely determined by JB? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high2 r.e. degree a is determined by JB, then R cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form (...) presented—by constructing pairs a0, a1 of distinct r.e. degrees such that JB=JB within any possible jump class {x:x'=c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity. (shrink)

We show that there is a limit lemma for enumeration reducibility to 0 e ', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X< e W such that W≤ e X' there is a set A such that X≤ e A< e W and A'=W'. (All jumps (...) are enumeration degree jumps.) The degrees of sets with good approximations include the Σ0 2 degrees and the n-CEA degrees. (shrink)

We show that every nontrivial interval in the recursively enumerable degrees contains an incomparable pair which have an infimum in the recursively enumerable degrees.

We give several new characterizations of the continuous enumeration degrees. The main one proves that an enumeration degree is continuous if and only if it is not half of a nontrivial relativized $\mathcal {K}$ -pair. This leads to a structural dichotomy in the enumeration degrees.

We define a class of finite partial lattices which admit a notion of rank compatible with embedding constructions, and present a necessary and sufficient condition for the embeddability of a finite ranked partial lattice into the computably enumerable degrees.

We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1-types.

The jump operator on the ω-enumeration degrees was introduced in [I.N. Soskov, The ω-enumeration degrees, J. Logic Computat. 17 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of and of the enumeration degrees (...) are isomorphic. In the second part of the paper we study the jumps of the ω-enumeration degrees below . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω-enumeration degrees below which are high n for some n and of the class L of the ω-enumeration degrees below which are low n for some n. (shrink)

We give several new characterizations of the continuous enumeration degrees. The main one proves that an enumeration degree is continuous if and only if it is not half a nontrivial relativized K-pair. This leads to a structural dichotomy in the enumeration degrees.