The Sacks density theorem and Σ2-bounding

1996 ◽  
Vol 61 (2) ◽  
pp. 450-467 ◽  
Author(s):  
Marcia J. Groszek ◽  
Michael E. Mytilinaios ◽  
Theodore A. Slaman
Keyword(s):  
Recursion Theory ◽  
Density Theorem ◽  
Turing Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Models Of Arithmetic

AbstractThe Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P− + BΣ2. The proof has two components: a lemma that in any model of P− + BΣ2, if B is recursively enumerable and incomplete then IΣ1 holds relative to B and an adaptation of Shore's [9] blocking technique in α-recursion theory to models of arithmetic.

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

1971 ◽  
Vol 36 (2) ◽  
pp. 193-215 ◽  
Author(s):  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Elementary Equivalence ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Nonstandard Models ◽  
Relational Systems ◽  
The Relationship ◽  
Models Of Arithmetic

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense

1984 ◽  
Vol 49 (2) ◽  
pp. 503-513 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Density Theorem ◽  
Turing Degrees ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursive Sequences ◽  
Recursively Enumerable Sets ◽  
Finite Set ◽  
Image Position ◽  
Density Problem

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ withProof. We enumerate an e-operator Θ so as to satisfy the list of conditions:Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

A representation of recursively enumerable sets through Horn formulas in higher recursion theory

2016 ◽  
Vol 73 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Juan A. Nido Valencia ◽  
Julio E. Solís Daun ◽  
Luis M. Villegas Silva
Keyword(s):  
Recursion Theory ◽  
Recursively Enumerable ◽  
Horn Formulas ◽  
Recursively Enumerable Sets

1-genericity in the enumeration degrees

1988 ◽  
Vol 53 (3) ◽  
pp. 878-887 ◽  
Author(s):  
Kate Copestake
Keyword(s):  
Partial Function ◽  
Partial Ordering ◽  
Turing Degree ◽  
Turing Degrees ◽  
Original Formulation ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Generic Set

The structure of the Turing degrees of generic and n-generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]).In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree.Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree ae which is minimal-like; that is, all functions in degrees below ae have partial recursive extensions.

scholarly journals On Transfinite Levels of the Ershov Hierarchy

2021 ◽  
Vol 27 (2) ◽  
pp. 220-221
Author(s):  
Cheng Peng
Keyword(s):  
Recursion Theory ◽  
Density Theorem ◽  
Turing Degrees ◽  
Elementary Substructure ◽  
Ordered Structures ◽  
Research Areas ◽  
Ershov Hierarchy ◽  
Sigma 1 ◽  
E Mail

AbstractIn this thesis, we study Turing degrees in the context of classical recursion theory. What we are interested in is the partially ordered structures $\mathcal {D}_{\alpha }$ for ordinals $\alpha <\omega ^2$ and $\mathcal {D}_{a}$ for notations $a\in \mathcal {O}$ with $|a|_{o}\geq \omega ^2$ .The dissertation is motivated by the $\Sigma _{1}$ -elementary substructure problem: Can one structure in the following structures $\mathcal {R}\subsetneqq \mathcal {D}_{2}\subsetneqq \dots \subsetneqq \mathcal {D}_{\omega }\subsetneqq \mathcal {D}_{\omega +1}\subsetneqq \dots \subsetneqq \mathcal {D(\leq \textbf {0}')}$ be a $\Sigma _{1}$ -elementary substructure of another? For finite levels of the Ershov hierarchy, Cai, Shore, and Slaman [Journal of Mathematical Logic, vol. 12 (2012), p. 1250005] showed that $\mathcal {D}_{n}\npreceq _{1}\mathcal {D}_{m}$ for any $n < m$ . We consider the problem for transfinite levels of the Ershov hierarchy and show that $\mathcal {D}_{\omega }\npreceq _{1}\mathcal {D}_{\omega +1}$ . The techniques in Chapters 2 and 3 are motivated by two remarkable theorems, Sacks Density Theorem and the d.r.e. Nondensity Theorem.In Chapter 1, we first briefly review the background of the research areas involved in this thesis, and then review some basic definitions and classical theorems. We also summarize our results in Chapter 2 to Chapter 4. In Chapter 2, we show that for any $\omega $ -r.e. set D and r.e. set B with $D<_{T}B$ , there is an $\omega +1$ -r.e. set A such that $D<_{T}A<_{T}B$ . In Chapter 3, we show that for some notation a with $|a|_{o}=\omega ^{2}$ , there is an incomplete $\omega +1$ -r.e. set A such that there are no a-r.e. sets U with $A<_{T}U<_{T}K$ . In Chapter 4, we generalize above results to higher levels (up to $\varepsilon _{0}$ ). We investigate Lachlan sets and minimal degrees on transfinite levels and show that for any notation a, there exists a $\Delta ^{0}_{2}$ -set A such that A is of minimal degree and $A\not \equiv _T U$ for all a-r.e. sets U.Abstract prepared by Cheng Peng.E-mail: [email protected]

Nowhere simple sets and the lattice of recursively enumerable sets

1978 ◽  
Vol 43 (2) ◽  
pp. 322-330 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Classification Scheme ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets ◽  
Invariant Properties

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.

Index sets in Ershov's hierarchy

1974 ◽  
Vol 39 (1) ◽  
pp. 97-104 ◽  
Author(s):  
Jacques Grassin
Keyword(s):  
Recursion Theory ◽  
Case Note ◽  
Boolean Combination ◽  
Open Set ◽  
Recursively Enumerable ◽  
Power Set ◽  
Index Sets ◽  
Recursively Enumerable Sets ◽  
Complete Set ◽  
Recursive Isomorphism

This work is an attempt to characterize the index sets of classes of recursively enumerable sets which are expressible in terms of open sets in the Baire topology on the power set of the set N of natural numbers, usual in recursion theory. Let be a class of subsets of N and be the set of indices of recursively enumerable sets Wх belonging to .A well-known theorem of Rice and Myhill (cf. [5, p. 324, Rice-Shapiro Theorem]) states that is recursively enumerable if and only ifis a r.e. open set. In this case, note that if is not empty and does not contain all recursively enumerable sets, is a complete set. This theorem will be partially extended to classes which are boolean combinations of open sets by the following:(i) There is a canonical boolean combination which represents, namely the shortest among boolean combinations which represent.(ii) The recursive isomorphism type of depends on the length n of this canonical boolean combination (and trivial properties of ); for instance, is recursively isomorphic (in the particular case where is a boolean combination of recursive open sets) to an elementary set combination Yn or Un, constructed from {х ∣ х Wх) and depending on the length n. We can say also that is a complete set in the sense of Ershov's hierarchy [1] (in this particular case).

scholarly journals The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

2005 ◽  
Vol 11 (3) ◽  
pp. 398-410
Author(s):  
Noam Greenberg
Keyword(s):  
Partially Ordered Sets ◽  
Recursion Theory ◽  
Ordered Sets ◽  
First Order ◽  
Recursively Enumerable ◽  
Order Language ◽  
Replacement Scheme ◽  
Admissible Ordinals ◽  
Models Of Arithmetic

AbstractWhen 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 α-finiteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the α-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 natural elementary differences between the structures of α-r.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the α-r.e. degrees is complicated, we get that for every admissible ordinal α, the α-r.e. degrees and the classical r.e. degrees are not elementarily equivalent.

A limit on relative genericity in the recursively enumerable sets

1989 ◽  
Vol 54 (2) ◽  
pp. 376-395 ◽  
Author(s):  
Steffen Lempp ◽  
Theodore A. Slaman
Keyword(s):  
Turing Degrees ◽  
Recursively Enumerable Set ◽  
Recursively Enumerable ◽  
Low Degree ◽  
Recursively Enumerable Sets ◽  
Nonzero Degree

AbstractWork in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X′, its Turing jump, is recursive in ∅′ and high if X′ computes ∅″. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of A ⊕ W is recursive in the jump of A. We prove that there are no deep degrees other than the recursive one.Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turing functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that (W ⊕ A)′ is forced to disagree with Φ(−;A′). The conversion has some ambiguity; in particular, A cannot be found uniformly from W.We also show that there is a “moderately” deep degree: There is a low nonzero degree whose join with any other low degree is not high.

An invariance notion in recursion theory

1982 ◽  
Vol 47 (1) ◽  
pp. 48-66 ◽  
Author(s):  
Robert E. Byerly
Keyword(s):  
Recursion Theory ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Partial Recursive Functions

AbstractA set of gödel numbers is invariant if it is closed under automorphisms of (ω, ·), where ω is the set of all gödel numbers of partial recursive functions and · is application (i.e., n · m ≃ φn(m)). The invariant arithmetic sets are investigated, and the invariant recursively enumerable sets and partial recursive functions are partially characterized.

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