A minimal degree not realizing least possible jump

1974 ◽  
Vol 39 (3) ◽  
pp. 571-574 ◽  
Author(s):  
Leonard P. Sasso
Keyword(s):  
Lebesgue Measure ◽  
Minimal Degree ◽  
Partial Ordering ◽  
Recursively Enumerable ◽  
Degree Constructions ◽  
A Priority ◽  
Combinatorial Device ◽  
Priority Argument ◽  
New System

The least possible jump for a degree of unsolvability a is its join a ∪ 0′ with 0′. Friedberg [1] showed that each degree b ≥ 0′ is the jump of a degree a realizing least possible jump (i.e., satisfying the equation a′ = a ∪ 0′). Sacks (cf. Stillwell [8]) showed that most (in the sense of Lebesgue measure) degrees realize least possible jump. Nevertheless, degrees not realizing least possible jump are easily found (e.g., any degree b ≥ 0′) even among the degrees <0′ (cf. Shoenfield [5]) and the recursively enumerable (r.e.) degrees (cf. Sacks [3]).A degree is called minimal if it is minimal in the natural partial ordering of degrees excluding least element 0. The existence of minimal degrees <0” was first shown by Spector [7]; Sacks [3] succeeded in replacing 0” by 0′ using a priority argument. Yates [9] asked whether all minimal degrees <0′ realize least possible jump after showing that some do by exhibiting minimal degrees below each r.e. degree. Cooper [2] subsequently showed that each degree b > 0′ is the jump of a minimal degree which, as corollary to his method of proof, realizes least possible jump. We show with the aid of a simple combinatorial device applied to a minimal degree construction in the manner of Spector [7] that not all minimal degrees realize least possible jump. We have observed in conjunction with S. B. Cooper and R. Epstein that the new combinatorial device may also be applied to minimal degree constructions in the manner of Sacks [3], Shoenfield [6] or [4] in order to construct minimal degrees <0′ not realizing least possible jump. This answers Yates' question in the negative. Yates [10], however, has been able to draw this as an immediate corollary of the weaker result by carrying out the proof in his new system of prioric games.

An α-finite injury method of the unbounded type

1976 ◽  
Vol 41 (1) ◽  
pp. 1-17
Author(s):  
C. T. Chong
Keyword(s):  
Fine Structure ◽  
Initial Segment ◽  
Fundamental Importance ◽  
Upper Semilattice ◽  
The Past ◽  
Recursively Enumerable ◽  
Background Material ◽  
A Priority ◽  
Admissible Ordinals ◽  
Priority Argument

Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα, much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

Minimal degrees and the jump operator

1973 ◽  
Vol 38 (2) ◽  
pp. 249-271 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Minimal Degree ◽  
Partial Ordering ◽  
Jump Operator ◽  
Upper Semilattice ◽  
Recursively Enumerable ◽  
Original Number ◽  
Nonzero Degree ◽  
The Relationship ◽  
Degrees Of Unsolvability

The jump a′ of a degree a is defined to be the largest degree recursively enumerable in a in the upper semilattice of degrees of unsolvability. We examine below some of the ways in which the jump operation is related to the partial ordering of the degrees. Fried berg [3] showed that the equation a = x′ is solvable if and only if a ≥ 0′. Sacks [13] showed that we can find a solution of a = x′ which is ≤ 0′ (and in fact is r.e.) if and only if a ≥ 0′ and is r.e. in 0′. Spector [16] constructed a minimal degree and Sacks [13] constructed one ≤ 0′. So far the only result concerning the relationship between minimal degrees and the jump operator is one due to Yates [17] who showed that there is a minimal predecessor for each non-recursive r.e. degree, and hence that there is a minimal degree with jump 0′. In §1, we obtain an analogue of Friedberg's theorem by constructing a minimal degree solution for a = x′ whenever a ≥ 0′. We incorporate Friedberg5s original number-theoretic device with a complicated sequence of approximations to the nest of trees necessary for the construction of a minimal degree. The proof of Theorem 1 is a revision of an earlier, shorter presentation, and incorporates many additions and modifications suggested by R. Epstein. In §2, we show that any hope for a result analogous to that of Sacks on the jumps of r.e. degrees cannot be fulfilled since 0″ is not the jump of any minimal degree below 0′. We use a characterization of the degrees below 0′ with jump 0″ similar to that found for r.e. degrees with jump 0′ by R. W. Robinson [12]. Finally, in §3, we give a proof that every degree a ≤ 0′ with a′ = 0″ has a minimal predecessor. Yates [17] has already shown that every nonzero r.e. degree has a minimal predecessor, but that there is a nonzero degree ≤ 0′ with no minimal predecessor (see [18]; or for the original unrelativized result see [10] or [4]).

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.

OVERCOMING POVERTY AS A PRIORITY IN MANAGING THE COUNTRY'S SOCIAL DEVELOPMENT

2021 ◽  
pp. 38-45
Author(s):  
Lyudmila L. Kalinina ◽  
Keyword(s):  
Economic Development ◽  
Social Development ◽  
Human Values ◽  
Social Contracts ◽  
Self Sufficiency ◽  
Socio Economic Development ◽  
Economic Reality ◽  
A Priority ◽  
New System

The article considers the poverty issue as that of a clear threat to the successful socio-economic development of Russia. It gives an assessment of measures aimed at overcoming and shows the role of social contracts not only as an opportunity for the formation of new sources of income for self-sufficiency, but also as a basis for the formation of a new system of human values, the ability to determine his own life in the new economic reality. The article also identifies the ways of solving that issue in the conditions of increased risks in the context of overcoming the consequences of coronavirus.

scholarly journals Uniform almost everywhere domination

2006 ◽  
Vol 71 (3) ◽  
pp. 1057-1072 ◽  
Author(s):  
Peter Cholak ◽  
Noam Greenberg ◽  
Joseph S. Miller
Keyword(s):  
Lebesgue Measure ◽  
Almost Everywhere ◽  
A Priority ◽  
Almost All

AbstractWe explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

On a question of G. E. Sacks

1966 ◽  
Vol 31 (1) ◽  
pp. 66-69 ◽  
Author(s):  
Donald A. Martin
Keyword(s):  
Enumerable Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree ◽  
Image Position ◽  
Combinatorial Device

In [1], p. 171, Sacks asks (question (Q5)) whether there is a recursively enumerable degree of unsolvability d such that for all n ≧ 0. Sacks points out that the set of conditions which d must satisfy is not arithmetical. For this reason he suggests that a proof of (Q5) might require some new combinatorial device. The purpose of this note is to show how (Q5) may be proved simply by extending the methods of [l].2

Σn sets which are Δn-incomparable (uniformly)

1974 ◽  
Vol 39 (2) ◽  
pp. 295-304 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
The Other ◽  
Direct Generalization ◽  
Natural Notion ◽  
A Priority ◽  
Definition Of ◽  
Admissible Ordinals ◽  
Priority Argument

In this paper we will present an application of generalized recursion theory to (noncombinatorial) set theory. More precisely we will combine a priority argument in α-recursion theory with a forcing construction to prove a theorem about the interdefinability of certain subsets of admissible ordinals.Our investigation was prompted by G. Sacks and S. Simpson asking [6] if it is obvious that there are, for each Σn-admissible α, Σn (over Lα) subsets of α which are Δn-incomparable. If one understands “B is Δn in C” to mean that there are Σn/Lα reduction procedures which put out B and when one feeds in C, then the answer is an unqualified “yes.” In this sense “Δn in” is a direct generalization of “α-recursive in” (replace Σ1 by Σn in the definition) and so amenable to the methods of [7, §§3, 5]. Indeed one simply chooses a complete Σn−1 set A and mimics the construction of [6] as modified in [7, §5] to produce two α-A-r.e. sets B and C neither of which is α-A-recursive in the other. By the remarks on translation [7, §3] this will immediately give the desired result for this definition of “Δn in.”There is, however, the more obvious and natural notion of “Δn in” to be considered: B is Δn in C iff there are Σn and Πn formulas of ⟨Lα, C⟩ which define B.

A minimal pair of Π10 classes

1971 ◽  
Vol 36 (1) ◽  
pp. 66-78 ◽  
Author(s):  
Carl G. Jockusch ◽  
Robert I. Soare
Keyword(s):  
Peano Arithmetic ◽  
Minimal Degree ◽  
Minimal Pair ◽  
Turing Degrees ◽  
Complete Extension ◽  
Recursively Enumerable

A pair of sets (A0, A1) forms a minimal pair if A0 and A1 are nonrecursive, and if whenever a set B is recursive both in A0 and in A1 then B is recursive. C. E. M. Yates [8] and independently A. H. Lachlan [4] proved the existence of a minima] pair of recursively enumerable (r.e.) sets thereby establishing a conjecture of G. E. Sacks [6]. We simplify Lachlan's construction, and then generalize this result by constructing two disjoint pairs of r.e. sets (A0, B0) and (A1B1) such that if C0 separates (A0, A1 and C1 separates (B0, B1), then C0 and C1 form a minimal pair. (We say that C separates (A0, A1) if A0 ⊂ C and C ∩ = ∅.) The question arose in our study of (Turing) degrees of members of certain classes, where we proved the weaker result [2, Theorem 4.1] that the above pairs may be chosen so that C0 and C2 are merely Turing incomparable. (There we used a variation of the weaker result to improve a result of Scott and Tennenbaum that no complete extension of Peano arithmetic has minimal degree.)

scholarly journals Rola NATO w regulacji światowego porządku. Wyzwania i zagrożenia dla bezpieczeństwa w XXI wieku

2019 ◽  
Vol 4 ◽  
pp. 71-83
Author(s):  
Raul Andrzej Kosta
Keyword(s):  
World Order ◽  
Muslim Community ◽  
Wrong Decision ◽  
Regional Systems ◽  
The World ◽  
A Priority ◽  
The One ◽  
New System

Wojny w Afganistanie czy Iraku to etapy większego zjawiska, którym jest wojna z terroryzmem. Wymaga ona cierpliwości, porozumienia i dialogu międzycywilizacyjnego, lecz nie wąsko pojętego w ujęciu Huntingtona, tylko szeroko rozumianego dialogu cywilizacji ludzkiej. Każda wojna jest do wygrania, jeżeli priorytetem w niej nie jest dążenie do hegemonii w świecie, ale do bezpieczeństwa własnych obywateli. Dlatego by możliwe było stworzenie nowego systemu bezpieczeństwa światowego, który poprzez sieć systemów regionalnych utworzyłby pewnego rodzaju zaporę chroniącą przed terrorystami, równocześnie ze stabilizacją należy prowadzić dialog ze społecznością muzułmańską i nie tylko. Stworzenie takiego systemu musi być priorytetem dla polityków i państw, które reprezentują. Polityk jak każdy człowiek ma prawo do błędnej decyzji. Jednakże przy jej podejmowaniu musi liczyć się z konsekwencjami. Tym większymi, im ranga decyzji jest wyższa, albowiem decyzja o wojnie jest jedną z najbardziej brzemiennych w skutki. NATO’s role in regulating the world order. Challenges and threats to security in the 21st centuryThe wars in Afghanistan or in Iraq are just stages of a wider phenomenon of war against terrorism. War requires patience, understanding, dialogue within a civilization, but not the one narrowly understood as Huntington sees it, but general human civilization. Each war can be won if its priority is not to try to conquer the world but to win security for the inhabitants of a state. Therefore, stabilization should be accompanied by dialogue with the Muslim community, but not only with that community if a new system of world security is to be created. A network of regional systems that would form a kind of anti-terrorist dam should be created. The appropriate formation of such a system must be a priority for politicians and the states they represent. Like any other man, a politician can sometimes make a wrong decision. However, he or she must be aware of its consequences when taking it. The more serious the consequences, the higher the rank of the decision. The decision to wage war is, however, the most burdened with consequences of all decisions.

Jump embeddings in the Turing degrees

1991 ◽  
Vol 56 (2) ◽  
pp. 563-591 ◽  
Author(s):  
Peter G. Hinman ◽  
Theodore A. Slaman
Keyword(s):  
Partial Ordering ◽  
Turing Degrees ◽  
First Order ◽  
Upper Semilattice ◽  
Recursively Enumerable ◽  
Locally Countable ◽  
Image Set ◽  
Turing Reducibility ◽  
Image Position ◽  
Order Structures

Since its introduction in [K1-Po], the upper semilattice of Turing degrees has been an object of fascination to practitioners of the recursion-theoretic art. Starting from relatively simple concepts and definitions, it has turned out to be a structure of enormous complexity and richness. This paper is a contribution to the ongoing study of this structure.Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if {P, ≤P) is a partial ordering of cardinality at most ℵ1 which is locally countable—each point has at most countably many predecessors—then there is an embeddingwhere D is the set of all Turing degrees and <T is Turing reducibility. If (P, ≤P) is a countable partial ordering, then the image of the embedding may be taken to be a subset of R, the set of recursively enumerable degrees. Without attempting to make the notion completely precise, we shall call embeddings of the first sort global, in contrast to local embeddings which impose some restrictions on the image set.

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