Σ2 Induction and infinite injury priority argument, Part I: Maximal sets and the jump operator

1998 ◽  
Vol 63 (3) ◽  
pp. 797-814 ◽  
Author(s):  
C. T. Chong ◽  
Yue Yang
Keyword(s):  
Recursion Theory ◽  
Central Position ◽  
Jump Operator ◽  
Concerted Effort ◽  
Injury Type ◽  
Recursively Enumerable ◽  
The Subject ◽  
Intensive Development ◽  
Priority Argument ◽  
Fragments Of Peano Arithmetic

The study of recursion theory on models of fragments of Peano arithmetic has hitherto been concentrated on recursively enumerable (r.e.) sets and their degrees (with a few exceptions, such as that in [2] on minimal degrees). The reason for such a concerted effort is clear: priority arguments have occupied a central position in post Friedberg-Muchnik recursion theory, and after almost forty years of intensive development in the subject, they are still the essential tools on which investigations of r.e. sets and their degrees depend. There are two possible approaches to the study within fragments of arithmetic: To give a general analysis of strategies, and identify their proof-theoretic strengths (for example in [6] on infinite injury priority methods), or to consider specific theorems in recursion theory, and, if possible, pinpoint the exact levels of induction provably equivalent to the theorems. The work reported in this paper belongs to the second approach. More precisely, we single out two infinitary injury type constructions of r.e. sets—one concerning maximal sets and the other based on the notion of the jump operator—to be the topics of study.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Friedberg Numbering in Fragments of Peano Arithmetic and α-Recursion Theory

2013 ◽  
Vol 78 (4) ◽  
pp. 1135-1163 ◽  
Author(s):  
Wei Li
Keyword(s):  
Recursion Theory ◽  
Peano Arithmetic ◽  
Fragments Of Peano Arithmetic

AbstractIn this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel's constructible hierarchy Lα, where α is Σ1 admissible. We prove that(1) Over P− + BΣ2, the existence of a Friedberg numbering is equivalent to IΣ2, and(2) For Lα, there is a Friedberg numbering if and only if the tame Σ2 projectum of α equals the Σ2 cofinality of α.

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.

scholarly journals Some Remarks in Defense of Immanuel Kant's Theory of Experience

2021 ◽  
Vol 2 (2) ◽  
pp. 0
Author(s):  
Balanovskiy Valentin
Keyword(s):  
Immanuel Kant ◽  
Scientific Information ◽  
Experiential Knowledge ◽  
Central Position ◽  
Pure Reason ◽  
Contemporary Philosophy ◽  
Critique Of Pure Reason ◽  
The Subject ◽  
Reliable Knowledge ◽  
Conceptual Nature

The author attempts to answer a question of whether the fact that Immanuel Kant’s theory of experience most likely has a conceptual nature decreases an importance of Kant’s ideas for contemporary philosophy, because if experience is conceptual by nature, then certain problems with the search for means to verify experiential knowledge arise. In particular, two approaches are proposed. According to the first approach, the exceptional conceptuality of Kant’s theory of experience may be a consequence of absence of some important chains in arguments contained in the Critique of Pure Reason, which could clarify a question of how the conceptual apparatus of the subject corresponds to the reality. The author puts a hypothesis that the missing chains are not a mistake, but Kant’s deliberate silence caused by the lack of accurate scientific information that could not have been available to humankind in Enlightenment epoch. According to the second approach even if Kant’s theory of experience is exclusively conceptual by nature, this cannot automatically lead to a conclusion that it is unsuitable for obtaining reliable knowledge about reality, since transcendental idealism has powerful internal tools for verifying data in the process of cognition. The central position among them is occupied by transcendental reflection.

scholarly journals O możliwym spojrzeniu na supersystem rozrywkowy z perspektywy wykorzystania wybranych osiągnięć językoznawstwa. Propozycja metodologiczna

2019 ◽  
Vol 24 ◽  
pp. 329-356
Author(s):  
Adam Mazurkiewicz
Keyword(s):  
Social Communication ◽  
Central Position ◽  
Language System ◽  
Transmedia Storytelling ◽  
System A ◽  
Symbolic Meanings ◽  
Integral Element ◽  
The Subject ◽  
Object Of Study ◽  
Time Culture

On the possible analysis of the entertainment supersystem from the perspective of employing selected achievements of linguistics: A methodological approachCurrently linguistics treats the subject of its study not so much as a tool for social communication, but as an integral element of culture. At the same time, culture is perceived as a system of symbolic meanings. However, the same position is occupied by “entertainment supersystems” whose role is transmedia storytelling. From the perspective of semiotics they are — just as the language system — a sign. Thus, perhaps employing the descriptive instrumentarium of language mechanisms, due to its peculiar character, will allow for a more adequate consideration of cultural phenomena in this case the entertainment supersystem than applying this methodology outside humanities. What is more, a transfer of focus from an ontological perspective seeking to answer the question of what an entertainment supersystem is or is not to an epistemological one an attempt to understand how it functions in society, that is, “how it is used” seems to be compliant with the transition from linguistic structuralism to the post-structuralistic paradigm. At the same time, considering methodological implications which derive from the analysis of mechanisms regulating the functioning of entertainment supersystems by means typical for the linguistics instrumentarium, one can easily reach the conviction that text as an object of study has been reinstated in its central position.

Hydraulic characteristics and calculation of the innovative systems of surface runoff disposal

2021 ◽  
pp. 46-51
Author(s):  
Ю.В. Брянская ◽  
А.Э. Тен ◽  
Н.Т. Джумагулова ◽  
Г.Н. Громов
Keyword(s):  
Cross Sections ◽  
Surface Runoff ◽  
Experimental Studies ◽  
Drainage System ◽  
Anthropogenic Pollution ◽  
Hydraulic Characteristics ◽  
Regulatory Requirements ◽  
The Subject ◽  
Intensive Development ◽  
Selection Of

В условиях интенсивного развития новых отечественных и зарубежных технологий, материалов и оборудования, применяемых для защиты окружающей природной среды от загрязнений техногенного происхождения, особую актуальность приобретают разработки новых систем отвода и очистки поверхностных сточных вод. Эти системы позволяют использовать последние достижения отраслевой науки и оптимизировать алгоритм выполнения операций и практических приемов их гидравлического расчета. Примером является инновационная система отвода поверхностных сточных вод АСО Qmax, которая относится к открытой системе каналов (лотков) для сбора и отведения поверхностных сточных вод, формирующихся при выпадении атмосферных осадков. Однако широкому применению данного вида конструкций в России препятствует отсутствие методики их гидравлического расчета, в том числе таблиц для подбора сечений (диаметров) каналов, которая бы удовлетворяла требованиям российской нормативно-методической базы проектирования систем отведения поверхностных сточных вод. В этой связи предметом данной статьи явилась оценка гидравлических характеристик трубопроводов, каналов (лотков) системы водоотвода АСО Qmax. Приведены результаты теоретических и экспериментальных исследований гидравлических характеристик системы АСО Qmaxс учетом адаптации для российских условий и нормативных требований, а также обоснование рекомендуемых параметров для их использования. In the context of the intensive development of new domestic and foreign technologies, materials and equipment used to protect the environment from anthropogenic pollution, the development of advanced systems for surface runoff removal and treatment is of special actuality. These systems provide for using the latest achievements of the sectoral science and optimizing the algorithm for performing operations and practical methods for the hydraulic calculations. An example of the innovative surface runoff disposal system is ASO Qmax, that refers to an open system of channels for the collection and disposal of surface runoff formed during precipitation. However, the widespread use of these facilities in Russia is hampered by the lack of a method for the hydraulic calculations, including tables for the selection of cross-sections (diameters) of channels that meet the requirements of the Russian guidelines and regulations for the design of surface runoff disposal systems. In this regard, the subject of this paper is the estimation of the hydraulic characteristics of pipelines, channels of ASO Qmax drainage system. The results of theoretical and experimental studies of the hydraulic characteristics of ASO Qmax system with account of the adaptation for the Russian conditions and regulatory requirements, as well as the justification of the recommended parameters for their use are presented.

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]).

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (3) ◽  
pp. 681-694
Author(s):  
Anne Leggett ◽  
Richard A. Shore
Keyword(s):  
Lattice Theory ◽  
Lattice Structure ◽  
Sufficient Conditions ◽  
Recursion Theory ◽  
Recursively Enumerable ◽  
Countable Ordinal ◽  
Congruence Relations ◽  
General Program ◽  
Simple Sets

One general program of α-recursion theory is to determine as much as possible of the lattice structure of (α), the lattice of α-r.e. sets under inclusion. It is hoped that structure results will shed some light on whether or not the theory of (α) is decidable with respect to a suitable language for lattice theory. Fix such a language ℒ.Many of the basic results about the lattice structure involve various sorts of simple α-r.e. sets (we use definitions which are definable in ℒ over (α)). It is easy to see that simple sets exist for all admissible α. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple α-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal α-r.e. sets. Lerman [6] has shown that maximal α-r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of (α). Here various sorts of simple sets have also proved to be vital tools. The importance of simple α-r.e. sets to the study of the lattice structure of (α) is hence obvious.Lerman [6, Q22] has posed the following problem: Find an admissible α for which all simple α-r.e. sets have the same 1-type with respect to the language ℒ. The structure of (α) for such an α would be much less complicated than that of (ω). Lerman [7] showed that such an α could not be a regular cardinal of L. We show that there is no such admissible α.

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
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