Constructive set theory

1975 ◽  
Vol 40 (3) ◽  
pp. 347-382 ◽  
Author(s):  
John Myhill
Keyword(s):  
Set Theory ◽  
Mathematical Practice ◽  
Partial Function ◽  
Advanced Mathematics ◽  
The Third ◽  
Formal Systems ◽  
Constructive Set Theory ◽  
Definition Of

This paper is the third in a series collectively entitled Formal systems of intuitionistic analysis. The first two are [4] and [5] in the bibliography; in them I attempted to codify Brouwer's mathematical practice. In the present paper, which is independent of [4] and [5], I shall do the same for Bishop's book [1]. There is a widespread current impression, due partly to Bishop himself (see [2]) and partly to Goodman and the author (see [3]) that the theory of Gödel functionals, with quantifiers and choice, is the appropriate formalism for [1]. That this is not so is seen as soon as one really tries to formalize the mathematics of [1] in detail. Even so simple a matter as the definition of the partial function 1/x on the nonzero reals is quite a headache, unless one is prepared either to distinguish nonzero reals from reals (a nonzero real being a pair consisting of a real x and an integer n with ∣x∣ > 1/n) or, to take the Dialectica interpretation seriously, by adjoining to the Gödel system an axiom saying that every formula is equivalent to its Dialectica interpretation. (See [1, p. 19], [2, pp. 57–60] respectively for these two methods.) In more advanced mathematics the complexities become intolerable.

Relevance and paraconsistency—a new approach

1990 ◽  
Vol 55 (2) ◽  
pp. 707-732 ◽  
Author(s):  
Arnon Avron
Keyword(s):  
Set Theory ◽  
Elementary Theory ◽  
Basic Relation ◽  
Binary Relations ◽  
New Approach ◽  
Von Neumann ◽  
The Third ◽  
Formal Systems ◽  
Levels Of Reality ◽  
Intuitive Idea

In this work we describe a new approach to the notions of relevance and paraconsistency. Unlike the works of Anderson and Belnap or da Costa (see [2], [8] and [7]) we shall mainly be guided in it by semantical intuitions. In the first two sections we introduce and investigate the algebraic structures that reflect those intuitions. The corresponding formal systems are briefly described in the third section (a more detailed treatment of these systems, including full proofs, will be given in another paper).Our basic intuitive idea is that of “domains of discourse” or “relevance domains”. Classical logic, so we think, is valid in as much as sentences get values inside one domain; limitations on its use can be imposed only with respect to inferences in which more than one domain is involved. There are two basic binary relations over the collection of domains. One is relevance. It is reflexive and symmetric (but not necessarily transitive). Under a given interpretation two sentences are relevant to each other when their values are in relevant domains. Another basic relation between domains, no less important, is that of grading according to “degrees of reality”. The idea behind it is not new. Gentzen, for example, divided in [9] the world of mathematics into three grades, representing three “levels of reality”. The elementary theory of numbers has the highest degree or level of reality; set theory has the smallest degree and mathematical analysis occupies the intermediate level. In the theory of types, or in the accumulative von Neumann universe for set theory, we can find indication of a richer hierarchy.

An intuitionistic proof of Tychonoff's theorem

1992 ◽  
Vol 57 (1) ◽  
pp. 28-32 ◽  
Author(s):  
Thierry Coquand
Keyword(s):  
Set Theory ◽  
Free Space ◽  
Direct Proof ◽  
Axiom Of Choice ◽  
Inductive Definition ◽  
Compact Spaces ◽  
Inductive Definitions ◽  
Constructive Set Theory ◽  
Definition Of ◽  
The Axiom Of Choice

Tychonoff's theorem states that a product of compact spaces is compact. In [3], P. Johnstone presents a proof of Tychonoff's theorem in a “localic” framework. The surprise is that the point-free formulation of Tychonoff's theorem is provable without the axiom of choice, whereas in the usual formulation it is equivalent to the axiom of choice (see Kelley [5]).The proof given in [3], however, is classical and seems to use the replacement axiom of Zermelo-Fraenkel. The aim of this paper is to present what we believe to be a more direct proof, which is intuitionistic and can be proved using as primitive only the notion of inductive definition, as it is for instance presented in Martin-Löf [6]. One main point of the paper is to show that the theory of locales can be developed rather naturally in the framework of inductive definitions. We think that our arguments can be presented in the constructive set theory of Aczel [2].The paper is organized as follows. In §1 we show the argument in the case of a product of two spaces. This proof has a direct generalisation to the case of a product over a set with a decidable equality.§1. Product of two spaces. We first recall a possible definition of a point-free space (see Johnstone [3] or Vickers [10]). It is a poset (X, ≤), together with a meet operation ab, written multiplicatively, and, for each a ∈ X, a set Cov(a) of subsets of {x ∈ X ∣ x ≤ a}. We ask that if M ∈ Cov(a) and b ≤ a, then {bs ∈ s ∈ M} ∈ Cov(b). This property of Cov will be called the axiom of covering. The elements of Cov(u) are called basic covers of u ∈ X.

ABOUT THE PROBLEM OF CLASSIFICATION OF THE CONCEPTS OF INFORMATICS STUDIED AT SECONDARY SCHOOL

2018 ◽  
pp. 4-7
Author(s):  
S. I. Zenko
Keyword(s):  
Secondary School ◽  
Foreign Language ◽  
Computer Science ◽  
Parts Of Speech ◽  
The Third ◽  
The Cross ◽  
Definition Of

The article raises the problem of classification of the concepts of computer science and informatics studied at secondary school. The efficiency of creation of techniques of training of pupils in these concepts depends on its solution. The author proposes to consider classifications of the concepts of school informatics from four positions: on the cross-subject basis, the content lines of the educational subject "Informatics", the logical and structural interrelations and interactions of the studied concepts, the etymology of foreign-language and translated words in the definition of the concepts of informatics. As a result of the first classification general and special concepts are allocated; the second classification — inter-content and intra-content concepts; the third classification — stable (steady), expanding, key and auxiliary concepts; the fourth classification — concepts-nouns, conceptsverbs, concepts-adjectives and concepts — combinations of parts of speech.

(A review). BRAUN-BLANQUETIA. Recueil de travaux de geobotanique / review of geobotanical monographs. Vol. 46. Centenaire de la phytosociologie. CAMERINO, 2010. 423 p.

2011 ◽  
pp. 143-147
Author(s):  
L. G. Naumova ◽  
V. B. Martynenko ◽  
S. M. Yamalov
Keyword(s):  
Plant Association ◽  
Northern Asia ◽  
The Third ◽  
International Botanical ◽  
International Botanical Congress ◽  
History Of ◽  
Definition Of

Date of «birth» of phytosociology (phytocenology) is considered to be 1910, when at the third International Botanical Congress in Brussels adopted the definition of plant association in the wording Including Flaó and K. Schröter (Flahault, Schröter, 1910; Alexandrov, 1969). The centenary of this momentous event in the history of phytocenology devoted to the 46th edition of the Yearbook «Braun-Blanquetia», which began to emerge in 1984 in Camerino (Italy) and it has a task to publish large geobotanical works. During the years of the publication of the Yearbook on its pages were published twice work of the Russian scientists — «The steppes of Mongolia» (Z. V. Karamysheva, V. N. Khramtsov. Vol. 17. 1995), and «Classification of continental hemiboreal forests of Northern Asia» (N. B. Ermakov in collaboration with English colleagues and J. Dring, J. Rodwell. Vol. 28. 2000).

Characteristics of the Metaphorical Image in the Poetry of Al-Sharif Al-Radhi

2019 ◽  
Vol 16 (2) ◽  
Author(s):  
Al-KhaierAmer Abdul Kareem
Keyword(s):  
The Third ◽  
Definition Of

Abstract The research started with an introduction containing the statement of the problem. The study was divided into four parts: a preamble and three sections. The preamble involved a definition of the metaphorical image and its importance. The first section covered the sources of the metaphorical image, the second dealt with the types of the metaphorical image, and the third discussed the functions of metaphorical representation as well as the main tools that contributed to the construction and formation of the metaphorical image. Finally, the study ended with a conclusion comprising the most significant findings of this research. keyword: metaphorical image, AL-Sharif Al-Radi.

Ildar Garipzanov, Caroline Goodson, and Henry Maguire, ed., Graphic Signs of Identity, Faith, and Power in Late Antiquity and the Early Middle Ages. Cursor Mundi, 27, Turnhout: Brepols, 2017, xviii, 392 p., 141 ill.

Mediaevistik ◽  
2020 ◽  
Vol 32 (1) ◽  
pp. 377-379
Author(s):  
Kriszta Kotsis
Keyword(s):  
Middle Ages ◽  
Interdisciplinary Research ◽  
Late Antiquity ◽  
Visual Communication ◽  
Group Identity ◽  
Religious Affiliation ◽  
Late Antique ◽  
The Third ◽  
Cursor Mundi ◽  
Definition Of

Late antique and early medieval graphic signs have traditionally been studied by narrowly focused specialists leading to the fragmentation and decontextualization of this important body of material. Therefore, the volume aims “to deepen interdisciplinary research on graphic signs” (7) of the third through tenth centuries, with contributions from archaeologists, historians, art historians, a philologist, and a paleographer. Ildar Garipzanov’s introduction defines the central terms (sign, symbol, graphicacy), calls for supplanting the text-image binary with “the concept of the visual-written continuum” (15), and argues that graphicacy was central to visual communication in this period. He emphasizes the agency of graphic signs and notes that their study can amplify our understanding of the definition of personal and group identity, the articulation of power, authority, and religious affiliation, and communication with the supernatural sphere.

The Limit Situation And The Narrative Of War (On The Material Of The Novel “My Lieutenant” By Daniil Granin)

2020 ◽  
Vol 6 (1) ◽  
pp. 90-104
Author(s):  
Anatoly S. Kuprin ◽  
Galina I. Danilina
Keyword(s):  
20Th Century ◽  
Scientific Work ◽  
Point Of View ◽  
Continuous Limit ◽  
The Novel ◽  
Term Limit ◽  
Abstract Author ◽  
The Third ◽  
Definition Of

The purpose of this study is the analysis of limit situation in the narrative of war. The material of the study is the novel of Daniil Granin “My Lieutenant” and related texts. In the first part of the paper, the authors explore existing approaches to the term “limit situation” and similar concepts into scientific and philosophical traditions; limits of its applicability in literary studies and its relation to the categories of “narrative instances” and “event”. Proposed a literary-theoretical definition of the limit situation, which can be used in the analysis of fiction texts. Existing approaches to the examination of the situation of war are analyzed: philosophical-existential, psychoanalytic, sociological, literary. In the second part of the paper, the authors propose their method for analyzing limit situations in texts about war, which basis on existing approaches and preserves the text-centric principle of studying the structure of the story. Two interrelated areas of research have been identified: the study of war as a continuous limit situation in the intertextual aspect (the discourse of war); the study of limit situations (death, suffering, guilt, accident) in the narrative of war as part of a specific text. In the third part of the scientific work,the analysis of war as a continuous limit situation results in the study of the concept of “limit” (border) in a fiction text. The role of “limit” (border) concept in the texts about the war is studied, the possible types of limits in the discourse of war are examined. Limit situations in the narrative of war are analyzed on the basis of the novel “My Lieutenant” by Daniil Granin. A review of journalistic and scientific works about the novel revealed both the continuity and the differences between the novel and the “lieutenant” prose of the 20th century. An analysis of the limit situations in the novel revealed their key position in the narrative. These situations are independent of the fiction time, of the fluctuation of the point of view’; the function of the abstract author is to build the narrative as a “directive” immersion of the hero and narrator in these situations.

Delving into Horror

2019 ◽  
Vol 29 (5) ◽  
pp. 135-150
Keyword(s):  
The Self ◽  
Time And Space ◽  
The Third ◽  
The World ◽  
The Subject ◽  
Perception Of Time ◽  
Definition Of

The springboard for this essay is the author’s encounter with the feeling of horror and her attempts to understand what place horror has in philosophy. The inquiry relies upon Leonid Lipavsky’s “Investigation of Horror” and on various textual plunges into the fanged and clawed (and possibly noumenal) abyss of Nick Land’s work. Various experiences of horror are examined in order to build something of a typology, while also distilling the elements characteristic of the experience of horror in general. The essay’s overall hypothesis is that horror arises from a disruption of the usual ways of determining the boundaries between external things and the self, and this leads to a distinction between three subtypes of horror. In the first subtype, horror begins with the indeterminacy at the boundaries of things, a confrontation with something that defeats attempts to define it and thereby calls into question the definition of the self. In the second subtype, horror springs from the inability to determine one’s own boundaries, a process opposed by the crushing determinacy of the world. In the third subtype, horror unfolds by means of a substitution of one determinacy by another which is unexpected and ungrounded. In all three subtypes of horror, the disturbance of determinacy deprives the subject, the thinking entity, of its customary foundation for thought, and even of an explanation of how that foundation was lost; at times this can lead to impairment of the perception of time and space. Understood this way, horror comes within a hair’s breadth of madness - and may well cross over into it.

After Chalcedon

2018 ◽  
Author(s):  
Brian E. Daley, SJ
Keyword(s):  
Eighth Century ◽  
Seventh Century ◽  
The Third ◽  
John Of Damascus ◽  
Perceived Impact ◽  
Orthodox Theology ◽  
Definition Of ◽  
Maximus The Confessor ◽  
Severus Of Antioch

The Council of Chalcedon’s definition of the terms in which Nicene orthodoxy should conceive of Christ’s person remained controversial. Leontius of Byzantium argued for the correctness of the Council’s formulation, especially against the arguments of Severus of Antioch, but suggested that more than academic issues were at stake: the debate concerned the lived, permanently dialectical unity between God and humanity. In the mid-seventh century, imperially sponsored efforts to lessen the perceived impact of Chalcedonian language by stressing that Christ’s two natures were activated by “a single, theandric energy,” also remained without effect: largely because of the monk Maximus “the Confessor”, who argued that two complete spheres of activity and two wills remained evident in Christ’s life. Maximus’s position was ratified at the Lateran Synod and at the Third Council of Constantinople. The eighth-century Palestinian monk John of Damascus incorporated these arguments into his own influential synthesis of orthodox theology.

Covert Action: United States Law in Substance, Process, and Practice

2010 ◽  
pp. 586-607 ◽  
Author(s):  
James E. Baker
Keyword(s):  
National Security ◽  
Executive Branch ◽  
Legal Practice ◽  
Legal Regime ◽  
Covert Action ◽  
The Third ◽  
Definition Of ◽  
Traditional Activity ◽  
The U.S

This article discusses covert action within the context of the U.S. law. The first section describes the main elements of the U.S. legal regime, including the definition of covert action and the “traditional activity” exceptions, the elements of a covert action finding, and the thresholds and requirements for congressional notification. The second section describes some of the significant limitations on the conduct of covert action. The third section discusses the nature of executive branch legal practice in this area of the law. And the last section draws conclusions about the role of national security law within the context of covert action.

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