scholarly journals The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information

2020 ◽  
Author(s):  
Vasil Penchev
Keyword(s):  
Set Theory ◽  
Peano Arithmetic ◽  
Theory Of Information ◽  
Relationship Of ◽  
The Relationship

scholarly journals The relationship of arithmetic as two twin Peano arithmetic(s) and set theory: A new glance from the theory of information

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Peano Arithmetic ◽  
Complex Hilbert Space ◽  
New Concepts ◽  
Four Color Theorem ◽  
The World ◽  
Mathematical Problems ◽  
Theory Of Information ◽  
Relationship Of ◽  
The Relationship ◽  
Mathematics And Physics

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested.

scholarly journals Towards a critical theory of information

2009 ◽  
Vol 7 (2) ◽  
pp. 243-292
Author(s):  
Christian Fuchs
Keyword(s):  
Information Theory ◽  
Critical Theory ◽  
Critical Information ◽  
Internet Economy ◽  
Theory Of Information ◽  
And Control ◽  
Relationship Of ◽  
The Relationship ◽  
Nancy Fraser ◽  
Class Play

Critical information theory is an endeavour that focuses ontologically on the analysis of information in the context of domination, asymmetrical power relations, exploitation, oppression, and control by employing epistemologically all theoretical and/or empirical means necessary for doing so in order to contribute at the praxeological level to the establishment of a participatory, co-operative society. Three foundational aspects of a critical theory of information are discussed in this paper: the relation of immanence and transcendence, the relation of base and superstructure, and ideology critique.The logical figure of immanent transcendence is based on the dialectic of essence and existence and poses a viable counterpart to positivistic and postmodern definitions of critique. As an example for the logic of immanent transcendence to critical information theory, a contradiction of the Internet economy is discussed.The debate on redistribution and recognition between critical theorists Nancy Fraser and Axel Honneth gives the opportunity to renew the discussion of the relationship of base and superstructure in critical social theory. Critical information theory needs to be aware of economic, political, and cultural demands that it needs to make in struggles for ending domination and oppression, and of the unifying role that the economy and class play in these demands and struggles. Objective and subjective information concepts are based on the underlying worldview of reification. Reification endangers human existence. Information as process and relation enables political and ethical alternatives that have radical implications for society.

Diatonic Connotations of Pitch-Class Sets

1997 ◽  
Vol 15 (1) ◽  
pp. 1-29 ◽  
Author(s):  
René Van Egmond ◽  
David Butler
Keyword(s):  
Set Theory ◽  
Theoretical Study ◽  
Class Relations ◽  
Tonal Music ◽  
Pitch Class ◽  
Tonal Pitch ◽  
Relationship Of ◽  
The Relationship ◽  
Pitch Class Sets

This is a music-theoretical study of the relationship of two-, three-, four-, five-, and six-member subsets of the major (pure minor), harmonic minor, and melodic (ascending) minor reference collections, using pitchclass set analytic techniques. These three collections will be referred to as the diatonic sets. Several new terms are introduced to facilitate the application of pitch-class set theory to descriptions of tonal pitch relations and to retain characteristic intervallic relationships in tonal music typically not found in discussions of atonal pitch-class relations. The description comprises three parts. First, pitch sets are converted to pitchclass sets. Second, the pitch- class sets are categorized by transpositional types. Third, the relations of these transpositional types are described in terms of their key center and modal references to the three diatonic sets. Further, it is suggested that the probability of a specific key interpretation by a listener may depend on the scale-degree functions of the tones.

Set theory

2018 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Georg Cantor ◽  
Late Nineteenth Century ◽  
The Status ◽  
Reverse Situation ◽  
Ernst Zermelo ◽  
Relationship Of ◽  
The Relationship ◽  
Infinite Set ◽  
Modern Mathematics

In the late nineteenth century, Georg Cantor created mathematical theories, first of sets or aggregates of real numbers (or linear points), and later of sets or aggregates of arbitrary elements. The relationship of element a to set A is written a∈A; it is to be distinguished from the relationship of subset B to set A, which holds if every element of B is also an element of A, and which is written B⊆A. Cantor is most famous for his theory of transfinite cardinals, or numbers of elements in infinite sets. A subset of an infinite set may have the same number of elements as the set itself, and Cantor proved that the sets of natural and rational numbers have the same number of elements, which he called ℵ0; also that the sets of real and complex numbers have the same number of elements, which he called c. Cantor proved ℵ0 to be less than c. He conjectured that no set has a number of elements strictly between these two. In the early twentieth century, in response to criticism of set theory, Ernst Zermelo undertook its axiomatization; and, with amendments by Abraham Fraenkel, his have been the accepted axioms ever since. These axioms help distinguish the notion of a set, which is too basic to admit of informative definition, from other notions of a one made up of many that have been considered in logic and philosophy. Properties having exactly the same particulars as instances need not be identical, whereas sets having exactly the same elements are identical by the axiom of extensionality. Hence for any condition Φ there is at most one set {x|Φ(x)} whose elements are all and only those x such that Φ(x) holds, and {x|Φ(x)}={x|Ψ(x)} if and only if conditions Φ and Ψ hold of exactly the same x. It cannot consistently be assumed that {x|Φ(x)} exists for every condition Φ. Inversely, the existence of a set is not assumed to depend on the possibility of defining it by some condition Φ as {x|Φ(x)}. One set x0 may be an element of another set x1 which is an element of x2 and so on, x0∈x1∈x2∈…, but the reverse situation, …∈y2∈y1∈y0, may not occur, by the axiom of foundation. It follows that no set is an element of itself and that there can be no universal set y={x|x=x}. Whereas a part of a part of a whole is a part of that whole, an element of an element of a set need not be an element of that set. Modern mathematics has been greatly influenced by set theory, and philosophies rejecting the latter must therefore reject much of the former. Many set-theoretic notations and terminologies are encountered even outside mathematics, as in parts of philosophy: pair {a,b} {x|x=a or x=b} singleton {a} {x|x=a} empty set ∅ {x|x≠x} union ∪X {a|a∈A for some A∈X} binary union A∪B {a|a∈A or a∈B} intersection ∩X {a|a∈A for all A∈X} binary intersection A∩B {a|a∈A and a∈B} difference A−B {a|a∈A and not a∈B} complement A−B power set ℘(A) {B|B⊆A} (In contexts where only subsets of A are being considered, A-B may be written -B and called the complement of B.) While the accepted axioms suffice as a basis for the development not only of set theory itself, but of modern mathematics generally, they leave some questions about transfinite cardinals unanswered. The status of such questions remains a topic of logical research and philosophical controversy.

scholarly journals The Noble Family as “Singular Multiplicity”? Redefining the Smoczynski–Zarycki’s Totemic Definition of Nobility through the Lenses of Alain Badiou’s Mathematical Ontology

2019 ◽  
Vol 9 (1) ◽  
pp. 53-69
Author(s):  
Urszula Idziak ◽  
Bartosz Piotr Bednarczyk
Keyword(s):  
Set Theory ◽  
Empirical Studies ◽  
The State ◽  
Mathematical Ontology ◽  
Definition Of ◽  
Generic Theory ◽  
Relationship Of ◽  
The Relationship ◽  
Theoretical Standpoint

Abstract In our paper, we redefine the category of “family” denoting the relationship of selected members of a post-noble/post-aristocratic milieu in Poland using Alain Badiou’s terminology. Badiou’s ontology based on a mathematical set theory and a generic theory is the most developed, complex, and revolutionary ontology of the 20th and 21st centuries. However, it is rarely adapted to new empirical studies probably because of its novelty and complexity. We do not intend to use the empirical case study made by Smoczynski–Zarycki to inform our argument but instead perform a translation of the Durkheim–Lacanian theoretical standpoint from “Totem…” into the category of “singularity” [singularité] in its relation to “the state of situation” [état de la situation] from “Being and Event” (Badiou 2005). This approach seeks to find a universalizing potential of nobility that will allow it to become a relevant subject for truth procedure analysis.

scholarly journals ФИЛОСОФСКИЕ СЛЕДСТВИЯ КВАНТОВОЙ ТЕОРИИ ИНФОРМАЦИИ К.Ф. ФОН ВАЙЦЗЕККЕРА

2020 ◽  
Vol 54 (4) ◽  
pp. 120-124
Author(s):  
А.В. Родина
Keyword(s):  
Quantum Theory ◽  
Semantic Content ◽  
Philosophy Of Physics ◽  
Initial Material ◽  
Theory Of Information ◽  
Relationship Of ◽  
The Relationship

Статья посвящена философским следствиям квантовой теории информации К.Ф. фон Вайцзеккера и нацелена на выявление смыслового содержания понятия «первоальтернатива», а также анализ соотношения материи, формы и информации в концепции исследователя. Автор приходит к выводу, что информация является числовой мерой субстанции или мерой множественности форм, форма служит основанием материи, а ур-альтернативы – исходным материалом для реконструкции объекта. Ключевые слова: квантовая теория информации, философия физики, К.Ф. фон Вайцзеккер, ур-альтернатива, кубит The article deals with the philosophical consequences of K.F. von Weizsäcker’s quantum theory of information. It is aimed at identifying the semantic content of the concept of «original alternative», as well as analyzing the relationship of matter, form and information in the concept of the researcher. The author comes to the conclusion that information is a numerical measure of a substance or a measure of a plurality of forms, a form serves as the basis of matter, and original alternatives are the initial material for the reconstruction of an object. Keywords: quantum theory of information, philosophy of physics, K.F. von Weizsäcker, original alternative, qubit

scholarly journals Towards a critical theory of information

2009 ◽  
Vol 7 (2) ◽  
pp. 243-292 ◽  
Author(s):  
Christian Fuchs
Keyword(s):  
Information Theory ◽  
Critical Theory ◽  
Critical Information ◽  
Internet Economy ◽  
Theory Of Information ◽  
And Control ◽  
Relationship Of ◽  
The Relationship ◽  
Nancy Fraser ◽  
Class Play

Critical information theory is an endeavour that focuses ontologically on the analysis of information in the context of domination, asymmetrical power relations, exploitation, oppression, and control by employing epistemologically all theoretical and/or empirical means necessary for doing so in order to contribute at the praxeological level to the establishment of a participatory, co-operative society. Three foundational aspects of a critical theory of information are discussed in this paper: the relation of immanence and transcendence, the relation of base and superstructure, and ideology critique.The logical figure of immanent transcendence is based on the dialectic of essence and existence and poses a viable counterpart to positivistic and postmodern definitions of critique. As an example for the logic of immanent transcendence to critical information theory, a contradiction of the Internet economy is discussed.The debate on redistribution and recognition between critical theorists Nancy Fraser and Axel Honneth gives the opportunity to renew the discussion of the relationship of base and superstructure in critical social theory. Critical information theory needs to be aware of economic, political, and cultural demands that it needs to make in struggles for ending domination and oppression, and of the unifying role that the economy and class play in these demands and struggles. Objective and subjective information concepts are based on the underlying worldview of reification. Reification endangers human existence. Information as process and relation enables political and ethical alternatives that have radical implications for society.

Granular Computing

2011 ◽  
pp. 774-780
Author(s):  
Georg Peters
Keyword(s):  
Probability Theory ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Sets ◽  
Granular Computing ◽  
Information Granularity ◽  
Relationship Of ◽  
The Relationship

It is well accepted that in many real life situations information is not certain and precise but rather uncertain or imprecise. To describe uncertainty probability theory emerged in the 17th and 18th century. Bernoulli, Laplace and Pascal are considered to be the fathers of probability theory. Today probability can still be considered as the prevalent theory to describe uncertainty. However, in the year 1965 Zadeh seemed to have challenged probability theory by introducing fuzzy sets as a theory dealing with uncertainty (Zadeh, 1965). Since then it has been discussed whether probability and fuzzy set theory are complementary or rather competitive (Zadeh, 1995). Sometimes fuzzy sets theory is even considered as a subset of probability theory and therefore dispensable. Although the discussion on the relationship of probability and fuzziness seems to have lost the intensity of its early years it is still continuing today. However, fuzzy set theory has established itself as a central approach to tackle uncertainty. For a discussion on the relationship of probability and fuzziness the reader is referred to e.g. Dubois, Prade (1993), Ross et al. (2002) or Zadeh (1995). In the meantime further ideas how to deal with uncertainty have been suggested. For example, Pawlak introduced rough sets in the beginning of the eighties of the last century (Pawlak, 1982), a theory that has risen increasing attentions in the last years. For a comparison of probability, fuzzy sets and rough sets the reader is referred to Lin (2002). Presently research is conducted to develop a Generalized Theory of Uncertainty (GTU) as a framework for any kind of uncertainty whether it is based on probability, fuzziness besides others (Zadeh, 2005). Cornerstones in this theory are the concepts of information granularity (Zadeh, 1979) and generalized constraints (Zadeh, 1986). In this context the term Granular Computing was first suggested by Lin (1998a, 1998b), however it still lacks of a unique and well accepted definition. So, for example, Zadeh (2006a) colorfully calls granular computing “ballpark computing” or more precisely “a mode of computation in which the objects of computation are generalized constraints”.

Intuitionistic modal logic and set theory

1991 ◽  
Vol 56 (2) ◽  
pp. 497-516 ◽  
Author(s):  
K. Lano
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Mathematical Treatment ◽  
Related Research ◽  
Applied Development ◽  
Partial Truth ◽  
Relationship Of ◽  
The Relationship ◽  
Intuitionistic Set Theory ◽  
Theory Of Fuzzy Sets

The mathematical treatment of the concepts of vagueness and approximation is of increasing importance in artificial intelligence and related research. The theory of fuzzy sets was created by Zadeh [Z] to allow representation and mathematical manipulation of situations of partial truth, and proceeding from this a large amount of theoretical and applied development of this concept has occurred. The aim of this paper is to develop a natural logic and set theory that is a candidate for the formalisation of the theory of fuzzy sets. In these theories the underlying logic of properties and sets is intuitionistic, but there is a subset of formulae that are ‘crisp’, classical and two-valued, which represent the certain information. Quantum logic or logics weaker than intuitionistic can also be adopted as the basis, as described in [L]. The relationship of this theory to the intensional set theory MZF of [Gd] and the global intuitionistic set theory GIZF of Takeuti and Titani [TT] is also treated.

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