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X A Xiao - One of the best experts on this subject based on the ideXlab platform.

  • mathematical significance and application perspectives of medium logic calculus ml and medium Axiomatic Set Theory ms
    Fuzzy Sets and Systems, 1990
    Co-Authors: X A Xiao
    Abstract:

    Abstract In the sense of basic Theory of mathematics, ML (Medium Logic) and MS (Medium Set Theory) have widened the foundation of logic and Set Theory of the precise classical mathematics and extended the research objects of mathematics from precision into fuzziness. Hence ML and MS, in a higher form, contain the whole of precise classical mathematics and its theoretical foundation. ML and MS have also resolved the problems of the revision of the comprehension principle left over historically, i.e. they have not only excluded all the paradoxes in the system, but also preserved all the reasonable content of the comprehension principle. ML and MS have provided a common theoretical foundation for both precise classical mathematics and future uncertain mathematics. The fifth generation of computers must lay its foundation on the acknowledgement of intelligent logic of medium states. Obviously ML will be this logic. ML and MS will also be the theoretical foundation of expert systems and advance the techniques of pattern recognition, graphic analysis, artificial intelligence, intelligent robots, computer vision, etc. to a new stage. In addition to the uncertain quantitative analysis, based on the theoretical foundation of ML and MS, the Theory of large systems will present extraordinarily new prospects.

Paul Bernays - One of the best experts on this subject based on the ideXlab platform.

  • A system of Axiomatic Set Theory—Part II
    Journal of Symbolic Logic, 2020
    Co-Authors: Paul Bernays
    Abstract:

    For the formulation of the remaining axioms we need the notions of a function and of a one-to-one correspondence.We define a function to be a class of pairs in which different elements always have different first members; or, in other words, a class F of pairs such that, to every element a of its domain there is a unique element b of its converse domain determined by the condition 〈a, b〉ηF. We shall call the Set b so determined the value of F for a, and denote it (following the mathematical usage) by F(a).A Set which represents a function—i.e., a Set of pairs in which different elements always have different first members—will be called a functional Set.If b is the value of the function F for a, we shall say that F assigns the Set b to the Set a; and if a functional Set f represents F, we shall say also that f assigns the Set b to the Set a.A class of pairs will be called a one-to-one correspondence if both it and its converse class are functions. We shall say that there exists a one-to-one correspondence between the classes A and B (or of A to B) if A and B are domain and converse domain of a one-to-one correspondence. Likewise we shall say that there exists a one-to-one correspondence between the Sets a and b (or of a to b) if a and b respectively represent the domain and the converse domain of a one-to-one correspondence. In the same fashion we speak of a one-to-one correspondence between a class and a Set, or a Set and a class.

  • A system of Axiomatic Set Theory - Part VII
    Journal of Symbolic Logic, 2020
    Co-Authors: Paul Bernays
    Abstract:

    The reader of Part VI will have noticed that among the Set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a Set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of infinity. Thereby it becomes possible to Set up the models on the basis of only I–III, and either VII or Va, a basis from which number Theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first Setting up number Theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic Set-theoretic system, and those which are relative to the model to be defined.

  • A system of Axiomatic Set Theory—Part I
    Journal of Symbolic Logic, 2020
    Co-Authors: Paul Bernays
    Abstract:

    Introduction. The system of axioms for Set Theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the Set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.The Theory is not Set up as a pure formalism, but rather in the usual manner of elementary axiom Theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the Theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell).

  • A System of Axiomatic Set Theory: Part IV. General Set Theory
    Journal of Symbolic Logic, 2020
    Co-Authors: Paul Bernays
    Abstract:

    11. Elementary one-to-one correspondences, fundamental theorems on power. Our task in the treatment of general Set Theory will be to give a survey for the purpose of characterizing the different stages and the principal theorems with respect to their Axiomatic requirements from the point of view of our system of axioms. The delimitation of "general Set Theory" which we have in view differs from that of Fraenkel's general Set Theory, and also from that of "standard logic" as understood by most logicians. It is adapted rather to the tendency of von Neumann's system of Set Theory-the von Neumann system having been the first in which the possibility appeared of separating the assumptions which are required for the conceptual formations from those which lead to the Cantor hierarchy of powers. Thus our intention is to obtain general Set Theory without use of the axioms V d, V c, VI. It will also be desirable to separate those proofs which can be made without the axiom of choice, and in doing this we shall have to use the axiom V* i.e., the theorem of replacement taken as an axiom. From V*, as we saw in ?4, we can immediately derive V a and V b as theorems, and also the theorem that a function whose domain is represented by a Set is itself represented by a functional Set; and on the other hand V* was found to be derivable from V a and V b in combination with the axiom of choice.39 (These statements on deducibility are of course all on the basis of the axioms I-III.) In the development of general Set Theory we shall always have V a at our disposal, in some contexts as an axiom, in others as a theorem. But, as we have seen, V a in connection with the axioms I-III enables us to obtain the fundamental theorems on ordinals, and also the iteration theorem and number Theory. Hence we shall be able to dispense with the axiom VII throughout our treatment of general Set Theory. For this first more elementary part of general Set Theory we introduce an axiom which we shall call the pair class axiom, and which asserts that the pair class A X B is represented by a Set if the classes A and B are represented by Sets. This was obtained in ?4 as a consequence of V b, c, and also as a consequence of V a, c, d.40 Later we shall show that it is also derivable from IV, V a, V b, and hence, on the basis of the axiom of choice, can be dispensed with as an axiom for general Set Theory.4'

  • Mathematics as a Domain of Theoretical Science and of Mental Experience
    Studies in logic and the foundations of mathematics, 2014
    Co-Authors: Paul Bernays
    Abstract:

    Publisher Summary This chapter highlights mathematics as a domain of theoretical science and mental experience. The Theory of algebraic functions has been a central domain; it includes function Theory, algebra, algebraic geometry, Theory of Riemann surfaces, and topology. There are various embracing theories, and Axiomatic Set Theory itself is extended by model Theory, where Set theoretic concepts are used independently from the axiomatization. The modified situation is especially clear in the general Theory of mappings, which is called “the Theory of categories.” The chapter explains that mathematicians have different opinions about the suitability of stronger or only weaker methods of idealizations.

J. C. Flores - One of the best experts on this subject based on the ideXlab platform.

  • Competitive Exclusion and Axiomatic Set-Theory: De Morgan’s Laws, Ecological Virtual Processes, Symmetries and Frozen Diversity
    Acta Biotheoretica, 2016
    Co-Authors: J. C. Flores
    Abstract:

    This work applies the competitive exclusion principle and the concept of potential competitors as simple Axiomatic tools to generalized situations in ecology. These tools enable apparent competition and its dual counterpart to be explicitly evaluated in poorly understood ecological systems. Within this Set-Theory framework we explore theoretical symmetries and invariances, De Morgan’s laws, frozen evolutionary diversity and virtual processes. In particular, we find that the exclusion principle compromises the geometrical growth of the number of species. By theoretical extending this principle, we can describe interspecific depredation in the dual case. This study also briefly considers the debated situation of intraspecific competition. The ecological consequences of our findings are discussed; particularly, the use of our framework to reinterpret coupled mathematical differential equations describing certain ecological processes.

  • competitive exclusion and Axiomatic Set Theory de morgan s laws ecological virtual processes symmetries and frozen diversity
    Acta Biotheoretica, 2016
    Co-Authors: J. C. Flores
    Abstract:

    This work applies the competitive exclusion principle and the concept of potential competitors as simple Axiomatic tools to generalized situations in ecology. These tools enable apparent competition and its dual counterpart to be explicitly evaluated in poorly understood ecological systems. Within this Set-Theory framework we explore theoretical symmetries and invariances, De Morgan’s laws, frozen evolutionary diversity and virtual processes. In particular, we find that the exclusion principle compromises the geometrical growth of the number of species. By theoretical extending this principle, we can describe interspecific depredation in the dual case. This study also briefly considers the debated situation of intraspecific competition. The ecological consequences of our findings are discussed; particularly, the use of our framework to reinterpret coupled mathematical differential equations describing certain ecological processes.

E. Kozakova - One of the best experts on this subject based on the ideXlab platform.

  • Ladislav Svante Rieger and his Work in Mathematical Logic and Set Theory
    2020
    Co-Authors: E. Kozakova
    Abstract:

    The present paper is dedicated to Ladislav Svante Rieger (1916-1963), a prominent Czech mathematician who worked in three principal domains: algebra, mathematical logic, and Axiomatic Set Theory. It is a free continuation of the article Kozakova (2005a) in which the personality of L.S. Rieger is introduced and his work in the field of algebra is evaluated. In the following text, we give a brief account on Rieger's professional life and overview of his research activities and describe his achievements in mathematical logic and Axiomatic Set Theory in more detail. At the end of the paper, we provide the list of Rieger's most significant publications.

  • Ladislav Svante Rieger and His Algebraic Work
    2020
    Co-Authors: E. Kozakova
    Abstract:

    This paper introduces the personality of the Czech mathematician of the first half of the 20th century, Ladislav Svante Rieger. Rieger's professional activities were devoted to three main mathematical areas; algebra, mathematical logic and Axiomatic Set Theory. First, we describe his personal and professional life and give a brief overview of his mathematical activities. Second, his major results reached in the domain of algebra are presented. At the end, a list of Rieger's most significant scientific publications is provided.

Wensheng Yu - One of the best experts on this subject based on the ideXlab platform.

  • a formal system of Axiomatic Set Theory in coq
    IEEE Access, 2020
    Co-Authors: Wensheng Yu
    Abstract:

    Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic Set Theory is a foundational system of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of Set Theory. In this paper, we present a formal system of Axiomatic Set Theory based on the Coq proof assistant. The Axiomatic system used in the formal system refers to Morse-Kelley Set Theory which is a relatively complete and concise Axiomatic Set Theory. In this formal system, we complete the formalization of the basic definitions of Sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano’s postulates are proved as theorems. According to the axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schroeder-Bernstein theorem. The whole formalization of the system includes eight axioms, one axiom schema, 62 definitions, and 148 corollaries or theorems. The “Axiomatic Set Theory” formal system is free from the more apparent paradoxes, and a complete Axiomatic system is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the system, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.

  • A Formal Proof in Coq of Cantor-Bernstein-Schroeder’s Theorem without axiom of choice
    2019 Chinese Automation Congress (CAC), 2019
    Co-Authors: Yaoshun Fu, Wensheng Yu
    Abstract:

    This paper describes a formal proof of Cantor-Bernstein-Schroeder’s Theorem based on Morse-Kelley Axiomatic Set Theory, which use inductive without axiom of choice in the proof assistant Coq. Firstly, we formalize a few definitions, axioms and theorems from the Axiomatic Set Theory, just needed for formal proof. Furthermore, some new definition were added to make the system self-closed. Then, some reusable lemmas and Integrated strategy are given to make the code simpler and more automated. At last, we give a formal proof of the theorem in detail. All the proofs are formally checked by Coq proof assistant. The formalizations embody that mechanical proving of mathematical theorem based on Coq has the characteristics of readability and interactivity. Every step proves to normalized, rigorous and credible. The Cantor-Bernstein-Schroeder’s Theorem is the key result in the Set Theory that allows comparison of infinite Sets, and its formalization lay a foundation for formal proof of lots of important theorems and puzzles in Set Theory. Index Terms–formal proof, Cantor-Bernstein-Schroeder’s theorem, Morse-Kelley Axiomatic Set Theory, Coq, reliable

  • Formalization of General Topology in Coq — A Formal Proof of Tychonoff's Theorem
    2019 Chinese Control Conference (CCC), 2019
    Co-Authors: Yaoshun Fu, Wensheng Yu
    Abstract:

    The order, algebra and topology of the Bourbaki group are the foundation of modern mathematics. On the basis of the proof assistant Coq, the formal system of these three major parental structures can be constructed completely. On the basis of the formalization of Axiomatic Set Theory, this paper present the formal framework of general topology. This paper completes the formalization of some basic definitions of general topology, including base, subbase, topology, and compact space. In addition, the formal proof of the Alexander subbase theorem is proposed on the basis of Tukey’s lemma. Finally we complete the formal proof of Tychonoff’s theorem, which is famous in general topology. All formal processes have been verified in Coq, which demonstrates the reliability, readability and rigor.