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Axiomatic Set Theory
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X A Xiao – One of the best experts on this subject based on the ideXlab platform.
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mathematical significance and application perspectives of medium logic calculus ml and medium Axiomatic Set Theory ms
Fuzzy Sets and Systems, 1990Co-Authors: X A XiaoAbstract: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.
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A system of Axiomatic Set Theory—Part II
Journal of Symbolic Logic, 2020Co-Authors: Paul BernaysAbstract: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.
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A system of Axiomatic Set Theory – Part VII
Journal of Symbolic Logic, 2020Co-Authors: Paul BernaysAbstract: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.
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A system of Axiomatic Set Theory—Part I
Journal of Symbolic Logic, 2020Co-Authors: Paul BernaysAbstract: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).
J. C. Flores – One of the best experts on this subject based on the ideXlab platform.
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Competitive Exclusion and Axiomatic Set–Theory: De Morgan’s Laws, Ecological Virtual Processes, Symmetries and Frozen Diversity
Acta Biotheoretica, 2016Co-Authors: J. C. FloresAbstract: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.
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competitive exclusion and Axiomatic Set Theory de morgan s laws ecological virtual processes symmetries and frozen diversity
Acta Biotheoretica, 2016Co-Authors: J. C. FloresAbstract: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.