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Published: 27 July 2024

Last updated: 29 July 2024

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1

Jiang, Jingying. "From Set Theory to the Axiomatization of Set Theory." Highlights in Science, Engineering and Technology 88 (March 29, 2024): 243–47. http://dx.doi.org/10.54097/nbmg4652.

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Axiomatic set theory was created by German mathematician Zermelo as a strategy for addressing and resolving paradoxes in the field of mathematical study. By adopting this strategy, the axiomatic technique will be applied to set theory. The person argues that Cantor's failure to impose restrictions on the idea of a set is the cause of the dilemma. They also claim that Cantor's definition of a set is unclear. Both of these arguments are predicated on the idea that Cantor neglected to place limitations on the idea of a set. Zermelo hypothesized that the condensed version of the axioms would make it easier to define a set and elaborate on its properties, and he was correct in his prediction. The creation of an axiomatization for set theory is the first of this research's main goals. Second, the investigation of various set theory development methods in comparison to one another. Even though there are intriguing puzzles in the field of set theory that have not yet been solved, the axiomatization of set theory is widely regarded as a significant accomplishment in the field.

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Cutolo, Raffaella, Ulderico Dardano, and Virginia Vaccaro. "Axiomatic set theory and unincreasable infinity." Applied Mathematical Sciences 8 (2014): 6725–32. http://dx.doi.org/10.12988/ams.2014.49687.

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3

DEISER, OLIVER. "AN AXIOMATIC THEORY OF WELL-ORDERINGS." Review of Symbolic Logic 4, no. 2 (March 4, 2011): 186–204. http://dx.doi.org/10.1017/s1755020310000390.

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We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gödel’s axiom of constructibility. In list theory there are strong arguments favoring Gödel’s axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is a natural and attractive modification of ALT where every object is constructible and countable. In order to substantiate our foundational interest in lists, we also compare sets and lists from the perspective of finite objects, arguing that lists are, from a certain point of view, conceptually simpler than sets.

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Vdovin, A. M. "FOUNDATIONS OF A NEW AXIOMATIC SET THEORY." Mathematics of the USSR-Izvestiya 37, no. 2 (April 30, 1991): 467–73. http://dx.doi.org/10.1070/im1991v037n02abeh002074.

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5

Vdovin, A. M. "EXTENSION OF A NEW AXIOMATIC SET THEORY." Russian Academy of Sciences. Izvestiya Mathematics 42, no. 3 (June 30, 1994): 615–19. http://dx.doi.org/10.1070/im1994v042n03abeh001548.

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6

Tamir, Dan E., Cao Zhi-Qiang, Abraham Kandel, and Joe L. Mott. "An axiomatic approach to fuzzy set theory." Information Sciences 52, no. 1 (October 1990): 75–83. http://dx.doi.org/10.1016/0020-0255(90)90036-a.

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7

Hintikka, Jaakko. "Independence-friendly logic and axiomatic set theory." Annals of Pure and Applied Logic 126, no. 1-3 (April 2004): 313–33. http://dx.doi.org/10.1016/j.apal.2003.11.006.

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8

Anacona, Maribel, Luis Carlos Arboleda, and F. Javier Pérez-Fernández. "On Bourbaki’s axiomatic system for set theory." Synthese 191, no. 17 (July 26, 2014): 4069–98. http://dx.doi.org/10.1007/s11229-014-0515-1.

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9

Roemer, John E. "Welfarism and Axiomatic Bargaining Theory." Recherches économiques de Louvain 56, no. 3-4 (1990): 287–301. http://dx.doi.org/10.1017/s0770451800043918.

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SummaryConsider the domain of economic environments E whose typical element is ξ = (U1, U2, Ω, ω*), where ui are Neumann-Morgenstern utility functions, Ω is a set of lotteries on a fixed finite set of alternatives, and ω* ∈ Ω. A mechanism f associates to each ξ a lottery f(ξ) in Ω. Formulate the natural version of Nash’s axioms, from his bargaining solution, for mechanisms on this domain. (e.g., IIA says that if ξ′ = (U1, U2, Δ, ω′), Δ ⊂ Ω, and f ∈ Δ then f(ξ′) = f(ξ).) It is shown that the Nash axioms (Pareto, symmetry, IIA, invariance w.r.t. cardinal transformations of the utility functions) hardly restrict the behavior of the mechanism at all. In particular, for any integer M, choose M environments ξi, i = 1, … , M, and choose a Pareto optimal lottery ωi ∈ Ωi, restricted only so that no axiom is directly contradicted by these choices. Then there is a mechanism f for which f(ξi) = ωi, which satisfies all the axioms, and is continuous on E.

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HUANG, GEORGE Q., and ZUHUA JIANG. "FuzzySTAR: Fuzzy set theory of axiomatic design review." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 16, no. 4 (September 2002): 291–302. http://dx.doi.org/10.1017/s0890060402164031.

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Product development involves multiple phases. Design review (DR) is an essential activity formally conducted to ensure a smooth transition from one phase to another. Such a formal DR is usually a multicriteria decision problem, involving multiple disciplines. This paper proposes a systematic framework for DR using fuzzy set theory. This fuzzy approach to DR is considered particularly relevant for several reasons. First, information available at early design phases is often incomplete and imprecise. Second, the relationships between the product design parameters and the review criteria cannot usually be exactly expressed by mathematical functions due to the enormous complexity. Third, DR is frequently carried out using subjective expert judgments with some degree of uncertainty. The DR is defined as the reverse mapping between the design parameter domain and design requirement (review criterion) domain, as compared with Suh's theory of axiomatic design. Fuzzy sets are extensively introduced in the definitions of the domains and the mapping process to deal with imprecision, uncertainty, and incompleteness. A simple case study is used to demonstrate the resulting fuzzy set theory of axiomatic DR.

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11

Hintikka, Jaakko. "Truth Definitions, Skolem Functions and Axiomatic Set Theory." Bulletin of Symbolic Logic 4, no. 3 (September 1998): 303–37. http://dx.doi.org/10.2307/421033.

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§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski (1935). In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore (1994, p. 635) asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable.

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12

Al-Odhari, Adel Mohammed. "Axiomatic of Neutrosophic Groups." مجلة جامعة صنعاء للعلوم التطبيقية والتكنولوجيا 2, no. 2 (May 5, 2024): 205–14. http://dx.doi.org/10.59628/jast.v2i2.793.

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The aim of this article is to; present a neutrosophic group according to axioms such as classical group theory and the neutrosophic set, and to study some properties and theorems related to the neutrosophic group. The new concept of the neutrosophic set is a new approach that is suitable for mathematical problems related to philosophical concepts, such as uncertainty and indeterminacy, in which human knowledge and human evaluation are necessary. Neutrosophic algebra is a branch of neutrosophic set theory, and in 2004, Kandasamy and Smarandache introduced basic neutrosophic algebraic structures and their applications to fuzzy and neutrosophic models; in 2006, Kandasamy and Smarandache presented neutrosophic algebraic structures and neutrosophic N-algebraic structures; in 2019, Smarandache introduced new fields of research in Neutrosophy, which he called Neutro-Structures and Anti-Structures. In 2020, Agboola presented an Introduction to the neutrosophic group by a different presentation.

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Lovyagin, Yuri N., and Nikita Yu Lovyagin. "Finite Arithmetic Axiomatization for the Basis of Hyperrational Non-Standard Analysis." Axioms 10, no. 4 (October 19, 2021): 263. http://dx.doi.org/10.3390/axioms10040263.

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The standard elementary number theory is not a finite axiomatic system due to the presence of the induction axiom scheme. Absence of a finite axiomatic system is not an obstacle for most tasks, but may be considered as imperfect since the induction is strongly associated with the presence of set theory external to the axiomatic system. Also in the case of logic approach to the artificial intelligence problems presence of a finite number of basic axioms and states is important. Axiomatic hyperrational analysis is the axiomatic system of hyperrational number field. The properties of hyperrational numbers and functions allow them to be used to model real numbers and functions of classical elementary mathematical analysis. However hyperrational analysis is based on well-known non-finite hyperarithmetic axiomatics. In the article we present a new finite first-order arithmetic theory designed to be the basis of the axiomatic hyperrational analysis and, as a consequence, mathematical analysis in general as a basis for all mathematical application including AI problems. It is shown that this axiomatics meet the requirements, i.e., it could be used as the basis of an axiomatic hyperrational analysis. The article in effect completes the foundation of axiomatic hyperrational analysis without calling in an arithmetic extension, since in the framework of the presented theory infinite numbers arise without invoking any new constants. The proposed system describes a class of numbers in which infinite numbers exist as natural objects of the theory itself. We also do not appeal to any “enveloping” set theory.

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14

Sun, Tianyu, and Wensheng Yu. "A Formal System of Axiomatic Set Theory in Coq." IEEE Access 8 (2020): 21510–23. http://dx.doi.org/10.1109/access.2020.2969486.

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15

Thompson, Simon. "Axiomatic recursion theory and the continuous functionals." Journal of Symbolic Logic 50, no. 2 (June 1985): 442–50. http://dx.doi.org/10.2307/2274232.

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AbstractWe define, in the spirit of Fenstad [2], a higher type computation theory, and show that countable recursion over the continuous functionals forms such a theory. We also discuss Hyland's proposal from [4] for a scheme with which to supplement S1–S9, and show that this augmented set of schemes fails to generate countable recursion. We make another proposal to which the methods of this section do not apply.

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Lupenko, Serhii, Andrii Zozulya, Christopher Chizoba, Nataliya Stadnyk, and Andrii Horkunenko. "METHOD OF SET AND TAXONOMY INDUCTION OF CYCLIC FUNCTIONAL RELATIONS CLASSES WITHIN THE FRAMEWORK OF AXIOMATIC-DEDUCTIVE STRATEGY OF ORGANIZATION OF CYCLIC FUNCTIONAL RELATIONS THEORY." Innovative Solution in Modern Science 4, no. 48 (September 24, 2021): 92. http://dx.doi.org/10.26886/2414-634x.4(48)2021.7.

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The paper defines the classes of cyclic functional relations and their taxonomy, which allowed to formalize and organize the theory according to the axiomatic-deductive strategy. A set of cyclic attributes, a set of domains of definition, a set of types of rhythm of cyclic functional relation are formed, which forms a taxonomy of models of cyclic signals, which in turn are a component of the theory of cyclic functional relations. A method for generating a set and taxonomy of classes of cyclic functional relations has been developed. The taxonomy of models’ classes, methods, algorithms and software for processing and simulation (generation) of cyclic signals within the theory of cyclic functional relations is developed.Keywords: induction method, class taxonomy, cyclic functional relations, axiomatic-deductive strategy.

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17

WEBER, ZACH. "TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY." Review of Symbolic Logic 3, no. 1 (January 14, 2010): 71–92. http://dx.doi.org/10.1017/s1755020309990281.

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This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

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18

Zhu, Wujia, Yi Lin, Guoping Du, and Ningsheng Gong. "Inconsistency of uncountable infinite sets under ZFC framework." Kybernetes 37, no. 3/4 (April 11, 2008): 453–57. http://dx.doi.org/10.1108/03684920810863417.

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PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

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19

Guo, Hongyue, Witold Pedrycz, and Xiaodong Liu. "Fuzzy time series forecasting based on axiomatic fuzzy set theory." Neural Computing and Applications 31, no. 8 (January 9, 2018): 3921–32. http://dx.doi.org/10.1007/s00521-017-3325-9.

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20

Alexandru, Andrei, and Gabriel Ciobanu. "A Topological Approach in the Extended Fraenkel-Mostowski Model of Set Theory." Annals of the Alexandru Ioan Cuza University - Mathematics 60, no. 2 (November 24, 2014): 261–77. http://dx.doi.org/10.2478/aicu-2013-0029.

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Abstract Lattices of subgroups are presented as algebraic domains. Given an arbitrary group, we define the Scott topology over the subgroups lattice of that group. A basis for this topology is expressed in terms of finitely generated subgroups. Several properties of the continuous functions with respect the Scott topology are obtained; they provide new order properties of groups. Finally there are expressed several properties of the group of permutations of atoms in a permutative model of set theory. We provide new properties of the extended interchange function by presenting some topological properties of its domain. Several order and topological properties of the sets in the Fraenkel-Mostowski model remains also valid in the Extended Fraenkel-Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel-Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel-Mostowski model.

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21

Hansson, Sven Ove. "Theory contraction and base contraction unified." Journal of Symbolic Logic 58, no. 2 (June 1993): 602–25. http://dx.doi.org/10.2307/2275221.

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AbstractOne way to construct a contraction operator for a theory (belief set) is to assign to it a base (belief base) and an operator of partial meet contraction for that base. Axiomatic characterizations are given of the theory contractions that are generated in this way by (various types of) partial meet base contractions.

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22

Changat, Manoj, Lekshmi Kamal K. Sheela, and Prasanth G. Narasimha-Shenoi. "Axiomatic characterizations of Ptolemaic and chordal graphs." Opuscula Mathematica 43, no. 3 (2023): 393–407. http://dx.doi.org/10.7494/opmath.2023.43.3.393.

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The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set \(V\) to the power set of \(V\) satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs.

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23

Seridi, Hamid, and Herman Akdag. "Approximate Reasoning for Processing Uncertainty." Journal of Advanced Computational Intelligence and Intelligent Informatics 5, no. 2 (March 20, 2001): 110–18. http://dx.doi.org/10.20965/jaciii.2001.p0110.

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In Bayesian networks as well as in knowledge-based systems, uncertainty in propositions can be represented by various degrees of belief encoded by qualitative values. In this paper, we present a qualitative approach of classical probability theory in the particular case where the set of probability degrees is replaced by a totally ordered set of symbolic values. We first define the four elementary operations (addition, subtraction, multiplication and division) allowing to manipulate these symbolic degrees of uncertainty, then we propose an axiomatic. The properties obtained from this axiomatic allows to show that our theory constitutes a qualitative approach for processing uncertain statements of natural language. The obtained results are usable in inferential processes as well as in Bayesian networks.

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Cabbolet, Marcoen J. T. F. "A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory." Axioms 10, no. 2 (June 14, 2021): 119. http://dx.doi.org/10.3390/axioms10020119.

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It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

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Li, Zedong, Xiaodong Duan, Qingling Zhang, Cunrui Wang, Yuangang Wang, and Wanquan Liu. "Multi-ethnic facial features extraction based on axiomatic fuzzy set theory." Neurocomputing 242 (June 2017): 161–77. http://dx.doi.org/10.1016/j.neucom.2017.02.070.

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26

Ramani, Lakshmi, and Padmaja Poosapati. "A Study of Notations and Illustrations of Axiomatic Fuzzy Set Theory." International Journal of Computer Applications 134, no. 11 (January 15, 2016): 7–12. http://dx.doi.org/10.5120/ijca2016907999.

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27

Hajnal, András, and László Kalmár. "An elementary combinatorial theorem with an application to axiomatic set theory." Publicationes Mathematicae Debrecen 4, no. 3-4 (July 1, 2022): 431–49. http://dx.doi.org/10.5486/pmd.1956.4.3-4.42.

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28

Dixon, Paul W. "Axiomatic Construction for Language Creativity and Self-Actualization." Perceptual and Motor Skills 77, no. 1 (August 1993): 203–6. http://dx.doi.org/10.2466/pms.1993.77.1.203.

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Piagetian theory views intellective development in children as the unfolding of the axioms which form the basis of symbolic logic through maturation and learning. The modern derivation of set theory from these axioms may be seen as a model of how we formulate valid inferences. The axiomatic construction of the aleph null as equal to 1, which may be derived from the Cantorian algebra, can be used to extend the axiomatic basis of propositional calculus seen as the epistemological root of human knowledge by Piaget. A central postulation regarding natural languages proposed by Chomsky is the creative aspect of language. This axiomatic construction for the Continuum Hypothesis of Gregor Cantor, which permits a nonconfirmatory decision regarding this hypothesis, may be generalized to account for the various aspects of creativity in personality theory, forming (as an extension to Piaget's theory) essential mechanisms of cognition, language behaviour, and self-actualization.

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Gholami, Atena, Reza Sheikh, Neda Mizani, and Shib Sankar Sana. "ABC analysis of the customers using axiomatic design and incomplete rough set." RAIRO - Operations Research 52, no. 4-5 (October 2018): 1219–32. http://dx.doi.org/10.1051/ro/2018022.

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Customer’s recognition, classification, and selecting the target market are the most important success factors of a marketing system. ABC classification of the customers based on axiomatic design exposes the behavior of the customer in a logical way in each class. Quite often, missing data is a common occurrence and can have a significant effect on the decision- making problems. In this context, this proposed article determines the customer’s behavioral rule by incomplete rough set theory. Based on the proposed axiomatic design, the managers of a firm can map the rules on designed structures. This study demonstrates to identify the customers, determine their characteristics, and facilitate the development of a marketing strategy.

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Koumou, Gilles Boevi, and Georges Dionne. "Coherent Diversification Measures in Portfolio Theory: An Axiomatic Foundation." Risks 10, no. 11 (October 26, 2022): 205. http://dx.doi.org/10.3390/risks10110205.

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We provide an axiomatic foundation for the measurement of correlation diversification in a one-period portfolio model. We propose a set of eight desirable axioms for this class of diversification measures. We name the measures satisfying these axioms coherent correlation diversification measures. We study the compatibility of our axioms with rank-dependent expected utility theory. We also test them against the two most frequently used methods for measuring correlation diversification in portfolio theory: portfolio variance and the diversification ratio. Lastly, we provide an example of a functional representation of a coherent correlation diversification measure.

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CAPOTORTI, ANDREA, and ANDREA FORMISANO. "Comparative uncertainty: theory and automation." Mathematical Structures in Computer Science 18, no. 1 (February 2008): 57–79. http://dx.doi.org/10.1017/s0960129507006561.

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In recent decades, qualitative approaches to probabilistic uncertainty have received more and more attention. We propose a characterisation of partial preference orders through a uniform axiomatic treatment of a variety of qualitative uncertainty notions. To this end, we prove a representation result that connects qualitative notions of partial uncertainty to their numerical counterparts. We describe an executable specification, in the declarative framework of Answer Set Programming, that constitutes the core engine for qualitative management of uncertainty. Some basic reasoning tasks are also identified.

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Tao, Lili, Yan Chen, Xiaodong Liu, and Xin Wang. "An integrated multiple criteria decision making model applying axiomatic fuzzy set theory." Applied Mathematical Modelling 36, no. 10 (October 2012): 5046–58. http://dx.doi.org/10.1016/j.apm.2011.12.042.

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Yakovlev, I. V. "Model Approach to Nonstandard Analysis in the Context of Axiomatic Set Theory." Mathematical Notes 79, no. 1-2 (January 2006): 122–28. http://dx.doi.org/10.1007/s11006-006-0012-z.

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Iakovlev, Serguei I. "On removal of one undetected paradox in foundations of probability theory." Journal of Physics: Conference Series 2373, no. 6 (December 1, 2022): 062014. http://dx.doi.org/10.1088/1742-6596/2373/6/062014.

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Abstract The paper points out a logical contradiction that arises in the modern approach to teaching the foundations of the probability theory. This contradiction is of the following form: 𝝎 = {𝝎}, where 𝝎 is an arbitrary elementary outcome. In modern mathematics the formation of expressions of the form 𝒙 = {𝒙} is unacceptable. Here {𝒙} is a unit set, i.e., a set whose sole element is some element 𝒙 of an arbitrary nature. This paradox 𝝎 = {𝝎} is a consequence of the fact that random events and between them also elementary outcomes 𝝎 are considered as some subsets in the set of all elementary outcomes. It is shown how this contradiction can be easily eliminated by applying the axiomatic method. It is based on the introduction of two simple axioms that impose restrictions on elementary outcomes and on the class of random events under consideration. In addition from these two axioms on the basis of strictly logical reasoning the authors derived the representation of an arbitrary random event in the form of a sum of elementary outcomes favorable to it. Moreover, another non-axiomatic approach to the elimination of this paradox is proposed. In the final part of the paper other variants of the second of the introduced axioms are considered as well.

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Andreev, P. V., and E. I. Gordon. "An axiomatics for nonstandard set theory, based on von Neumann–Bernays–Gödel Theory." Journal of Symbolic Logic 66, no. 3 (September 2001): 1321–41. http://dx.doi.org/10.2307/2695109.

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AbstractWe present an axiomatic framework for nonstandard analysis—the Nonstandard Class Theory (NCT) which extends von Neumann–Gödel–Bernays Set Theory (NBG) by adding a unary predicate symbol St to the language of NBG (St(X) means that the class X is standard) and axioms—related to it—analogs of Nelson's idealization, standardization and transfer principles. Those principles are formulated as axioms, rather than axiom schemes, so that NCT is finitely axiomatizable. NCT can be considered as a theory of definable classes of Bounded Set Theory by V. Kanovei and M. Reeken. In many aspects NCT resembles the Alternative Set Theory by P. Vopenka. For example there exist semisets (proper subclasses of sets) in NCT and it can be proved that a set has a standard finite cardinality iff it does not contain any proper subsemiset. Semisets can be considered as external classes in NCT. Thus the saturation principle can be formalized in NCT.

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Ramírez, Juan. "A New Set Theory for Analysis." Axioms 8, no. 1 (March 6, 2019): 31. http://dx.doi.org/10.3390/axioms8010031.

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We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

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37

Nasso, Mauro Di. "An axiomatic presentation of the nonstandard methods in mathematics." Journal of Symbolic Logic 67, no. 1 (March 2002): 315–25. http://dx.doi.org/10.2178/jsl/1190150046.

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AbstractA nonstandard set theory *ZFC is proposed that axiomatizes the nonstandard embedding *. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. *ZFC is a conservative extension of ZFC.

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38

Zhang, Luyao, and Dan Levin. "Bounded Rationality and Robust Mechanism Design: An Axiomatic Approach." American Economic Review 107, no. 5 (May 1, 2017): 235–39. http://dx.doi.org/10.1257/aer.p20171030.

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We propose an axiomatic approach to study the superior performance of mechanisms with obviously dominant strategies to those with only dominant strategies. Guided by the psychological inability to reason state-by-state, we develop Obvious Preference as a weakening of Subjective Expected Utility Theory. We show that a strategy is an obviously dominant if and only if any Obvious Preference prefer it to any deviating strategy at any reachable information set. Applying the concept of Nash Equilibrium to Obvious Preference, we propose Obvious Nash Equilibrium to identify a set of mechanisms that are more robust than mechanisms with only Nash Equilibria.

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39

Biradar, Gagan, Yacine Izza, Elita Lobo, Vignesh Viswanathan, and Yair Zick. "Axiomatic Aggregations of Abductive Explanations." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 10 (March 24, 2024): 11096–104. http://dx.doi.org/10.1609/aaai.v38i10.28986.

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The recent criticisms of the robustness of post hoc model approximation explanation methods (like LIME and SHAP) have led to the rise of model-precise abductive explanations. For each data point, abductive explanations provide a minimal subset of features that are sufficient to generate the outcome. While theoretically sound and rigorous, abductive explanations suffer from a major issue --- there can be several valid abductive explanations for the same data point. In such cases, providing a single abductive explanation can be insufficient; on the other hand, providing all valid abductive explanations can be incomprehensible due to their size. In this work, we solve this issue by aggregating the many possible abductive explanations into feature importance scores. We propose three aggregation methods: two based on power indices from cooperative game theory and a third based on a well-known measure of causal strength. We characterize these three methods axiomatically, showing that each of them uniquely satisfies a set of desirable properties. We also evaluate them on multiple datasets and show that these explanations are robust to the attacks that fool SHAP and LIME.

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40

BRUNI, RICCARDO. "A NOTE ON THEORIES FOR QUASI-INDUCTIVE DEFINITIONS." Review of Symbolic Logic 2, no. 4 (December 2009): 684–99. http://dx.doi.org/10.1017/s175502030909025x.

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This paper introduces theories for arithmeticalquasi-inductive definitions(Burgess, 1986) as it has been done for first-order monotone and nonmonotone inductive ones. After displaying the basic axiomatic framework, we provide some initial result in the proof theoretic bounds line of research (the upper one being given in terms of a theory of sets extending Kripke–Platek set theory).

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Karni, Edi, and Marie-Louise Vierø. "“Reverse Bayesianism”: A Choice-Based Theory of Growing Awareness." American Economic Review 103, no. 7 (December 1, 2013): 2790–810. http://dx.doi.org/10.1257/aer.103.7.2790.

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This article introduces a new approach to modeling the expanding universe of decision makers in the wake of growing awareness, and invokes the axiomatic approach to model the evolution of decision makers' beliefs as awareness grows. The expanding universe is accompanied by extension of the set of acts, the preference relations over which are linked by a new axiom, invariant risk preferences, asserting that the ranking of lotteries is independent of the set of acts under consideration. The main results are representation theorems and rules for updating beliefs over expanding state spaces and events that have the flavor of “reverse Bayesianism.” (JEL D81, D83)

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42

Rychkov, S. V. "The splitting problem for pure extensions of Abelian groups and axiomatic set theory." Russian Mathematical Surveys 40, no. 2 (April 30, 1985): 230–31. http://dx.doi.org/10.1070/rm1985v040n02abeh003574.

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43

Wang, Yuangang, Xiaodong Duan, Xiaodong Liu, Cunrui Wang, and Zedong Li. "A spectral clustering method with semantic interpretation based on axiomatic fuzzy set theory." Applied Soft Computing 64 (March 2018): 59–74. http://dx.doi.org/10.1016/j.asoc.2017.12.004.

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44

Ballard, David, and Karel Hrbacek. "Standard foundations for nonstandard analysis." Journal of Symbolic Logic 57, no. 2 (June 1992): 741–48. http://dx.doi.org/10.2307/2275304.

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In the thirty years since its invention by Abraham Robinson, nonstandard analysis has become a useful tool for research in many areas of mathematics. It seems fair to say, however, that the search for practically satisfactory foundations for the subject is not yet completed. New proposals, intended to remedy various shortcomings of older approaches, continue to be put forward. The objective of this paper is to show that nonstandard concepts have a natural place in the usual (more or less “standard”) set theory, and to argue that this approach improves upon various aspects of hitherto considered systems, while retaining most of their attractive features. We do this by working in Zermelo-Fraenkel set theory with non-well-founded sets. It has always been clear that the axiom of regularity may fail for external sets. The previous approaches either avoid non-well-foundedness by considering only that fragment of nonstandard set theory that is well-founded (over individuals; enlargements of Robinson and Zakon [17]) or reluctantly live with it (various axiomatic nonstandard set theories). Ballard and Davidon [2] were the first to propose constructive use for non-well-foundedness in the foundations of nonstandard analysis. In the present paper we adopt a very strong anti-foundation axiom. In the resulting more or less “usual” set theory, the (to the “standard” mathematician) unfamiliar concepts of standard, external and internal sets can be defined and their requisite properties proved (rather than postulated, as is the case in axiomatic nonstandard set theories).

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Avigad, Jeremy. "Forcing in Proof Theory." Bulletin of Symbolic Logic 10, no. 3 (September 2004): 305–33. http://dx.doi.org/10.2178/bsl/1102022660.

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AbstractPaul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.

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46

de Swart, Harrie de. "Logic, Game Theory, and Social Choice: What Do They Have in Common?" Axioms 11, no. 10 (September 30, 2022): 518. http://dx.doi.org/10.3390/axioms11100518.

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The answer to the question above is that in all these domains axiomatic characterizations are given of, respectively, mathematical reasoning, certain notions from game theory, and certain social choice rules. The meaning of the completeness theorem in logic is that mathematical reasoning can be characterized by a handful of certain (logical) axioms and rules. If we apply mathematical reasoning to elementary arithmetic, i.e., the addition and multiplication of natural numbers, it turns out that almost all true arithmetical statements, for instance, ∀x∀y[x+y=y+x], can be logically deduced from the axioms of Peano. However, in 1931 Kurt Godel showed that the axioms of Peano do not (fully) characterize the addition and multiplication of the natural numbers, more precisely, that there are certain special self-referential arithmetical sentences that, although true, cannot be deduced from Peano’s axioms. There are axiomatic characterizations of several social choice and ranking rules that say that a given rule is the only one satisfying a particular set of axioms. Arrow’s impossibility theorem in social choice theory tells us that a certain set of, at first sight, quite reasonable axioms for a social ranking rule turns out to be inconsistent. Consequently, a social ranking rule that satisfies the axioms in question cannot exist. Finally, many notions from game theory, such as the Shapley–Shubik and the Banzhaf index, may also be characterized by a set of axioms.

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47

Baur, Michael. "On the Aim of Scientific Theories in Relating to the World: A Defence of the Semantic Account." Dialogue 29, no. 3 (1990): 323–34. http://dx.doi.org/10.1017/s001221730001310x.

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According to the received view of scientific theories, a scientific theory is an axiomatic-deductive linguistic structure which must include some set of guidelines (“correspondence rules”) for interpreting its theoretical terms with reference to the world of observable phenomena. According to the semantic view, a scientific theory need not be formulated as an axiomatic-deductive structure with correspondence rules, but need only specify models which are said to be “isomorphic” with actual phenomenal systems. In this paper, I consider both the received and semantic views as they bear on the issue of how a theory relates to the world (Section 1). Then I offer a critique of some arguments frequently put forth in support of the semantic view (Section 2). Finally, I suggest a more convincing “meta-methodological” argument (based on the thought of Bernard Lonergan) in favour of the semantic view (Section 3).

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Alkema, Abigaël. "Lines and Semi-Countably Differentiable Primes." Mathematical Statistician and Engineering Applications 70, no. 2 (February 26, 2021): 90–98. http://dx.doi.org/10.17762/msea.v70i2.17.

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Let l(u)⊃ |G|. A central problem in higher non-linear graph theoryis the construction of projective numbers. We show that Recent developments in axiomatic set theory [6] have raised the questionof whetherEis not dominated byl. On the other hand, the work in [6, 24] did not consider the hyper-real case.

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ANCONA, DAVIDE, and ELENA ZUCCA. "A theory of mixin modules: algebraic laws and reduction semantics." Mathematical Structures in Computer Science 12, no. 6 (December 2002): 701–37. http://dx.doi.org/10.1017/s0960129502003687.

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Mixins are modules that may contain deferred components, that is, components not defined in the module itself; moreover, in contrast to parameterised modules (like ML functors), they can be mutually dependent and allow their definitions to be overridden. In a preceding paper we defined a syntax and denotational semantics of a kernel language of mixin modules. Here, we take instead an axiomatic approach, giving a set of algebraic laws expressing the expected properties of a small set of primitive operators on mixins. Interpreting axioms as rewriting rules, we get a reduction semantics for the language and prove the existence of normal forms. Moreover, we show that the model defined in the earlier paper satisfies the given axiomatisation.

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50

Wu, Hongbo. "The Set Theory and the Description of Infinity: the Nature of Infinity Perceived from Metaphysics and Ontology." Journal of Education, Humanities and Social Sciences 8 (February 7, 2023): 421–26. http://dx.doi.org/10.54097/ehss.v8i.4282.

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Infinity is possibly one of the most complicated conceptions confronting mathematics and philosophy. People could always conceive it via intuition, but find it hard to describe in the logical realm. In the 19th century, German mathematician Cantor proposed the set theory, for the first time, people could use mathematical tools to explain the nature of infinity. However, such an explanation still lies on the axiomatic foundation, which does not directly fit our intuition about infinity. Therefore, this study will attempt to establish a 'bridge' between the infinity that fits intuition and the infinity that is described by set theory, hence arguing why the set theory could describe infinity universally in the senses of metaphysics and ontology.

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