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  1. Up with Categories, Down with Sets; Out with Categories, In with Sets!Jonathan Kirby - 2024 - Philosophia Mathematica 32 (2):216-227.
    Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for ‘looking down’ or ‘in’ at subsets and the category-theoretic approach is the most practical for ‘looking up’ or ‘out’ at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory.
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  2. Natural Formalization: Deriving the Cantor-Bernstein Theorem in Zf.Wilfried Sieg & Patrick Walsh - 2021 - Review of Symbolic Logic 14 (1):250-284.
    Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame (...)
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  3. From Pictures to Employments: Later Wittgenstein on 'the Infinite'.Philip Bold - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that (...)
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  4. The Turing Degrees and Keisler’s Order.Maryanthe Malliaris & Saharon Shelah - 2024 - Journal of Symbolic Logic 89 (1):331-341.
    There is a Turing functional $\Phi $ taking $A^\prime $ to a theory $T_A$ whose complexity is exactly that of the jump of A, and which has the property that $A \leq _T B$ if and only if $T_A \trianglelefteq T_B$ in Keisler’s order. In fact, by more elaborate means and related theories, we may keep the complexity at the level of A without using the jump.
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  5. The Origin and Significance of Zero: An Interdisciplinary Perspective.Peter Gobets & Robert Lawrence Kuhn (eds.) - 2024 - Leiden: Brill.
    Zero has been axial in human development, but the origin and discovery of zero has never been satisfactorily addressed by a comprehensive, systematic and above all interdisciplinary research program. In this volume, over 40 international scholars explore zero under four broad themes: history; religion, philosophy & linguistics; arts; and mathematics & the sciences. Some propose that the invention/discovery of zero may have been facilitated by the prior evolution of a sophisticated concept of Nothingness or Emptiness (as it is understood in (...)
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  6. Cantor's Illusion.Hudson Richard L. - manuscript
    This analysis shows Cantor's diagonal definition in his 1891 paper was not compatible with his horizontal enumeration of the infinite set M. The diagonal sequence was a counterfeit which he used to produce an apparent exclusion of a single sequence to prove the cardinality of M is greater than the cardinality of the set of integers N.
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  7. Peter Schroeder-Heister on Proof-Theoretic Semantics.Thomas Piecha & Kai F. Wehmeier (eds.) - 2024 - Springer.
    This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...)
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  8. Sizes of Countable Sets.Kateřina Trlifajová - 2024 - Philosophia Mathematica 32 (1):82-114.
    The paper introduces the notion of size of countable sets, which preserves the Part-Whole Principle. The sizes of the natural and the rational numbers, their subsets, unions, and Cartesian products are algorithmically enumerable as sequences of natural numbers. The method is similar to that of Numerosity Theory, but in comparison it is motivated by Bolzano’s concept of infinite series, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree, but some differ, (...)
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  9. Identity and Extensionality in Boffa Set Theory.Nuno Maia & Matteo Nizzardo - 2024 - Philosophia Mathematica 32 (1):115-123.
    Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger contends that this theory cannot be a faithful description of set-theoretic reality. He argues that, even after granting that there are non-well-founded sets, ‘the extensional nature of sets’ precludes numerically distinct Quine atoms. In this paper we uncover important similarities between Rieger’s argument and how non-rigid structures are conceived within mathematical structuralism. This opens the way for an objection against Rieger, whilst (...)
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  10. The MultiAlist System of Thought (philosophical essay).Florentin Smarandache - 2023 - Neutrosophic Sets and Systems 61:598-605.
    The goal of this short note is to expand the concepts of ‘pluralism’, ‘neutrosophy’, ‘refined neutrosophy’, ‘refined neutrosophic set’, ‘multineutrosophic set’, and ‘plithogeny’ (Smarandache 2002, 2013, 2017, 2019, 2021, 2023a, 2023b, 2023c), into a larger category that I will refer to as MultiAlism (or MultiPolar). As a straightforward generalization, I propose the conceptualization of a MultiPolar System (different from a PluriPolar System), which is formed not only by multiple elements that might be random, or contradictory, or adjuvant, but also by (...)
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  11. Advancements in plithogenics exploring new dimensions of soft sets.Sima Das, Monojit Manna & Subrata Modak - 2024 - In Florentin Smarandache, Leyva Vázquez & Maikel Yelandi (eds.), Plithogenics and new types of soft sets. Hershey PA: Engineering Science Reference.
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  12. (1 other version)Development of some new hybrid structures of hypersoft set with possibility-degree settings.Atiqe Ur Rahman, Florentin Smarandache, Muhammad Saeed & Khuram Ali Khan - 2024 - In Florentin Smarandache, Leyva Vázquez & Maikel Yelandi (eds.), Plithogenics and new types of soft sets. Hershey PA: Engineering Science Reference.
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  13. Plithogenics and new types of soft sets.Florentin Smarandache, Leyva Vázquez & Maikel Yelandi (eds.) - 2024 - Hershey PA: Engineering Science Reference.
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  14. The Palgrave Companion to the Philosophy of Set Theory.Carolin Antos, Neil Barton & Giorgio Venturi (eds.) - 2023 - Palgrave.
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  15. The hidden use of new axioms.Deborah Kant - 2023 - In Carolin Antos, Neil Barton & Giorgio Venturi (eds.), The Palgrave Companion to the Philosophy of Set Theory. Palgrave.
    This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of the set-theoretic community---set-theoretic practitioners with (...)
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  16. Operator Counterparts of Types of Reasoning.Urszula Wybraniec-Skardowska - 2023 - Logica Universalis 17 (4):511-528.
    Logical and philosophical literature provides different classifications of reasoning. In the Polish literature on the subject, for instance, there are three popular ones accepted by representatives of the Lvov-Warsaw School: Jan Łukasiewicz, Tadeusz Czeżowski and Kazimierz Ajdukiewicz (Ajdukiewicz in Logika pragmatyczna [Pragmatic Logic]. PWN, Warsaw (1965, 2nd ed. 1974). Translated as: Pragmatic Logic. Reidel & PWN, Dordrecht, 1975). The author of this paper, having modified those classifications, distinguished the following types of reasoning: (1) deductive and (2) non-deductive, and additionally two (...)
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  17. Against ‘Interpretation’: Quantum Mechanics Beyond Syntax and Semantics.Raoni Wohnrath Arroyo & Gilson Olegario da Silva - 2022 - Axiomathes 32 (6):1243-1279.
    The question “what is an interpretation?” is often intertwined with the perhaps even harder question “what is a scientific theory?”. Given this proximity, we try to clarify the first question to acquire some ground for the latter. The quarrel between the syntactic and semantic conceptions of scientific theories occupied a large part of the scenario of the philosophy of science in the 20th century. For many authors, one of the two currents needed to be victorious. We endorse that such debate, (...)
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  18. Expanding the notion of inconsistency in mathematics: the theoretical foundations of mutual inconsistency.Carolin Antos - forthcoming - From Contradiction to Defectiveness to Pluralism in Science: Philosophical and Formal Analyses.
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  19. (1 other version)Are Large Cardinal Axioms Restrictive?Neil Barton - 2023 - Philosophia Mathematica 31 (3):372-407.
    The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of (...)
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  20. Extensional Realizability and Choice for Dependent Types in Intuitionistic Set Theory.Emanuele Frittaion - 2023 - Journal of Symbolic Logic 88 (3):1138-1169.
    In [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$. We then show that adding (...)
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  21. An overview of Conceptual Analysis and Design.Dmitry E. Borisoglebsky - 2023 - Knowledge - International Journal 57 (3):353–365.
    Conceptual Analysis and Design (mCAD) is an information and cognitive technology for knowledge and systems engineering. A conceptual system for a complex knowledge domain contains thousands of linked concepts, necessary in the engineering and management of big and complex systems. Naturally evolved conceptual systems usually contain conceptual gaps and have multiple logical fallacies. mCAD addresses these issues by axiomatic deduction of concepts. This article is a concise overview of Conceptual Analysis and Design, covering its foundations, technological aspects, and notable applications.
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  22. A beginner's guide to mathematical logic.Raymond M. Smullyan - 2014 - Mineola, New York: Dover Publications.
    Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts. He highlights each subject with inventive explanations and unique problems. Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs. Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and graduate (...)
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  23. Concise introduction to logic and set theory.Iqbal H. Jebril - 2022 - Boca Raton: CRC Press, Taylor & Francis Group. Edited by Hemen Dutta & Ilwoo Cho.
    This book deals with two important branches of mathematics, namely, logic and set theory. Logic and set theory are closely related and play very crucial roles in the foundation of mathematics, and together produce several results in all of mathematics. The topics of logic and set theory are required in many areas of physical sciences, engineering, and technology. The book offers solved examples and exercises, and provides reasonable details to each topic discussed, for easy understanding. The book is designed for (...)
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  24. The notion of number and the notion of class.Richard Allen Arms - 1917 - Philadelphia,: Palala Press.
    This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...)
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  25. Précis de philosophie: classes de mathématiques et de mathématiques-technique.Armand Cuvillier - 1954 - Paris,: A. Colin.
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  26. Almene begreber fra logik, mængdelære og algebra.Bent Christiansen - 1964 - [Copenhagen]: Munksgaard. Edited by Jonas Lichtenberg, Pedersen, Johs & [From Old Catalog].
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  27. What makes a `good' modal theory of sets?Neil Barton - manuscript
    I provide an examination and comparison of modal theories for underwriting different non-modal theories of sets. I argue that there is a respect in which the `standard' modal theory for set construction---on which sets are formed via the successive individuation of powersets---raises a significant challenge for some recently proposed `countabilist' modal theories (i.e. ones that imply that every set is countable). I examine how the countabilist can respond to this issue via the use of regularity axioms and raise some questions (...)
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  28. Book 5 — ӝ : A Treatise Towards a Robust Set Theoretic Ontology.Jacob Parr - manuscript
    The author solves open problems in ontology , set theory , mathematics , modal logic . The author details the ontological commitments and implications of said found solutions and proves the need for and provides the solution to a new fundamental theory of algebra ... The author also provides the correct pronunciation of "prima facie" , a means for remembering the correct pronunciation of "prima facie" — and suggests others adopt the correct pronunciation of -- it's fun to say the (...)
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  29. Set Theory with Urelements.Bokai Yao - 2023 - Dissertation, University of Notre Dame
    This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also explored. In Chapter (...)
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  30. Schopenhauers Logikdiagramme in den Mathematiklehrbüchern Adolph Diesterwegs.Jens Lemanski - 2022 - Siegener Beiträge Zur Geschichte Und Philosophie der Mathematik 16:97-127.
    Ein Beispiel für die Rezeption und Fortführung der schopenhauerschen Logik findet man in den Mathematiklehrbüchern Friedrich Adolph Wilhelm Diesterwegs (1790–1866), In diesem Aufsatz werden die historische und systematische Dimension dieser Anwendung von Logikdiagramme auf die Mathematik skizziert. In Kapitel 2 wird zunächst die frühe Rezeption der schopenhauerschen Logik und Philosophie der Mathematik vorgestellt. Dabei werden einige oftmals tradierte Vorurteile, die das Werk Schopenhauers betreffen, in Frage gestellt oder sogar ausgeräumt. In Kapitel 3 wird dann die Philosophie der Mathematik und der (...)
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  31. Reduction games, provability and compactness.Damir D. Dzhafarov, Denis R. Hirschfeldt & Sarah Reitzes - 2022 - Journal of Mathematical Logic 22 (3).
    Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [math] principles over [math]-models of [math]. They also introduced a version of this game that similarly captures provability over [math]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [math] between two (...)
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  32. Sigma-Prikry forcing II: Iteration Scheme.Alejandro Poveda, Assaf Rinot & Dima Sinapova - 2022 - Journal of Mathematical Logic 22 (3):2150019.
    In Part I of this series [A. Poveda, A. Rinot and D. Sinapova, Sigma-Prikry forcing I: The axioms, Canad. J. Math. 73(5) (2021) 1205–1238], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We showed that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a non-reflecting (...)
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  33. (1 other version)Mathematical Modality: An Investigation in Higher-order Logic.Andrew Bacon - forthcoming - Journal of Philosophical Logic.
    An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency about the (...)
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  34. Lower and Upper Estimates of the Quantity of Algebraic Numbers.Yaroslav Sergeyev - 2023 - Mediterranian Journal of Mathematics 20:12.
    It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using ①-based infinite numbers is applied to measure the set A (where the number ① is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable (...)
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  35. Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set.Florentin Smarandache - 2018 - Neutrosophic Sets and Systems 22 (1):168-170.
    In this paper, we generalize the soft set to the hypersoft set by transforming the function F into a multi-attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.
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  36. Introduction to the Complex Refined Neutrosophic Set.Florentin Smarandache - 2017 - Critical Review: A Journal of Politics and Society 14 (1):5-9.
    In this paper, one extends the single-valued complex neutrosophic set to the subsetvalued complex neutrosophic set, and afterwards to the subset-valued complex refined neutrosophic set.
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  37. Realizing realizability results with classical constructions.Asaf Karagila - 2019 - Bulletin of Symbolic Logic 25 (4):429-445.
    J. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.
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  38. MORRIS, Sean: Quine, New Foundations, and the Philosophy of Set Theory.Ádám Tamas Tuboly - 2019 - Filozofia 74 (6).
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  39. RETRACTED ARTICLE: The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic (I).Janusz Czelakowski - 2023 - Studia Logica 111 (2):357-358.
    The paper is concerned with the old conjecture that there are infinitely many twin primes. In the paper we show that this conjecture is true, that is, it is true in the standard model of arithmetic. The proof is based on Rasiowa–Sikorski Lemma. The key role are played by the derived notion of a Rasiowa–Sikorski set and the method of forcing adjusted to arbitrary first–order languages. This approach was developed in the papers Czelakowski [ 4, 5 ]. The central idea (...)
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  40. An axiomatic approach to forcing in a general setting.Rodrigo A. Freire & Peter Holy - 2022 - Bulletin of Symbolic Logic 28 (3):427-450.
    The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which (...)
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  41. Logic and discrete mathematics: a concise introduction.Willem Conradie - 2015 - Hoboken, NJ, USA: Wiley. Edited by Valentin Goranko.
    A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical (...)
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  42. Four cardinals and their relations in ZF.Lorenz Halbeisen, Riccardo Plati, Salome Schumacher & Saharon Shelah - 2023 - Annals of Pure and Applied Logic 174 (2):103200.
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  43. A first course in mathematical logic and set theory.Michael L. O'Leary - 2016 - Hoboken, New Jersey: Wiley.
    Highlighting the applications and notations of basic mathematical concepts within the framework of logic and set theory, A First Course in Mathematical Logic and Set Theory introduces how logic is used to prepare and structure proofs and solve more complex problems.
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  44. Basic discrete mathematics: logic, set theory, & probability.Richard Kohar - 2016 - New Jersey: World Scientific.
    This lively introductory text exposes the student in the humanities to the world of discrete mathematics. A problem-solving based approach grounded in the ideas of George Pólya are at the heart of this book. Students learn to handle and solve new problems on their own. A straightforward, clear writing style and well-crafted examples with diagrams invite the students to develop into precise and critical thinkers. Particular attention has been given to the material that some students find challenging, such as proofs. (...)
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  45. Principles of mathematics: a primer.Vladimir Lepetic - 2016 - Hoboken, New Jersey: Wiley.
    Set theory -- Logic -- Proofs -- Functions -- Group theory -- Linear algebra.
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  46. Neurosophic optimization and it application on structural designs.Mridula Sarkar - 2018 - Brussels: Pons. Edited by Tapan Kumar Roy & Florentin Smarandache.
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  47. Advances of standard and nonstandard neutrosophic theories.Florentin Smarandache (ed.) - 2019 - Brussels, Belgium: Pons.
    In this book, we approach different topics related to neutrosophics, such as: Neutrosophic Set, Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set, Picture Fuzzy Set, Ternary Fuzzy Set, Pythagorean Fuzzy Set, Atanassov’s Intuitionistic Fuzzy Set of second type, Spherical Fuzzy Set, n-HyperSpherical Neutrosophic Set, q-Rung Orthopair Fuzzy Set, truth-membership, indeterminacy-membership, falsehood-nonmembership, Regret Theory, Grey System Theory, Three-Ways Decision, n-Ways Decision, Neutrosophy, Neutrosophication, Neutrosophic Probability, Refined Neutrosophy, Refined Neutrosophication, Nonstandard Analysis; (Theory, NeutroTheory, AntiTheory), S-denying an Axiom, Multispace with Multistructure, and so on.
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  48. Neutrosophic Triplet Structures. Volume I.Florentin Smarandache & Memet Şahin (eds.) - 2019 - Brussels, Belgium, EU: Pons editions.
    Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory was firstly proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, (...)
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  49. Generalized interval neutrosophic Choquet aggregation operators and their applications.Xin Li, Xiaohong Zhang & Choonkil Park - 2018 - In Florentin Smarandache, Xiaohong Zhang & Mumtaz Ali (eds.), Algebraic structures of neutrosophic triplets, neutrosophic duplets, or neutrosophic multisets. Basel: MDPI.
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  50. Ritratti dell'infinito: dodici primi piani e tre foto di gruppo.Piergiorgio Odifreddi - 2020 - [Milan]: Rizzoli.
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1 — 50 / 2829