Set Theory

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  1. Ontology and Arbitrariness.David Builes - 2022 - Australasian Journal of Philosophy 100 (3):485-495.
    In many different ontological debates, anti-arbitrariness considerations push one towards two opposing extremes. For example, in debates about mereology, one may be pushed towards a maximal ontology (mereological universalism) or a minimal ontology (mereological nihilism), because any intermediate view seems objectionably arbitrary. However, it is usually thought that anti-arbitrariness considerations on their own cannot decide between these maximal or minimal views. I will argue that this is a mistake. Anti-arbitrariness arguments may be used to motivate a certain popular thesis in (...)
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  2. Algunos tópicos de Lógica matemática y los Fundamentos de la matemática.Franklin Galindo - manuscript
    En este trabajo matemático-filosófico se estudian cuatro tópicos de la Lógica matemática: El método de construcción de modelos llamado Ultraproductos, la Propiedad de Interpolación de Craig, las Álgebras booleanas y los Órdenes parciales separativos. El objetivo principal del mismo es analizar la importancia que tienen dichos tópicos para el estudio de los fundamentos de la matemática, desde el punto de vista del platonismo matemático. Para cumplir con tal objetivo se trabajará en el ámbito de la Matemática, de la Metamatemática y (...)
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  3. Un teorema sobre el Modelo de Solovay.Franklin Galindo - 2020 - Divulgaciones Matematicas 21 (1-2): 42–46.
    The objective of this article is to present an original proof of the following theorem: Thereis a generic extension of the Solovay’s model L(R) where there is a linear order of P(N)/fin that extends to the partial order (P(N)/f in), ≤*). Linear orders of P(N)/fin are important because, among other reasons, they allow constructing non-measurable sets, moreover they are applied in Ramsey's Theory .
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  4. Tópicos de Ultrafiltros.Franklin Galindo - 2020 - Divulgaciones Matematicas 21 (1-2):54-77.
    Ultrafilters are very important mathematical objects in mathematical research [6, 22, 23]. There are a wide variety of classical theorems in various branches of mathematics where ultrafilters are applied in their proof, and other classical theorems that deal directly with ultrafilters. The objective of this article is to contribute (in a divulgative way) to ultrafilter research by describing the demonstrations of some such theorems related (uniquely or in combination) to topology, Measure Theory, algebra, combinatorial infinite, set theory and first-order logic, (...)
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  5. Dos Tópicos de Lógica Matemática y sus Fundamentos.Franklin Galindo - 2014 - Episteme NS: Revista Del Instituto de Filosofía de la Universidad Central de Venezuela 34 (1):41-66..
    El objetivo de este artículo es presentar dos tópicos de Lógica matemática y sus fundamentos: El primer tópico es una actualización de la demostración de Alonzo Church del Teorema de completitud de Gödel para la Lógica de primer orden, la cual aparece en su texto "Introduction to Mathematical Logic" (1956) y usa el procedimientos efectivos de Forma normal prenexa y Forma normal de Skolem; y el segundo tópico es una demostración de que la propiedad de partición (tipo Ramsey) del espacio (...)
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  6. El Teorema de Completitud de Gödel, el Teorema del Colapso Transitivo de Mostowski y el Principio de Reflexión.Franklin Galindo - manuscript
    Es conocido que el Teorema de Completitud de Gödel, el Teorema del Colapso Transitivo de Mostowski y el Principio de Reflexión son resultados muy útiles en las investigaciones de Lógica matemática y/o los Fundamentos de la matemática. El objetivo de este trabajo es presentar algunas demostraciones clásicas de tales resultados: Dos del Teorema de Completitud de Gödel, una del Teorema del Colapso Transitivo de Mostowski y una del Principio de Reflexión. Se aspira que estas notas sean de utilidad para estudiar (...)
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  7. El Método de Forcing: Algunas aplicaciones y una aproximación a sus fundamentos metamatemáticos.Franklin Galindo - manuscript
    Es conocido que el método de forcing es una de las técnicas de construcción de modelos más importantes de la Teoría de conjuntos en la actualidad, siendo el mismo muy útil para investigar problemas de matemática y/o de fundamentos de la matemática. El destacado matemático Joan Bagaria afirma lo siguiente sobre el método de forcing en su artículo "Paul Cohen y la técnica del forcing" (Gaceta de la Real Sociedad Matemática Española, Vol. 2, Nº 3, 1999, págs 543-553) : "Aunque (...)
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  8. Algunas notas introductorias sobre la Teoría de Conjuntos.Franklin Galindo - 2019 - Apuntes Filosóficos: Revista Semestral de la Escuela de Filosofía 18 (55):201-232.
    The objective of this document is to present three introductory notes on set theory: The first note presents an overview of this discipline from its origins to the present, in the second note some considerations are made about the evaluation of reasoning applying the first-order Logic and Löwenheim's theorems, Church Indecidibility, Completeness and Incompleteness of Gödel, it is known that the axiomatic theories of most commonly used sets are written in a specific first-order language, that is, they are developed within (...)
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  9. Mathematics is Ontology? A Critique of Badiou's Ontological Framing of Set Theory.Roland Bolz - 2020 - Filozofski Vestnik 2 (41):119-142.
    This article develops a criticism of Alain Badiou’s assertion that “mathematics is ontology.” I argue that despite appearances to the contrary, Badiou’s case for bringing set theory and ontology together is problematic. To arrive at this judgment, I explore how a case for the identification of mathematics and ontology could work. In short, ontology would have to be characterised to make it evident that set theory can contribute to it fundamentally. This is indeed how Badiou proceeds in Being and Event. (...)
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  10. Fermat’s Last Theorem Proved in Hilbert Arithmetic. II. Its Proof in Hilbert Arithmetic by the Kochen-Specker Theorem with or Without Induction.Vasil Penchev - 2022 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 14 (10):1-52.
    The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The (...)
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  11. What the Tortoise Said to Achilles: Lewis Carroll’s Paradox in Terms of Hilbert Arithmetic.Vasil Penchev - 2021 - Logic and Philosophy of Mathematics eJournal (Elsevier: SSRN) 13 (22):1-32.
    Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with (...)
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  12. Reinterpreting the Universe-Multiverse Debate in Light of Inter-Model Inconsistency in Set Theory.Daniel Kuby - manuscript
    In this paper I apply the concept of _inter-Model Inconsistency in Set Theory_ (MIST), introduced by Carolin Antos (this volume), to select positions in the current universe-multiverse debate in philosophy of set theory: I reinterpret H. Woodin’s _Ultimate L_, J. D. Hamkins’ multiverse, S.-D. Friedman’s hyperuniverse and the algebraic multiverse as normative strategies to deal with the situation of de facto inconsistency toleration in set theory as described by MIST. In particular, my aim is to situate these positions on the (...)
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  13. Against ‘Interpretation’: Quantum Mechanics Beyond Syntax and Semantics.Raoni Wohnrath Arroyo & Gilson Olegario da Silva - forthcoming - Axiomathes:1-37.
    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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  14. A General Framework for a Second Philosophy Analysis of Set-Theoretic Methodology.Carolin Antos & Deborah Kant - manuscript
    Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the procedure and (...)
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  15. Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I).Jaykov Foukzon - 2021 - Journal of Advances in Mathematics and Computer Science 36 (2):73-88.
    In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.
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  16. Forcing and the Universe of Sets: Must We Lose Insight?Neil Barton - 2020 - Journal of Philosophical Logic 49 (4):575-612.
    A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems (...)
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  17. Set-Theoretic Pluralism and the Benacerraf Problem.Justin Clarke-Doane - 2020 - Philosophical Studies 177 (7):2013-2030.
    Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...)
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  18. Inner-Model Reflection Principles.Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Joel David Hamkins, Jonas Reitz & Ralf Schindler - 2020 - Studia Logica 108 (3):573-595.
    We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...)
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  19. On Classical Set-Compatibility.Luis Felipe Bartolo Alegre - 2020 - El Jardín de Senderos Que Se Bifurcan y Confluyen: Filosofía, Lógica y Matemáticas.
    In this paper, I generalise the logical concept of compatibility into a broader set-theoretical one. The basic idea is that two sets are incompatible if they produce at least one pair of opposite objects under some operation. I formalise opposition as an operation ′ ∶ E → E, where E is the set of opposable elements of our universe U, and I propose some models. From this, I define a relation ℘U × ℘U × ℘U^℘U, which has (mutual) logical compatibility (...)
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  20. About Multidimensional Spaces.Alexander Klimets - 2004 - Physics of Consciousness and Life,Cosmology and Astrophysics 4 (3):41-44.
    In the article, based on the philosophical analysis of the concept of "three-dimensional space", a model of multidimensional space is constructed, reflecting the properties of intersections of multidimensional spaces. The model reveals some unusual aspects of multidimensional spaces.
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  21. From Art to Information System.Miro Brada - 2021 - AGI Laboratory.
    This insight to art came from chess composition concentrating art in a very dense form. To identify and mathematically assess the uniqueness is the key applicable to other areas eg. computer programming. Maximization of uniqueness is minimization of entropy that coincides as well as goes beyond Information Theory (Shannon, 1948). The reusage of logic as a universal principle to minimize entropy, requires simplified architecture and abstraction. Any structures (e.g. plugins) duplicating or dividing functionality increase entropy and so unreliability (eg. British (...)
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  22. Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems.Yaroslav Sergeyev - 2017 - EMS Surveys in Mathematical Sciences 4 (2):219–320.
    In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...)
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  23. Metaphysical and Absolute Possibility.Justin Clarke-Doane - 2019 - Synthese 198 (Suppl 8):1861-1872.
    It is widely alleged that metaphysical possibility is “absolute” possibility Conceivability and possibility, Clarendon, Oxford, 2002, p 16; Stalnaker, in: Stalnaker Ways a world might be: metaphysical and anti-metaphysical essays, Oxford University Press, Oxford, 2003, pp 201–215; Williamson in Can J Philos 46:453–492, 2016). Kripke calls metaphysical necessity “necessity in the highest degree”. Van Inwagen claims that if P is metaphysically possible, then it is possible “tout court. Possible simpliciter. Possible period…. possib without qualification.” And Stalnaker writes, “we can agree (...)
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  24. Infinite Value and the Best of All Possible Worlds.Nevin Climenhaga - 2018 - Philosophy and Phenomenological Research 97 (2):367-392.
    A common argument for atheism runs as follows: God would not create a world worse than other worlds he could have created instead. However, if God exists, he could have created a better world than this one. Therefore, God does not exist. In this paper I challenge the second premise of this argument. I argue that if God exists, our world will continue without end, with God continuing to create value-bearers, and sustaining and perfecting the value-bearers he has already created. (...)
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  25. Can Modalities Save Naive Set Theory?Peter Fritz, Harvey Lederman, Tiankai Liu & Dana Scott - 2018 - Review of Symbolic Logic 11 (1):21-47.
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  26. There is No Set of All Truths.Patrick Grim - 1984 - Analysis 44 (4):206-208.
    A Cantorian argument that there is no set of all truths. There is, for the same reason, no possible world as a maximal set of propositions. And omniscience is logically impossible.
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  27. Ortega y Gasset on Georg Cantor’s Theory of Transfinite Numbers.Lior Rabi - 2016 - Kairos (15):46-70.
    Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and articulated a (...)
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  28. Las reglas de Irving Copi y Carl Cohen son una condición necesaria y suficiente de la validez en los silogismos categóricos de forma estándar.Franklin Galindo & Kris Martins - 2005 - Episteme 25 (1):123-148.
    Resumen: En la actualidad uno de los libros más usados para dar lógica elemental es el de Irving Copi y Carl Cohen (Introducción a la lógica, 2001), allí se presentan unas reglas para decidir la validez de los silogismos categóricos de forma estándar. Pero en tal texto ni en ninguno que nosotros conozcamos se ofrece una fundamentación de las mismas. Es decir, una demostración de que ellas son realmente una condición necesaria y suficiente de la validez de un silogismo categórico (...)
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  29. Groups, Sets, and Wholes.Barry Smith - 2003 - Rivista di Estetica 43 (24):126-127.
    As he recalls in his book Naive Physics, Paolo Bozzi’s experiments on naïve or phenomenological physics were partly inspired by Aristotle’s spokesman Simplicio in Galileo’s Dialogue. Aristotle’s ‘naïve’ views of physical reality reflect the ways in which we are disposed perceptually to organize the physical reality we see. In what follows I want to apply this idea to the notion of a group, a term which I shall apply as an umbrella expression embracing ordinary visible collections (of pieces of fruit (...)
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  30. ""Lambda Theory: Introduction of a Constant for" Nothing" Into Set Theory, a Model of Consistency and Most Noticeable Conclusions.Laurent Dubois - 2013 - Logique Et Analyse 56 (222):165-181.
    The purpose of this article is to present several immediate consequences of the introduction of a new constant called Lambda in order to represent the object ``nothing" or ``void" into a standard set theory. The use of Lambda will appear natural thanks to its role of condition of possibility of sets. On a conceptual level, the use of Lambda leads to a legitimation of the empty set and to a redefinition of the notion of set. It lets also clearly appear (...)
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  31. Defending the Axioms-On the Philosophical Foundations of Set Theory, Penelope Maddy. [REVIEW]Eduardo Castro - 2012 - Teorema: International Journal of Philosophy 31 (1):147-150.
    Review of Maddy, Penelope "Defending the Axioms".
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  32. Glossary of Ontology.Juan José Luetich - 2012 - Identification Transactions of The Luventicus Academy (ISSN 1666-7581) 1 (2):4.
    The previous issue provided an introduction to ontology in which the word “being,” for example, was used in two different ways. Here we introduce a list that includes the following terms with their precise definitions and comments regarding the use they should be given: to be, as a verb with or without meaning, being as a noun; essence and existence; chaos, demiurge and cosmos; ontology and semiology; ontological tables and Carroll diagrams.
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  33. On the Infinite in Mereology with Plural Quantification.Massimiliano Carrara & Enrico Martino - 2011 - Review of Symbolic Logic 4 (1):54-62.
    In Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification (...)
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  34. Individuals, Universals, Collections: On the Foundational Relations of Ontology.Thomas Bittner, Maureen Donnelly & Barry Smith - 2004 - In Achille Varzi Laure Vieu (ed.), ”, Formal Ontology in Information Systems. Proceedings of the Third International Conference. Amsterdam: IOS Press. pp. 37–48.
    This paper provides an axiomatic formalization of a theory of foundational relations between three categories of entities: individuals, universals, and collections. We deal with a variety of relations between entities in these categories, including the is-a relation among universals and the part-of relation among individuals as well as cross-category relations such as instance-of, member-of, and partition-of. We show that an adequate understanding of the formal properties of such relations – in particular their behavior with respect to time – is critical (...)
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  35. On Sets and Worlds: A Reply to Menzel.Patrick Grim - 1986 - Analysis 46 (4):186 - 191.
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  36. On Omniscience and a 'Set of All Truths': A Reply to Bringsjord.Patrick Grim - 1990 - Analysis 50 (4):271 - 276.
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  37. Review of Oppy's Philosophical Perspectives on Infinity. [REVIEW]Anne Newstead - 2007 - Australasian Journal of Philosophy 85 (4):679-695.
    This is a book review of Oppy's "Philosophical Perspectives on Infinity", which is of interest to those in metaphysics, epistemology, philosophy of science, mathematics, and philosophy of religion.
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  38. Worlds by Supervenience: Some Further Problems.Patrick Grim - 1997 - Analysis 57 (2):146-51.
    Allen s has proposed a new approach to possible worlds, designed explicitly to overcome Cantorian difficulties for possible worlds construed as maximal consistent set of propositions. I emphasize some of the distinctive features of Hazenworlds, some of their weaknesses, and some further Cantorian problems for worlds against which they seem powerless.
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  39. On Infinite Number and Distance.Jeremy Gwiazda - 2012 - Constructivist Foundations 7 (2):126-130.
    Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue that (...)
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  40. Finite Level Borel Games and a Problem Concerning the Jump Hierarchy.Harold T. Hodes - 1984 - Journal of Symbolic Logic 49 (4):1301-1318.
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  41. Relevance, Relatedness and Restricted Set Theory.Barry Smith - 1991 - In Georg Schurz & Georg Jakob Wilhelm Dorn (eds.), Advances in Scientific Philosophy. Amsterdam: Rodopi. pp. 45-56.
    Relevance logic has become ontologically fertile. No longer is the idea of relevance restricted in its application to purely logical relations among propositions, for as Dunn has shown in his (1987), it is possible to extend the idea in such a way that we can distinguish also between relevant and irrelevant predications, as for example between “Reagan is tall” and “Reagan is such that Socrates is wise”. Dunn shows that we can exploit certain special properties of identity within the context (...)
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The Nature of Sets
  1. Undaunted Sets.Varol Akman - 1992 - ACM SIGACT News 23 (1):47-48.
    This is a short piece of humor (I hope) on nonstandard set theories. An earlier version appeared in Bull. EATCS 45: 146-147 (1991).
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  2. Do Abstract Mathematical Axioms About Infinite Sets Apply To The Real, Physical Universe?Roger Granet - manuscript
    Suppose one has a system, the infinite set of positive integers, P, and one wants to study the characteristics of a subset (or subsystem) of that system, the infinite subset of odd positives, O, relative to the overall system. In mathematics, this is done by pairing off each odd with a positive, using a function such as O=2P+1. This puts the odds in a one-to-one correspondence with the positives, thereby, showing that the subset of odds and the original set of (...)
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  3. The Many and the One: A Philosophical Study of Plural Logic.Salvatore Florio & Øystein Linnebo - 2021 - Oxford, England: Oxford University Press.
    Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between (...)
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  4. Note on the Significance of the New Logic.Frederique Janssen-Lauret - 2018 - The Reasoner 6 (12):47-48.
    Brief note explaining the content, importance, and historical context of my joint translation of Quine's The Significance of the New Logic with my single-authored historical-philosophical essay 'Willard Van Orman Quine's Philosophical Development in the 1930s and 1940s'.
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  5. Cantor’s Proof in the Full Definable Universe.Laureano Luna & William Taylor - 2010 - Australasian Journal of Logic 9:10-25.
    Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the (...)
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  6. Relevant First-Order Logic LP# and Curry’s Paradox Resolution.Jaykov Foukzon - 2015 - Pure and Applied Mathematics Journal Volume 4, Issue 1-1, January 2015 DOI: 10.11648/J.Pamj.S.2015040101.12.
    In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗n^C.
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  7. Groundedness - Its Logic and Metaphysics.Jönne Kriener - 2014 - Dissertation, Birkbeck College, University of London
    In philosophical logic, a certain family of model constructions has received particular attention. Prominent examples are the cumulative hierarchy of well-founded sets, and Kripke's least fixed point models of grounded truth. I develop a general formal theory of groundedness and explain how the well-founded sets, Cantor's extended number-sequence and Kripke's concepts of semantic groundedness are all instances of the general concept, and how the general framework illuminates these cases. Then, I develop a new approach to a grounded theory of proper (...)
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  8. What Constitutes a Formal Analogy?Kenneth Olson & Gilbert Plumer - 2002 - In Hans V. Hansen, Christopher W. Tindale, J. Anthony Blair, Ralph H. Johnson & Robert C. Pinto (eds.), Argumentation and its Applications [CD-ROM]. Ontario Society for the Study of Argumentation. pp. 1-8.
    There is ample justification for having analogical material in standardized tests for graduate school admission, perhaps especially for law school. We think that formal-analogy questions should compare different scenarios whose structure is the same in terms of the number of objects and the formal properties of their relations. The paper deals with this narrower question of how legitimately to have formal analogy test items, and the broader question of what constitutes a formal analogy in general.
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  9. Retrieving the Mathematical Mission of the Continuum Concept From the Transfinitely Reductionist Debris of Cantor’s Paradise. Extended Abstract.Edward G. Belaga - forthcoming - International Journal of Pure and Applied Mathematics.
    What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...)
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