Set Theory

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  1. The Ontological Import of Adding Proper Classes.Alfredo Roque Freire & Rodrigo de Alvarenga Freire - 2019 - Manuscrito 42 (2):85-112.
    In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of (...)
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  2. 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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  3. Metaphysical and Absolute Possibility.Justin Clarke-Doane - forthcoming - Synthese (special issue):1-12.
    It is widely alleged that metaphysical possibility is “absolute” possibility (Kripke [1980], Lewis [1986], Rosen [2006, 16], Stalnaker [2005, 203], Williamson [2016, 460]). Indeed, this is arguably its metaphysical significance. Kripke calls metaphysical necessity “necessity in the highest degree” ([1980, 99]). Williamson calls metaphysical possibility the “maximal objective modality” [2016, 459]. Rosen says that “metaphysical possibility is the [most inclusive] sort of real possibility” ([2006, 16]). And Stalnaker writes, “we can agree with Frank Jackson, David Chalmers, Saul Kripke, David Lewis, (...)
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  4. Set-Theoretic Pluralism and the Benacerraf Problem.Justin Clarke-Doane - forthcoming - Philosophical Studies:1-18.
    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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  5. 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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  6. 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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  7. ""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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  8. 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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  9. 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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  10. 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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  11. 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. 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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  12. 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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  13. 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. 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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  2. 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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  3. 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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  4. 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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  5. 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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  6. 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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The Iterative Conception of Set
  1. Against the Iterative Conception of Set.Edward Ferrier - 2019 - Philosophical Studies 176 (10):2681-2703.
    According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...)
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  2. Quantification and Paradox.Edward Ferrier - 2018 - Dissertation, University of Massachusetts Amherst
    I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...)
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  3. The Search for New Axioms in the Hyperuniverse Programme.Claudio Ternullo & Sy-David Friedman - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Berlin: Springer. pp. 165-188.
    The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...)
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  4. Ipotesi del Continuo.Claudio Ternullo - 2017 - Aphex 16.
    L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...)
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  5. Modal Set Theory.Christopher Menzel - forthcoming - In Otávio Bueno & Scott Shalkowski (eds.), The Routledge Handbook of Modality. London and New York: Routledge.
    This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.
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  6. The Argument From Collections.Christopher Menzel - 2018 - In J. Walls & T. Dougherty (eds.), Two Dozen (or so) Arguments for God: The Plantinga Project. New York: Oxford University Press. pp. 29-58.
    Very broadly, an argument from collections is an argument that purports to show that our beliefs about sets imply — in some sense — the existence of God. Plantinga (2007) first sketched such an argument in “Two Dozen” and filled it out somewhat in his 2011 monograph Where the Conflict Really Lies: Religion, Science, and Naturalism. In this paper I reconstruct what strikes me as the most plausible version of Plantinga’s argument. While it is a good argument in at least (...)
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  7. Indefinite Divisibility.Jeffrey Sanford Russell - 2016 - Inquiry: An Interdisciplinary Journal of Philosophy 59 (3):239-263.
    Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of (...)
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  8. Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory.Valérie Lynn Therrien - 2012 - Ithaque 10:43-65.
    In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...)
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  9. Wide Sets, ZFCU, and the Iterative Conception.Christopher Menzel - 2014 - Journal of Philosophy 111 (2):57-83.
    The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...)
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Ontology of Sets
  1. To Reduce Nothingness Into a Reference by Falsity.Hazhir Roshangar - manuscript
    Assuming the absolute nothingness as the most basic object of thought, I present a way to refer to this object, by reducing it onto a primitive object that supersedes and comes right after the absolute nothingness. The new primitive object that is constructed can be regarded as a formal system that can generate some infinite variety of symbols. [The PDF here is outdated, for a recent draft please contact me.].
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  2. What We Talk About When We Talk About Numbers.Richard Pettigrew - manuscript
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
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  3. The Argument From Collections.Christopher Menzel - 2018 - In J. Walls & T. Dougherty (eds.), Two Dozen (or so) Arguments for God: The Plantinga Project. New York: Oxford University Press. pp. 29-58.
    Very broadly, an argument from collections is an argument that purports to show that our beliefs about sets imply — in some sense — the existence of God. Plantinga (2007) first sketched such an argument in “Two Dozen” and filled it out somewhat in his 2011 monograph Where the Conflict Really Lies: Religion, Science, and Naturalism. In this paper I reconstruct what strikes me as the most plausible version of Plantinga’s argument. While it is a good argument in at least (...)
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  4. A Note on Gabriel Uzquiano’s “Varieties of Indefinite Extensibility”.Simon Hewitt - unknown - Notre Dame Journal of Formal Logic 59 (3):455-459.
    It is argued that Gabriel Uzquiano's approach to set-theoretic indefinite extensibility is a version of in rebus structuralism, and therefore suffers from a vacuity problem.
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  5. Wittgenstein And Labyrinth Of ‘Actual Infinity’: The Critique Of Transfinite Set Theory.Valérie Lynn Therrien - 2012 - Ithaque 10:43-65.
    In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...)
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  6. Structuralism and Its Ontology.Marc Gasser - 2015 - Ergo: An Open Access Journal of Philosophy 2 (1):1-26.
    A prominent version of mathematical structuralism holds that mathematical objects are at bottom nothing but "positions in structures," purely relational entities without any sort of nature independent of the structure to which they belong. Such an ontology is often presented as a response to Benacerraf's "multiple reductions" problem, or motivated on hermeneutic grounds, as a faithful representation of the discourse and practice of mathematics. In this paper I argue that there are serious difficulties with this kind of view: its proponents (...)
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  7. Wide Sets, ZFCU, and the Iterative Conception.Christopher Menzel - 2014 - Journal of Philosophy 111 (2):57-83.
    The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...)
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  8. To Be or to Be Not, That is the Dilemma.Juan José Luetich - 2012 - Identification Transactions of The Luventicus Academy (ISSN 1666-7581) 1 (1):4.
    A set is precisely defined. A given element either belongs or not to a set. However, since all of the elements being considered belong to the universe, if the element does not belong to the set, it belongs to its complement, that is, what remains after all of the elements from the set are removed from the universe.
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  9. Aristotle and Modern Mathematical Theories of the Continuum.Anne Newstead - 2001 - In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.
    This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...)
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  10. Cantor on Infinity in Nature, Number, and the Divine Mind.Anne Newstead - 2009 - American Catholic Philosophical Quarterly 83 (4):533-553.
    The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Cantor’s thought with (...)
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  11. Biometaphysics.Barry Smith - 2009 - In Robin Le Poidevin, Peter Simons, Andrew McGonigal & Ross Cameron (eds.), The Routledge Companion to Metaphysics. Routledge. pp. 537-544.
    While Darwin is commonly supposed to have demonstrated the inapplicability of the Aristotelian ontology of species to biological science, recent developments, especially in the wake of the Human Genome Project, have given rise to a new golden age of classification in which ontological ideas -- as for example in the Gene Ontology, the Cell Ontology, the Protein Ontology, and so forth -- are once again playing an important role. In regard to species, on the other hand, matters are more complex. (...)
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The Nature of Sets, Misc
  1. The Search for New Axioms in the Hyperuniverse Programme.Claudio Ternullo & Sy-David Friedman - 2016 - In Andrea Sereni & Francesca Boccuni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Berlin: Springer. pp. 165-188.
    The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...)
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  2. What We Talk About When We Talk About Numbers.Richard Pettigrew - manuscript
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
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  3. Some Recent Existential Appeals to Mathematical Experience.Michael J. Shaffer - 2006 - Principia: An International Journal of Epistemology 10 (2):143-170.
    Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a number of (...)
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  4. Maximally Consistent Sets of Instances of Naive Comprehension.Luca Incurvati & Julien Murzi - 2017 - Mind 126 (502).
    Paul Horwich (1990) once suggested restricting the T-Schema to the maximally consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...)
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  5. Tiny Proper Classes.Laureano Luna - 2016 - The Reasoner 10 (10):83-83.
    We propose certain clases that seem unable to form a completed totality though they are very small, finite, in fact. We suggest that the existence of such clases lends support to an interpretation of the existence of proper clases in terms of availability, not size.
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  6. Gödel's Cantorianism.Claudio Ternullo - 2015 - In Eva-Maria Engelen & Gabriella Crocco (eds.), Kurt Gödel: Philosopher-Scientist. Presses Universitaires de Provence. pp. 417-446.
    Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.
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  7. Three Concepts of Decidability for General Subsets of Uncountable Spaces.Matthew W. Parker - 2003 - Theoretical Computer Science 351 (1):2-13.
    There is no uniquely standard concept of an effectively decidable set of real numbers or real n-tuples. Here we consider three notions: decidability up to measure zero [M.W. Parker, Undecidability in Rn: Riddled basins, the KAM tori, and the stability of the solar system, Phil. Sci. 70(2) (2003) 359–382], which we abbreviate d.m.z.; recursive approximability [or r.a.; K.-I. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991]; and decidability ignoring boundaries [d.i.b.; W.C. Myrvold, The decision problem for entanglement, in: R.S. (...)
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  8. To Be or to Be Not, That is the Dilemma.Juan José Luetich - 2012 - Identification Transactions of The Luventicus Academy (ISSN 1666-7581) 1 (1):4.
    A set is precisely defined. A given element either belongs or not to a set. However, since all of the elements being considered belong to the universe, if the element does not belong to the set, it belongs to its complement, that is, what remains after all of the elements from the set are removed from the universe.
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  9. A Taxonomy of Composition Operations.Jan Westerhoff - 2004 - Logique and Analyse 2004 (47):375-393.
    A set of parameters for classifying composition operations is introduced. These parameters determine whether a composition operation is 1) universal, 2) determinate, 3) whether there is a difference between possible and actual compositions, 4) whether there can be singleton compositions, 5) whether they give rise to a hierarchy, and 6) whether components of compositions can be repeated. Philosophical implications of these parameters (in particular in relation to set theory) and mereology are discussed.
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Axioms of Set Theory
  1. Modal Set Theory.Christopher Menzel - forthcoming - In Otávio Bueno & Scott Shalkowski (eds.), The Routledge Handbook of Modality. London and New York: Routledge.
    This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.
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  2. Epistemology of Logic - Logic-Dialectic or Theory of the Knowledge.Epameinondas Xenopoulos - 1998 - Dissertation,
    1994.Επιστημολογία της Λογικής. Συγγραφέας Επαμεινώνδας Ξενόπουλος Μοναδική μελέτη και προσέγγιση της θεωρίας της γνώσης, για την παγκόσμια βιβλιογραφία, της διαλεκτικής πορείας της σκέψης από την λογική πλευρά της και της μελλοντικής μορφής που θα πάρουν οι διαλεκτικές δομές της, στην αδιαίρετη ενότητα γνωσιοθεωρίας, λογικής και διαλεκτικής, με την «μέθοδο του διαλεκτικού υλισμού». Έργο βαρύ με θέμα εξαιρετικά δύσκολο διακατέχεται από πρωτοτυπία και ζωντάνια που γοητεύει τον κάθε ανήσυχο στοχαστή από τις πρώτες γραμμές.
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