The Nature of Sets

Edited by Rafal Urbaniak (Uniwersytetu Gdanskiego, Uniwersytetu Gdanskiego)
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  1. Explanation and Plenitude in Non-Well-Founded Set Theories.Ross Cameron - forthcoming - Philosophia Mathematica.
  2. Cantor, Choice, and Paradox.Nicholas DiBella - forthcoming - The Philosophical Review.
    I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that is (...)
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  3. Peano, Frege and Russell’s Logical Influences.Kevin C. Klement - forthcoming - Forthcoming.
    This chapter clarifies that it was the works Giuseppe Peano and his school that first led Russell to embrace symbolic logic as a tool for understanding the foundations of mathematics, not those of Frege, who undertook a similar project starting earlier on. It also discusses Russell’s reaction to Peano’s logic and its influence on his own. However, the chapter also seeks to clarify how and in what ways Frege was influential on Russell’s views regarding such topics as classes, functions, meaning (...)
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  4. Replies.Øystein Linnebo - 2023 - Theoria 89 (3):393-406.
    Thin Objects has two overarching ambitions. The first is to clarify and defend the idea that some objects are ‘thin’, in the sense that their existence does not make a substantive demand on reality. The second is to develop a systematic and well-motivated account of permissible abstraction, thereby solving the so-called ‘bad company problem’. Here I synthesise the book by briefly commenting on what I regard as its central themes.
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  5. On the Plurality of Parts of Classes.Daniel Nolan - forthcoming - Dialectica.
    The ontological pictures underpinning David Lewis's Parts of Classes and On the Plurality of Worlds are in some tension. One tension concerns whether the sets and classes of Parts of Classes can be found in Lewis's modal space, since they cannot in general be parts of any possible world. The second is that the atoms that are the mathematical ontology of Parts of Classes seem to meet the criteria for being possible worlds themselves, and so fail to be the material (...)
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  6. Executing Gödel's Programme in Set Theory.Neil Barton - 2017 - Dissertation, Birkbeck, University of London
  7. Introducción a los conjuntos IndetermSoft e IndetermHyperSoft.Florentin Smarandache - 2022 - Neutrosophic Computing and Machine Learning 22 (1):1-22.
    This paper presents for the first time the IndetermSoft Set, as an extension of the classical (determinate) Soft Set, which operates on indeterminate data, and similarly the HyperSoft Set extended to the IndetermHyperSoft Set, where 'Indeterm' means 'Indeterminate' (uncertain, conflicting, non-unique result). They are built on an IndetermSoft Algebra which is an algebra dealing with IndetermSoft Operators resulting from our real world. Subsequently, the IndetermSoft and IndetermHyperSoft Sets and their Fuzzy/Fuzzy Intuitionistic/Neutrosophic and other fuzzy extensions and their applications are presented.
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  8. 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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  9. Numerical infinities and infinitesimals in optimization.Yaroslav D. Sergeyev & Renato De Leone - 2022 - 93413 Cham, Germania: Springer.
    From the Publisher: -/- This book presents a new powerful supercomputing paradigm introduced by Yaroslav D. Sergeyev -/- It gives a friendly introduction to the paradigm and proposes a broad panorama of a successful usage of numerical infinities -/- The volume covers software implementations of the Infinity Computer -/- Abstract -/- This book provides a friendly introduction to the paradigm and proposes a broad panorama of killing applications of the Infinity Computer in optimization: radically new numerical algorithms, great theoretical insights, (...)
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  10. On Number-Set Identity: A Study.Sean C. Ebels-Duggan - 2022 - Philosophia Mathematica 30 (2):223-244.
    Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.
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  11. 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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  12. Critical Plural Logic.Salvatore Florio & Øystein Linnebo - 2020 - Philosophia Mathematica 28 (2):172-203.
    What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.
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  13. 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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  14. J. I. Friedman. Proper classes as members of extended sets. Mathematische Annalen, vol. 83 , pp. 232–240.J. B. Paris - 1975 - Journal of Symbolic Logic 40 (3):462.
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  15. Mathematical Knowledge.Mary Leng, Alexander Paseau & Michael D. Potter (eds.) - 2007 - Oxford, England: Oxford University Press.
    What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions.
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  16. 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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  17. Infinite lotteries, large and small sets.Luc Lauwers - 2017 - Synthese 194 (6):2203-2209.
    One result of this note is about the nonconstructivity of countably infinite lotteries: even if we impose very weak conditions on the assignment of probabilities to subsets of natural numbers we cannot prove the existence of such assignments constructively, i.e., without something such as the axiom of choice. This is a corollary to a more general theorem about large-small filters, a concept that extends the concept of free ultrafilters. The main theorem is that proving the existence of large-small filters requires (...)
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  18. What is a Higher Level Set?Dimitris Tsementzis - 2016 - Philosophia Mathematica:nkw032.
    Structuralist foundations of mathematics aim for an ‘invariant’ conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. I argue in favor of the former over the latter. First, I explain why to choose between them we need to ask the question of what is the correct ‘categorified’ version of a set. Second, I argue in favor of groupoids over categories as ‘categorified’ sets by introducing a pre-formal understanding of groupoids as (...)
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  19. Foundations of Set Theory.Abraham Adolf Fraenkel & Yehoshua Bar-Hillel - 1973 - Atlantic Highlands, NJ, USA: Elsevier.
    Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...)
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  20. Understanding the Infinite I: Niceness, Robustness, and Realism†: Articles.David Corfield - 2010 - Philosophia Mathematica 18 (3):253-275.
    This paper treats the situation where a single mathematical construction satisfies a multitude of interesting mathematical properties. The examples treated are all infinitely large entities. The clustering of properties is termed ‘niceness’ by the mathematician Michiel Hazewinkel, a concept we compare to the ‘robustness’ described by the philosopher of science William Wimsatt. In the final part of the paper, we bring our findings to bear on the question of realism which concerns not whether mathematical entities exist as abstract objects, but (...)
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  21. Set‐Theories as Algebras.Paul Fjelstad - 1968 - Mathematical Logic Quarterly 14 (25-29):383-411.
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  22. Some Notions of Random Sequence and Their Set-Theoretic Foundations.Arthur H. Kruse - 1967 - Mathematical Logic Quarterly 13 (19-20):299-322.
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  23. 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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  24. Cantor’s Absolute in Metaphysics and Mathematics.Kai Hauser - 2013 - International Philosophical Quarterly 53 (2):161-188.
    This paper explores the metaphysical roots of Cantor’s conception of absolute infinity in order to shed some light on two basic issues that also affect the mathematical theory of sets: the viability of Cantor’s distinction between sets and inconsistent multiplicities, and the intrinsic justification of strong axioms of infinity that are studied in contemporary set theory.
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  25. Gurwitsch’s Theory of the Constitution of the Ordinal Numbers.William Mckenna - 1974 - Graduate Faculty Philosophy Journal 4 (1):36-40.
  26. Hennie Frederick C.. Analysis of bilateral iterative networks. Transactions of the I.R.E. Professional Group on Circuit Theory, vol. CT-6 no. 1 , pp. 35–45. [REVIEW]Edward F. Moore - 1959 - Journal of Symbolic Logic 24 (3):259-260.
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  27. Sets and supersets.Toby Meadows - 2016 - Synthese 193 (6):1875-1907.
    It is a commonplace of set theory to say that there is no set of all well-orderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe. In this paper, I am going to challenge this claim by taking seriously the idea that we can (...)
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  28. 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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  29. Transfinite recursion and computation in the iterative conception of set.Benjamin Rin - 2015 - Synthese 192 (8):2437-2462.
    Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative (...)
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  30. Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences.Jody Azzouni - 1994 - New York: Cambridge University Press.
    Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...)
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  31. 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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  32. Analysis of Bilateral Iterative Networks.Frederick C. Hennie - 1959 - Journal of Symbolic Logic 24 (3):259-260.
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  33. Explicit mathematics and operational set theory: Some ontological comparisons.Gerhard Jäger & Rico Zumbrunnen - 2017 - Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.
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  34. Encoded pilots for iterative receiver improvement.Hyuck M. Kwon, Khurram Hassan, Ashutosh Goyal, Mi-Kyung Oh, Dong-Jo Park & Yong Hoon Lee - 2005 - Complexity 4:5.
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  35. Review of some iterative root-finding methods from a dynamical point of view. [REVIEW]Sergio Amat, Sonia Busquier & Sergio Plaza - 2004 - Scientia 10:3-35.
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  36. The Elusiveness of Sets.Max Black - 1971 - Review of Metaphysics 24 (4):614-636.
    NOWADAYS, even schoolchildren babble about "null sets" and "singletons" and "one-one correspondences," as if they knew what they were talking about. But if they understand even less than their teachers, which seems likely, they must be using the technical jargon with only an illusion of understanding. Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should (...)
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  37. Iterativity vs. habituality: On the iterative interpretation of perfective sentences.Alessandro Lenci & Pier Marco Bertinetto - 2000 - In Achille Varzi, James Higginbotham & Fabio Pianesi (eds.), Speaking of Events. Oxford University Press.
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  38. Free sets and reverse mathematics.Carl G. Jockusch Jr - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 104.
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  39. NIIA: nonparametric iterative imputation algorithm.Shichao Zhang, Zhi Jin & Xiaofeng Zhu - 2008 - In Tu-Bao Ho & Zhi-Hua Zhou (eds.), Pricai 2008: Trends in Artificial Intelligence. Springer. pp. 544--555.
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  40. Discovery of Clusters from Proximity Data: An Approach Using Iterative Adjustment of Binary Classifications.Shoji Hirano & Shusaku Tsumoto - 2008 - In S. Iwata, Y. Oshawa, S. Tsumoto, N. Zhong, Y. Shi & L. Magnani (eds.), Communications and Discoveries From Multidisciplinary Data. Springer. pp. 251--268.
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  41. Indefinite Extensibility—Dialetheic Style.Graham Priest - 2013 - Studia Logica 101 (6):1263-1275.
    In recent years, many people writing on set theory have invoked the notion of an indefinitely extensible concept. The notion, it is usually claimed, plays an important role in solving the paradoxes of absolute infinity. It is not clear, however, how the notion should be formulated in a coherent way, since it appears to run into a number of problems concerning, for example, unrestricted quantification. In fact, the notion makes perfectly good sense if one endorses a dialetheic solution to the (...)
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  42. Philosophy of Mathematics in the Warsaw Mathematical School.Roman Murawski - 2010 - Axiomathes 20 (2-3):279-293.
    The aim of this paper is to present and discuss the philosophical views concerning mathematics of the founders of the so called Warsaw Mathematical School, i.e., Wacław Sierpiński, Zygmunt Janiszewski and Stefan Mazurkiewicz. Their interest in the philosophy of mathematics and their philosophical papers will be considered. We shall try to answer the question whether their philosophical views influenced their proper mathematical investigations. Their views towards set theory and its rôle in mathematics will be emphasized.
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  43. Review: Frederick C. Hennie, Analysis of Bilateral Iterative Networks. [REVIEW]Edward F. Moore - 1959 - Journal of Symbolic Logic 24 (3):259-260.
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  44. Review: Frederick C. Hennie III, Iterative Arrays of Logical Circuits. [REVIEW]Albert A. Mullin - 1962 - Journal of Symbolic Logic 27 (1):106-107.
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  45. Implementing Mathematical Objects in Set Theory.Thomas Forster - 2007 - Logique Et Analyse 50 (197):79-86.
    In general little thought is given to the general question of how to implement mathematical objects in set theory. It is clear that—at various times in the past—people have gone to considerable lengths to devise implementations with nice properties. There is a litera- ture on the evolution of the Wiener-Kuratowski ordered pair, and a discussion by Quine of the merits of an ordered-pair implemen- tation that makes every set an ordered pair. The implementation of ordinals as Von Neumann ordinals is (...)
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  46. Inscribing nonmeasurable sets.Szymon Żeberski - 2011 - Archive for Mathematical Logic 50 (3-4):423-430.
    Our main inspiration is the work in paper (Gitik and Shelah in Isr J Math 124(1):221–242, 2001). We will prove that for a partition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}$$\end{document} of the real line into meager sets and for any sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}_n}$$\end{document} of subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}$$\end{document} one can find a sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} (...)
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  47. Zermelo and Russell's Paradox: Is There a Universal set?G. Landini - 2013 - Philosophia Mathematica 21 (2):180-199.
    Zermelo once wrote that he had anticipated Russell's contradiction of the set of all sets that are not members of themselves. Is this sufficient for having anticipated Russell's Paradox — the paradox that revealed the untenability of the logical notion of a set as an extension? This paper argues that it is not sufficient and offers criteria that are necessary and sufficient for having discovered Russell's Paradox. It is shown that there is ample evidence that Russell satisfied the criteria and (...)
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  48. Enlarging One's Stall or How Did All of These Sets Get in Here?M. Wilson - 2013 - Philosophia Mathematica 21 (2):157-179.
    Following historical developments, this article traces two basic motives for employing sets within a physical setting and discusses whether they truly pose a problem for ‘mathematical naturalism’.
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  49. The hyperuniverse program.Tatiana Arrigoni & Sy-David Friedman - 2013 - Bulletin of Symbolic Logic 19 (1):77-96.
    The Hyperuniverse Program is a new approach to set-theoretic truth which is based on justifiable principles and leads to the resolution of many questions independent from ZFC. The purpose of this paper is to present this program, to illustrate its mathematical content and implications, and to discuss its philosophical assumptions.
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  50. Splitting the Μονάς.Claudio Majolino - 2011 - New Yearbook for Phenomenology and Phenomenological Philosophy 11:187-213.
    This paper assesses the philosophical heritage of Jacob Klein’s thought through an analysis of the key tenets of his Greek Mathematical Thought and theOrigin of Algebra. Threads of Klein’s thought are distinguished and subsequently singled out (phenomenological, epistemological, and anti-ontological; historical, ontological, and critical), and the peculiar way in which Klein’s project brings together ontology and history of mathematics is investigated. Plato’s theoretical logistic and Klein’s understanding thereof are questioned—especially the claim that the Platonic distinction between practical and theoretical logistic (...)
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