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  1. added 2020-03-27
    Finitist Set Theory in Ontological Modeling.Avril Styrman & Aapo Halko - 2018 - Applied Ontology 13 (2):107-133.
    This article introduces finitist set theory (FST) and shows how it can be applied in modeling finite nested structures. Mereology is a straightforward foundation for transitive chains of part-whole relations between individuals but is incapable of modeling antitransitive chains. Traditional set theories are capable of modeling transitive and antitransitive chains of relations, but due to their function as foundations of mathematics they come with features that make them unnecessarily difficult in modeling finite structures. FST has been designed to function as (...)
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  2. added 2019-09-05
    Ineffability Within the Limits of Abstraction Alone.Stewart Shapiro & Gabriel Uzquiano - 2016 - In Philip A. Ebert & Marcus Rossberg (eds.), Abstractionism: Essays in Philosophy of Mathematics. Oxford University Press.
    The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that (...)
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  3. added 2019-06-06
    Frege Meets Zermelo: A Perspective on Ineffability and Reflection: A Perspective on Ineffability and Reflection.Stewart Shapiro - 2008 - Review of Symbolic Logic 1 (2):241-266.
    _1. Philosophical background: iteration, ineffability, reflection._ There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice : the iterative conception and limitation of size. Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.
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  4. added 2019-03-01
    Maddy, Penelope, Defending the Axioms: On the Philosophical Foundations of Set Theory, Oxford: Oxford University Press, 2011, Pp. X + 150, £29/Us$45. [REVIEW]Jeffrey W. Roland - 2013 - Australasian Journal of Philosophy 91 (4):809-812.
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  5. added 2018-05-11
    What is Mathematics: Gödel's Theorem and Around (Edition 2015).Karlis Podnieks - manuscript
    Introduction to mathematical logic, part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.
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  6. added 2017-08-08
    Why is the Universe of Sets Not a Set?Zeynep Soysal - 2017 - Synthese:1-23.
    According to the iterative conception of sets, standardly formalized by ZFC, there is no set of all sets. But why is there no set of all sets? A simple-minded, though unpopular, “minimal” explanation for why there is no set of all sets is that the supposition that there is contradicts some axioms of ZFC. In this paper, I first explain the core complaint against the minimal explanation, and then argue against the two main alternative answers to the guiding question. I (...)
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  7. added 2016-09-09
    Russell’s Method of Analysis and the Axioms of Mathematics.Lydia Patton - 2017 - In Sandra Lapointe Christopher Pincock (ed.), Innovations in the History of Analytical Philosophy. London: Palgrave-Macmillan. pp. 105-126.
    In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s (...)
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  8. added 2016-05-22
    Die Axiome der Arithmetik mit besonderer Berücksichtigung der Beziehungen zur Mengenlehre.Kurt Grelling - 1910 - Dissertation, Georg-Augusts-Universität Göttingen
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  9. added 2016-04-27
    Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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  10. added 2015-08-24
    Boolos on the Justification of Set Theory.Alexander Paseau - 2007 - Philosophia Mathematica 15 (1):30-53.
    George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question.
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  11. added 2015-01-26
    Objectivity in Ethics and Mathematics.Justin Clarke-Doane - 2015 - Proceedings of the Aristotelian Society: The Virtual Issue 3.
    How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.
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  12. added 2014-03-25
    Fraenkel's Axiom of Restriction: Axiom Choice, Intended Models and Categoricity.Georg Schiemer - 2010 - In Benedikt L.öwe & Thomas Müller (eds.), PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. College Publications. pp. 307{340.
  13. added 2014-03-06
    Too Naturalist and Not Naturalist Enough: Reply to Horsten.Luca Incurvati - 2008 - Erkenntnis 69 (2):261 - 274.
    Leon Horsten has recently claimed that the class of mathematical truths coincides with the class of theorems of ZFC. I argue that the naturalistic character of Horsten’s proposal undermines his contention that this claim constitutes an analogue of a thesis that Daniel Isaacson has advanced for PA. I argue, moreover, that Horsten’s defence of his claim against an obvious objection makes use of a distinction which is not available to him given his naturalistic approach. I suggest a way out of (...)
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  14. added 2013-05-08
    What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - Noûs 47 (3):467-481.
    It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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  15. added 2011-06-08
    Models of Second-Order Zermelo Set Theory.Gabriel Uzquiano - 1999 - Bulletin of Symbolic Logic 5 (3):289-302.
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  16. added 2010-08-17
    A Reduction of the Axioms for the Theory of Propositional Types.Peter Andrews - 1963 - Fundamenta Mathematicae 52:345-350.
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