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    1. Countabilism and Maximality Principles.Neil Barton & Sy-David Friedman - manuscript
      It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that (...)
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    2. Independence Results for Finite Set Theories in Well-Founded Locally Finite Graphs.Funmilola Balogun & Benedikt Löwe - 2024 - Studia Logica 112 (5):1181-1200.
      We consider all combinatorially possible systems corresponding to subsets of finite set theory (i.e., Zermelo-Fraenkel set theory without the axiom of infinity) and for each of them either provide a well-founded locally finite graph that is a model of that theory or show that this is impossible. To that end, we develop the technique of _axiom closure of graphs_.
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    3. Observation and Intuition.Justin Clarke-Doane & Avner Ash - 2023 - In Carolin Antos, Neil Barton & Giorgio Venturi (eds.), The Palgrave Companion to the Philosophy of Set Theory. Palgrave.
      The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated (...)
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    4. In defense of Countabilism.David Builes & Jessica M. Wilson - 2022 - Philosophical Studies 179 (7):2199-2236.
      Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons (2000) and developed by Linnebo (2010, 2013) and Linnebo and Shapiro (2019), should also be applied to CT, in a way that (...)
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    5. Modal Pluralism and Higher‐Order Logic.Justin Clarke-Doane & William McCarthy - 2022 - Philosophical Perspectives 36 (1):31-58.
      In this article, we discuss a simple argument that modal metaphysics is misconceived, and responses to it. Unlike Quine's, this argument begins with the simple observation that there are different candidate interpretations of the predicate ‘could have been the case’. This is analogous to the observation that there are different candidate interpretations of the predicate ‘is a member of’. The argument then infers that the search for metaphysical necessities is misguided in much the way the ‘set-theoretic pluralist’ claims that the (...)
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    6. El Axioma de elección en el quehacer matemático contemporáneo.Franklin Galindo & Randy Alzate - 2022 - Aitías 2 (3):49-126.
      Para matemáticos interesados en problemas de fundamentos, lógico-matemáticos y filósofos de la matemática, el axioma de elección es centro obligado de reflexión, pues ha sido considerado esencial en el debate dentro de las posiciones consideradas clásicas en filosofía de la matemática (intuicionismo, formalismo, logicismo, platonismo), pero también ha tenido una presencia fundamental para el desarrollo de la matemática y metamatemática contemporánea. Desde una posición que privilegia el quehacer matemático, nos proponemos mostrar los aportes que ha tenido el axioma en varias (...)
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    7. Descriptivism about the Reference of Set-Theoretic Expressions: Revisiting Putnam’s Model-Theoretic Arguments.Zeynep Soysal - 2020 - The Monist 103 (4):442-454.
      Putnam’s model-theoretic arguments for the indeterminacy of reference have been taken to pose a special problem for mathematical languages. In this paper, I argue that if one accepts that there are theory-external constraints on the reference of at least some expressions of ordinary language, then Putnam’s model-theoretic arguments for mathematical languages don’t go through. In particular, I argue for a kind of descriptivism about mathematical expressions according to which their reference is “anchored” in the reference of expressions of ordinary language. (...)
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    8. A naturalistic justification of the generic multiverse with a core.Matteo de Ceglie - 2018 - Contributions of the Austrian Ludwig Wittgenstein Society 26:34-36.
      In this paper, I argue that a naturalist approach in philosophy of mathematics justifies a pluralist conception of set theory. For the pluralist, there is not a Single Universe, but there is rather a Multiverse, composed by a plurality of universes generated by various set theories. In order to justify a pluralistic approach to sets, I apply the two naturalistic principles developed by Penelope Maddy (cfr. Maddy (1997)), UNIFY and MAXIMIZE, and analyze through them the potential of the set theoretic (...)
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    9. Maximality and ontology: how axiom content varies across philosophical frameworks.Sy-David Friedman & Neil Barton - 2017 - Synthese 197 (2):623-649.
      Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face (...)
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    10. 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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    11. (2 other versions)The Search for New Axioms in the Hyperuniverse Programme.Claudio Ternullo & Sy-David Friedman - 2016 - In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Cham, Switzerland: Springer International Publishing. 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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    12. What is Absolute Undecidability?†.Justin Clarke-Doane - 2012 - Noûs 47 (3):467-481.
      It is often supposed 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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    13. The set-theoretic multiverse.Joel David Hamkins - 2012 - Review of Symbolic Logic 5 (3):416-449.
      The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...)
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    14. Set Theory: Boolean-Valued Models and Independence Proofs.John L. Bell - 2011 - Oxford University Press.
      This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice.
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    15. Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies.Juliette Kennedy & Roman Kossak (eds.) - 2011 - Cambridge University Press.
      Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James H. (...)
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    16. Constructibilidad relativizada y el Axioma de elección.Franklin Galindo & Carlos Di Prisco - 2010 - Mixba'al. Revista Metropolitana de Matemáticas 1 (1):23-40.
      El objetivo de este trabajo es presentar en un solo cuerpo tres maneras de relativizar (o generalizar) el concepto de conjunto constructible de Gödel que no suelen aparecer juntas en la literatura especializada y que son importantes en la Teoría de Conjuntos, por ejemplo para resolver problemas de consistencia o independencia. Presentamos algunos modelos resultantes de las diferentes formas de relativizar el concepto de constructibilidad, sus propiedades básicas y algunas formas débiles del Axioma de Elección válidas o no válidas en (...)
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    17. The foundations of arithmetic in finite bounded Zermelo set theory.Richard Pettigrew - 2010 - Cahiers du Centre de Logique 17:99-118.
      In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of ZFin0.<br><br>An equivalent (...)
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    18. On the Consistency of ZF Set Theory and Its Large Cardinal Extensions.Luca Bellotti - 2006 - Epistemologia 29 (1):41-60.
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    19. John L. BELL. Set theory: Boolean-valued models and independence proofs. Oxford: Clarendon press, 2005. Oxford logic guides, no. 47. pp. XXII + 191. ISBN 0-19-856852-5, 987-0-19-856852-0 (pbk). [REVIEW]Patricia Marino - 2006 - Philosophia Mathematica 14 (3):392-394.
      This is the third edition of a book originally published in the 1970s; it provides a systematic and nicely organized presentation of the elegant method of using Boolean-valued models to prove independence results. Four things are new in the third edition: background material on Heyting algebras, a chapter on ‘Boolean-valued analysis’, one on using Heyting algebras to understand intuitionistic set theory, and an appendix explaining how Boolean and Heyting algebras look from the perspective of category theory. The book presents results (...)
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    20. Independence and justification in mathematics.Krzysztof Wójtowicz - 2006 - Poznan Studies in the Philosophy of the Sciences and the Humanities 91 (1):349-373.
      In the article the problem of independence in mathematics is discussed. The status of the continuum hypothesis, large cardinal axioms and the axiom of constructablility is presented in some detail. The problem whether incompleteness is really relevant for ordinary mathematics and for empirical science is investigated. Another aim of the article is to give some arguments for the thesis that the problem of reliability and justification of new axioms is well-posed and worthy of attention. In my opinion, investigations concerning the (...)
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    21. Arithmetical independence results using higher recursion theory.Andrew Arana - 2004 - Journal of Symbolic Logic 69 (1):1-8.
      We extend an independence result proved in our earlier paper "Solovay's Theorem Cannot Be Simplified" (Annals of Pure and Applied Logic 112 (2001)). Our method uses the Barwise.
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    22. Primitive independence results.Harvey M. Friedman - 2003 - Journal of Mathematical Logic 3 (1):67-83.
      We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality. They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC. It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without (...)
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    23. Some independence results for control structures in complete numberings.Sanjay Jain & Jochen Nessel - 2001 - Journal of Symbolic Logic 66 (1):357-382.
      Acceptable programming systems have many nice properties like s-m-n-Theorem, Composition and Kleene Recursion Theorem. Those properties are sometimes called control structures, to emphasize that they yield tools to implement programs in programming systems. It has been studied, among others by Riccardi and Royer, how these control structures influence or even characterize the notion of acceptable programming system. The following is an investigation, how these control structures behave in the more general setting of complete numberings as defined by Mal'cev and Eršov.
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    24. Sobre una consecuencia del teorema de Lindström en teoría de conjuntos.Franklin Galindo - 2000 - Apuntes Filosóficos 16.
      El método de forcing usado (junto con el Método de "los constructibles" de Gödel) para probar la independencia de la hipótesis del continuo respecto de la axiomática de Zermelo-Fraenkel en los textos "Set Theory" de Kunen, y "Set Theory" de Jech, tiene entre sus fundamentos lógicos principales las propiedades de completitud y de Lowenheim -Skolem (hacia abajo). Por otro lado, se sabe por Lindström que no hay una lógica de mayor capacidad expresiva que la lógica de primer orden que satisfaga (...)
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    25. Interpolation theorems, lower Bounds for proof systems, and independence results for bounded arithmetic.Jan Krajíček - 1997 - Journal of Symbolic Logic 62 (2):457-486.
      A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1) Feasible (...)
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    26. Some independence results related to the Kurepa tree.Renling Jin - 1991 - Notre Dame Journal of Formal Logic 32 (3):448-457.
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    27. Some independence results in interpretability logic.Vítězslav Švejdar - 1991 - Studia Logica 50 (1):29 - 38.
      A Kripke-style semantics developed by de Jongh and Veltman is used to investigate relations between several extensions of interpretability logic, IL.
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    28. Axioms of symmetry: Throwing darts at the real number line.Chris Freiling - 1986 - Journal of Symbolic Logic 51 (1):190-200.
      We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show (...)
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    29. Further consistency and independence results in NF obtained by the permutation method.T. E. Forster - 1983 - Journal of Symbolic Logic 48 (2):236-238.
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    30. Independence results.Saharon Shelah - 1980 - Journal of Symbolic Logic 45 (3):563-573.
      We prove independence results concerning the number of nonisomorphic models (using the S-chain condition and S-properness) and the consistency of "ZCF + 2 ℵ 0 = ℵ 2 + there is a universal linear order of power ℵ 1 ". Most of these results were announced in [Sh 4], [Sh 5]. In subsequent papers we shall prove an analog f MA for forcing which does not destroy stationary subsets of ω 1 , investigate D-properness for various filters and prove the (...)
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    31. Independence results for class forms of the axiom of choice.Paul E. Howard, Arthur L. Rubin & Jean E. Rubin - 1978 - Journal of Symbolic Logic 43 (4):673-684.
      Let NBG be von Neumann-Bernays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.
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    32. The independence results of set theory: An informal exposition.Michael E. Levin & Margarita R. Levin - 1978 - Synthese 38 (1):1 - 34.
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    33. Some independence results for peano arithmetic.J. B. Paris - 1978 - Journal of Symbolic Logic 43 (4):725-731.
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    34. Non-classical logics and the independence results of set theory.Melvin Fitting - 1972 - Theoria 38 (3):133-142.
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    35. (1 other version)Fraenkel's addition to the axioms of Zermelo.Richard Montague - 1961 - In Bar-Hillel, Yehoshua & [From Old Catalog] (eds.), Essays on the Foundations of Mathematics. Jerusalem,: Magnes Press. pp. 662-662.
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    36. Discrete independence results.Harvey Friedman - manuscript
      A bi-infinite approximate fixed point of type (n,k) is an approximate fixed point of type (n,k) whose terms are biinfinite; i.e., contain infin-itely many positive and infinitely many negative elements.
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    37. New borel independence results.Harvey Friedman - manuscript
      S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo, C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes. C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A. Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P. de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S. Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore, M.G. Nadkarni, C. Nebbia, A.L.T. Patterson, U. Krengel, A.J. Kuntz, J.-P. Serre, S.D. Sinel'shchikov, T. Slaman, Solecki, R. Spatzier, J. Steel, D. Sullivan, S. (...)
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    38. Un problema abierto de independencia en la teoría de conjuntos relacionado con ultrafiltros no principales sobre el conjunto de los números naturales N, y con Propiedades Ramsey.Franklin Galindo - manuscript
      En el ámbito de la lógica matemática existe un problema sobre la relación lógica entre dos versiones débiles del Axioma de elección (AE) que no se ha podido resolver desde el año 2000 (aproximadamente). Tales versiones están relacionadas con ultrafiltros no principales y con Propiedades Ramsey (Bernstein, Polarizada, Subretículo, Ramsey, Ordinales flotantes, etc). La primera versión débil del AE es la siguiente (A): “Existen ultrafiltros no principales sobre el conjunto de los números naturales (ℕ)”. Y la segunda versión débil del (...)
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