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  1. added 2020-03-13
    Why Logical Pluralism?Colin R. Caret - forthcoming - Synthese:1-22.
    This paper scrutinizes the debate over logical pluralism. I hope to make this debate more tractable by addressing the question of motivating data: what would count as strong evidence in favor of logical pluralism? Any research program should be able to answer this question, but when faced with this task, many logical pluralists fall back on brute intuitions. This sets logical pluralism on a weak foundation and makes it seem as if nothing pressing is at stake in the debate. The (...)
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  2. added 2020-02-04
    A Dilemma for Mathematical Constructivism.Samuel Kahn - 2020 - Axiomathes:01-10.
    In this paper I argue that constructivism in mathematics faces a dilemma. In particular, I maintain that constructivism is unable to explain (i) the application of mathematics to nature and (ii) the intersubjectivity of mathematics unless (iii) it is conjoined with two theses that reduce it to a form of mathematical Platonism. The paper is divided into five sections. In the first section of the paper, I explain the difference between mathematical constructivism and mathematical Platonism and I outline my argument. (...)
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  3. added 2020-01-20
    Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?Nicolas Gisin - forthcoming - Erkenntnis:1-13.
    It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...)
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  4. added 2020-01-03
    Considerações de Brouwer sobre espaço e infinitude: O idealismo de Brouwer Diante do Problema Apresentado por Dummett Quanto à Possibilidade Teórica de uma Infinitude Espacial.Paulo Júnio de Oliveira - 2019 - Kinesis:94-108.
    Resumo Neste artigo, será discutida a noção de “infinitude cardinal” – a qual seria predicada de um “conjunto” – e a noção de “infinitude ordinal” – a qual seria predicada de um “processo”. A partir dessa distinção conceitual, será abordado o principal problema desse artigo, i.e., o problema da possibilidade teórica de uma infinitude de estrelas tratado por Dummett em sua obra Elements of Intuitionism. O filósofo inglês sugere que, mesmo diante dessa possibilidade teórica, deveria ser possível predicar apenas infinitude (...)
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  5. added 2019-12-29
    Constructive Mathematics and Equality.Bruno Bentzen - 2018 - Dissertation, Sun Yat-Sen University
    The aim of the present thesis is twofold. First we propose a constructive solution to Frege's puzzle using an approach based on homotopy type theory, a newly proposed foundation of mathematics that possesses a higher-dimensional treatment of equality. We claim that, from the viewpoint of constructivism, Frege's solution is unable to explain the so-called ‘cognitive significance' of equality statements, since, as we shall argue, not only statements of the form 'a = b', but also 'a = a' may contribute to (...)
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  6. added 2019-10-14
    The Entanglement of Logic and Set Theory, Constructively.Laura Crosilla - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    ABSTRACT Theories of sets such as Zermelo Fraenkel set theory are usually presented as the combination of two distinct kinds of principles: logical and set-theoretic principles. The set-theoretic principles are imposed ‘on top’ of first-order logic. This is in agreement with a traditional view of logic as universally applicable and topic neutral. Such a view of logic has been rejected by the intuitionists, on the ground that quantification over infinite domains requires the use of intuitionistic rather than classical logic. In (...)
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  7. added 2019-10-14
    Conservativity of Transitive Closure Over Weak Operational Set Theory.Laura Crosilla & Andrea Cantini - 2012 - In Ulrich Berger, Hannes Diener, Peter Schuster & Monika Seisenberger (eds.), Logic, Construction, Computation. De Gruyter.
    Constructive set theory a' la Myhill-Aczel has been extended in (Cantini and Crosilla 2008, Cantini and Crosilla 2010) to incorporate a notion of (partial, non--extensional) operation. Constructive operational set theory is a constructive and predicative analogue of Beeson's Inuitionistic set theory with rules and of Feferman's Operational set theory (Beeson 1988, Feferman 2006, Jaeger 2007, Jaeger 2009, Jaeger 1009b). This paper is concerned with an extension of constructive operational set theory (Cantini and Crosilla 2010) by a uniform operation of Transitive (...)
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  8. added 2019-09-09
    Eta-Rules in Martin-Löf Type Theory.Ansten Klev - 2019 - Bulletin of Symbolic Logic 25 (3):333-359.
    The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher-order eta rule is part of that type theory. The main aim of this paper is to clarify this somewhat puzzling situation. It will be argued that lower-order eta rules do not, whereas the (...)
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  9. added 2019-08-23
    Wittgenstein on Cantor's Proof.Chrysoula Gitsoulis - 2018 - In Gabriele M. Mras, Paul Weingartner & Bernhard Ritter (eds.), Philosophy of Logic and Mathematics: Contributions of the 41st International Wittgenstein Symposium. pp. 67-69.
    Cantor’s proof that the reals are uncountable forms a central pillar in the edifices of higher order recursion theory and set theory. It also has important applications in model theory, and in the foundations of topology and analysis. Due partly to these factors, and to the simplicity and elegance of the proof, it has come to be accepted as part of the ABC’s of mathematics. But even if as an Archimedean point it supports tomes of mathematical theory, there is a (...)
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  10. added 2019-07-04
    Ludwig Wittgenstein, Dictating Philosophy To Francis Skinner: The Wittgenstein-Skinner Manuscripts. Transcribed and Edited, with an Introduction, Introductory Chapters and Notes by Arthur Gibson.Arthur Gibson & Niamh O'Mahony (eds.) - forthcoming - Berlin, Germany: Springer.
  11. added 2019-06-06
    Brouwer’s Weak Counterexamples and Testability: Further Remarks: Brouwer’s Weak Counterexamples and Testability: Further Remarks.Charles Mccarty - 2013 - Review of Symbolic Logic 6 (3):513-523.
    Straightforwardly and strictly intuitionistic inferences show that the Brouwer– Heyting–Kolmogorov interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form ¬ f V ¬¬ f is valid. Therefore, the BHK and recognition, as described here, are inconsistent with the axioms both of intuitionistic mathematics and of Markovian constructivism. This finding also implies that, if the BHK and recognition are suitably formulated, then Brouwer’s original weak counterexample reasoning was fallacious. (...)
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  12. added 2019-06-06
    Antirealism and Constructivism: Brouwer’s Weak Counterexamples: Antirealism and Constructivism: Brouwer’s Weak Counterexamples.Charles Mccarty - 2013 - Review of Symbolic Logic 6 (1):147-159.
    Strictly intuitionistic inferences are employed to demonstrate that three conditions—the existence of Brouwerian weak counterexamples to _Test_, the recognition condition, and the _BHK_ interpretation of the logical signs—are together inconsistent. Therefore, if the logical signs in mathematical statements governed by the recognition condition are constructive in that they satisfy the clauses of the _BHK_, then every relevant instance of the classical principle _Test_ is true intuitionistically, and the antirealistic critique of conventional logic, once thought to yield such weak counterexamples, is (...)
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  13. added 2019-06-06
    Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on Finitism, Constructivity and Hilbert's Program.Solomon Feferman - 2008 - Dialectica 62 (2):179-203.
    This is a survey of Gödel's perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program, using his published and unpublished articles and lectures as well as the correspondence between Bernays and Gödel on these matters. There is also an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end.
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  14. added 2019-06-06
    From Sets and Types to Topology and Analysis—Towards Practicable Foundations for Constructive Mathematics. [REVIEW]Jaap van Oosten - 2006 - Bulletin of Symbolic Logic 12 (4):611-612.
  15. added 2019-06-06
    A Defence of Mathematical Pluralism †We Should Like to Thank D. Bridges for Helpful Comments.E. Brian Davies - 2005 - Philosophia Mathematica 13 (3):252-276.
    We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.
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  16. added 2019-06-06
    The Banach-Steinhaus Theorem for the Space [Mathematical Script Capital D] in Constructive Analysis.Satoru Yoshida - 2003 - Mathematical Logic Quarterly 49 (3):305-315.
    We prove the Banach-Steinhaus theorem for distributions on the space [MATHEMATICAL SCRIPT CAPITAL D] within Bishop's constructive mathematics. To this end, we investigate the constructive sequential completion equation image of [MATHEMATICAL SCRIPT CAPITAL D].
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  17. added 2019-06-06
    What Finitism Could Not Be.Matthias Schirn & Karl-Georg Niebergall - 2003 - Critica 35 (103):43-68.
    In his paper "Finitism", W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by him, (...)
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  18. added 2019-06-06
    Intuitionism As A Kuhnian Revolution In Mathematics.Bruce Pourciau - 2000 - Studies in History and Philosophy of Science Part A 31 (2):297-329.
    In this paper it is argued, firstly, that Kuhnian revolutions in mathematics are logically possible, in the sense of not being inconsistent with the nature of mathematics; and, secondly, that Kuhnian revolutions are actually possible, in the sense that a Kuhnian paradigm for mathematics can be exhibited which would, if accepted by the mathematical community, produce a full Kuhnian revolution. These two arguments depend on first proving that a shift from a classical conception of mathematics to an intuitionist conception would (...)
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  19. added 2019-06-06
    Concepts of General Topology in Constructive Mathematics and in Sheaves, II.R. J. Grayson - 1982 - Annals of Pure and Applied Logic 23 (1):55.
  20. added 2019-06-06
    Concepts of General Topology in Constructive Mathematics and in Sheaves.R. J. Grayson - 1981 - Annals of Pure and Applied Logic 20 (1):1.
  21. added 2019-06-06
    Principles of Continuous Choice and Continuity of Functions in Formal Systems for Constructive Mathematics.Michael J. Beeson - 1977 - Annals of Pure and Applied Logic 12 (3):249.
  22. added 2019-06-06
    Elements of Intuitionism.Michael Dummett - 1977 - Oxford University Press.
    This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.
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  23. added 2019-05-08
    Dennis E. Hesseling. Gnomes in the Fog. The Reception of Brouwer's Intuitionism in the 1920s. Science Networks. Historical Studies, Vol. 28. Birkhäuser, Boston, 2003, Xxiii + 447 Pp. [REVIEW]Mark van Atten - 2004 - Bulletin of Symbolic Logic 10 (3):423-427.
  24. added 2019-03-26
    Rumfitt on the Logic of Set Theory.Øystein Linnebo - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy 62 (7):826-841.
    ABSTRACTAccording to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's argument is contrasted with my own preferred alternative.
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  25. added 2019-03-04
    Inference Rules and the Meaning of the Logical Constants.Hermógenes Oliveira - 2019 - Dissertation, Eberhard Karls Universität Tübingen
    The dissertation provides an analysis and elaboration of Michael Dummett's proof-theoretic notions of validity. Dummett's notions of validity are contrasted with standard proof-theoretic notions and formally evaluated with respect to their adequacy to propositional intuitionistic logic.
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  26. added 2019-02-12
    Logical Revision by Counterexamples: A Case Study of the Paraconsistent Counterexample to Ex Contradictione Quodlibet.Seungrak Choi - 2019 - In Proceedings of the 14th and 15th Asian Logic Conferences. pp. 141-167.
    It is often said that a correct logical system should have no counterexample to its logical rules and the system must be revised if its rules have a counterexample. If a logical system (or theory) has a counterexample to its logical rules, do we have to revise the system? In this paper, focussing on the role of counterexamples to logical rules, we deal with the question. -/- We investigate two mutually exclusive theories of arithmetic - intuitionistic and paraconsistent theories. The (...)
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  27. added 2019-01-25
    Institutionism, Pluralism, and Cognitive Command.Stewart Shapiro & William W. Taschek - 1996 - Journal of Philosophy 93 (2):74.
  28. added 2018-11-20
    Intuitionistic Mereology.Paolo Maffezioli & Achille C. Varzi - forthcoming - Synthese:1-26.
    Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.
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  29. added 2018-11-09
    On the Intuitionistic Background of Gentzen's 1935 and 1936 Consistency Proofs and Their Philosophical Aspects.Yuta Takahashi - 2018 - Annals of the Japan Association for Philosophy of Science 27:1-26.
    Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer (...)
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  30. added 2018-10-08
    Constructivism for Philosophers (Be It a Remark on Realism).Ofer Gal - 2002 - Perspectives on Science 10 (4):523-549.
    : Bereft of the illusion of an epistemic vantage point external to science, what should be our commitment towards the categories, concepts and terms of that very science? Should we, despaired of the possibility to found these concepts on rock bottom, adopt empiricist skepticism? Or perhaps the inexistence of external foundations implies, rather, immunity for scientific ontology from epistemological criticism? Philosophy's "realism debate" died out without providing a satisfactory answer to the dilemma, which was taken over by the neighboring disciplines. (...)
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  31. added 2018-09-29
    Actual and Potential Infinity.Øystein Linnebo & Stewart Shapiro - 2019 - Noûs 53 (1):160-191.
    The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.
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  32. added 2018-06-06
    Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Springer. pp. 19-37.
    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...)
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  33. added 2018-02-18
    Structuralism, Invariance, and Univalence.Steve Awodey - 2014 - Philosophia Mathematica 22 (1):1-11.
    The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...)
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  34. added 2018-02-18
    A Proof System for Fork Algebras and its Applications to Reasoning in Logics Based on Intuitionism.M. Frias & E. Orlowska - 1995 - Logique Et Analyse 150:151-152.
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  35. added 2018-01-20
    Logic of Probability and Conjecture.Harry Crane - unknown
    I introduce a formalization of probability which takes the concept of 'evidence' as primitive. In parallel to the intuitionistic conception of truth, in which 'proof' is primitive and an assertion A is judged to be true just in case there is a proof witnessing it, here 'evidence' is primitive and A is judged to be probable just in case there is evidence supporting it. I formalize this outlook by representing propositions as types in Martin-Lof type theory (MLTT) and defining a (...)
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  36. added 2017-10-25
    Intuitionistic Logic and its Philosophy.Panu Raatikainen - 2013 - Al-Mukhatabat. A Trilingual Journal For Logic, Epistemology and Analytical Philosophy (6):114-127.
  37. added 2017-09-03
    The Justification of Identity Elimination in Martin-Löf’s Type Theory.Ansten Klev - 2019 - Topoi 38 (3):577-590.
    On the basis of Martin-Löf’s meaning explanations for his type theory a detailed justification is offered of the rule of identity elimination. Brief discussions are thereafter offered of how the univalence axiom fares with respect to these meaning explanations and of some recent work on identity in type theory by Ladyman and Presnell.
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  38. added 2017-02-27
    Hilbert’s Program.Richard Zach - 2003 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University.
    In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...)
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  39. added 2017-02-16
    Non-Realism, Nominalism and Strict Finitism the Sheer Complexity of It All.Jean Paul Van Bendegem - 2006 - Poznan Studies in the Philosophy of the Sciences and the Humanities 90:343-365.
  40. added 2017-02-15
    Hilbert's Finitism and the Notion of Infinity.Karl-Georg Niebergall & Matthias Schirn - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press.
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  41. added 2017-02-14
    Undecidable Theories and Reverse Mathematics.James H. Schmerl - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 21--349.
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  42. added 2017-02-14
    Free Sets and Reverse Mathematics.Carl G. Jockusch Jr - 2005 - In Stephen Simpson (ed.), Reverse Mathematics 2001. pp. 104.
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  43. added 2017-02-14
    Dirk van Dalen. Mystic, Geometer, and Intuitionist-The Life of LEJ Brouwer, Volume 1, The Dawning Revolution. Clarendon Press, Oxford, 1999. Pp. Xv+ 440. [REVIEW]Hiroshi Kaneko - 2002 - Annals of the Japan Association for Philosophy of Science 11 (1):51-56.
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  44. added 2017-02-14
    On Maximal Continuity Regulators for Constructive Functions.A. O. Slisenko - 1969 - In Studies in Constructive Mathematics and Mathematical Logic. New York: Consultants Bureau. pp. 82--84.
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  45. added 2017-02-13
    Topics in Reverse Mathematics.Mariagnese Giusto - 2003 - In Benedikt Löwe, Thoralf Räsch & Wolfgang Malzkorn (eds.), Foundations of the Formal Sciences Ii. Kluwer Academic Publishers. pp. 63--87.
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  46. added 2017-02-13
    Response Sequences Following “Wrongs” in a Concept Task.Jean L. Bresnahan & Martin M. Shapiro - 1973 - Bulletin of the Psychonomic Society 2 (4):193-195.
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  47. added 2017-02-12
    Reverse-Engineering Reverse Mathematics.Sam Sanders - 2013 - Annals of Pure and Applied Logic 164 (5):528-541.
    An important open problem in Reverse Mathematics is the reduction of the first-order strength of the base theory from IΣ1IΣ1 to IΔ0+expIΔ0+exp. The system ERNA, a version of Nonstandard Analysis based on the system IΔ0+expIΔ0+exp, provides a partial solution to this problem. Indeed, weak Königʼs lemma and many of its equivalent formulations from Reverse Mathematics can be ‘pushed down’ into ERNA, while preserving the equivalences, but at the price of replacing equality with ‘≈’, i.e. infinitesimal proximity . The logical principle (...)
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  48. added 2017-02-12
    Sequential, Pointwise, and Uniform Continuity: A Constructive Note.Douglas S. Bridges - 1993 - Mathematical Logic Quarterly 39 (1):55-61.
    The main result of this paper is a weak constructive version of the uniform continuity theorem for pointwise continuous, real-valued functions on a convex subset of a normed linear space. Recursive examples are given to show that the hypotheses of this theorem are necessary. The remainder of the paper discusses conditions which ensure that a sequentially continuous function is continuous. MSC: 03F60, 26E40, 46S30.
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  49. added 2017-02-12
    Constructive Well‐Orderings.Robin J. Grayson - 1982 - Mathematical Logic Quarterly 28 (33‐38):495-504.
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  50. added 2017-02-11
    Reverse Mathematics and Properties of Finite Character.Damir D. Dzhafarov & Carl Mummert - 2012 - Annals of Pure and Applied Logic 163 (9):1243-1251.
1 — 50 / 332