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  1. Foundations of Set Theory.J. R. Shoenfield - 1964 - Journal of Symbolic Logic 29 (3):141-141.
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  • Computability & Unsolvability.Clifford Spector - 1958 - Journal of Symbolic Logic 23 (4):432-433.
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  • Formalization, Primitive Concepts, and Purity: Formalization, Primitive Concepts, and Purity.John T. Baldwin - 2013 - Review of Symbolic Logic 6 (1):87-128.
    We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are (...)
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  • Second Philosophy: A Naturalistic Method.Penelope Maddy - 2007 - Oxford, England and New York, NY, USA: Oxford University Press.
    Many philosophers claim to be naturalists, but there is no common understanding of what naturalism is. Maddy proposes an austere form of naturalism called 'Second Philosophy', using the persona of an idealized inquirer, and she puts this method into practice in illuminating reflections on logical truth, philosophy of mathematics, and metaphysics.
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  • Abstract Elementary Classes and Infinitary Logics.David W. Kueker - 2008 - Annals of Pure and Applied Logic 156 (2):274-286.
    In this paper we study abstract elementary classes using infinitary logics and prove a number of results relating them. For example, if is an a.e.c. with Löwenheim–Skolem number κ then is closed under L∞,κ+-elementary equivalence. If κ=ω and has finite character then is closed under L∞,ω-elementary equivalence. Analogous results are established for . Galois types, saturation, and categoricity are also studied. We prove, for example, that if is finitary and λ-categorical for some infinite λ then there is some σLω1,ω such (...)
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  • Über eine bisher noch nicht benützte erweiterung Des finiten standpunktes.Von Kurt Gödel - 1958 - Dialectica 12 (3‐4):280-287.
    ZusammenfassungP. Bernays hat darauf hingewiesen, dass man, um die Widerspruchs freiheit der klassischen Zahlentheorie zu beweisen, den Hilbertschen flniter Standpunkt dadurch erweitern muss, dass man neben den auf Symbole sich beziehenden kombinatorischen Begriffen gewisse abstrakte Begriffe zulässt, Die abstrakten Begriffe, die bisher für diesen Zweck verwendet wurden, sinc die der konstruktiven Ordinalzahltheorie und die der intuitionistischer. Logik. Es wird gezeigt, dass man statt deesen den Begriff einer berechenbaren Funktion endlichen einfachen Typs über den natürlichen Zahler benutzen kann, wobei keine anderen (...)
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  • On Extensions of Elementary Logic.Per Lindström - 1969 - Theoria 35 (1):1-11.
  • Poincaré: Mathematics & Logic & Intuition.Colin Mclarty - 1997 - Philosophia Mathematica 5 (2):97-115.
    often insisted existence in mathematics means logical consistency, and formal logic is the sole guarantor of rigor. The paper joins this to his view of intuition and his own mathematics. It looks at predicativity and the infinite, Poincaré's early endorsement of the axiom of choice, and Cantor's set theory versus Zermelo's axioms. Poincaré discussed constructivism sympathetically only once, a few months before his death, and conspicuously avoided committing himself. We end with Poincaré on Couturat, Russell, and Hilbert.
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  • Godel on Computability.W. Sieg - 2006 - Philosophia Mathematica 14 (2):189-207.
    The identification of an informal concept of ‘effective calculability’ with a rigorous mathematical notion like ‘recursiveness’ or ‘Turing computability’ is still viewed as problematic, and I think rightly so. I analyze three different and conflicting perspectives Gödel articulated in the three decades from 1934 to 1964. The significant shifts in Gödel's position underline the difficulties of the methodological issues surrounding the Church-Turing Thesis.
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  • What Does Gödel's Second Theorem Say?Michael Detlefsen - 2001 - Philosophia Mathematica 9 (1):37-71.
    We consider a seemingly popular justification (we call it the Re-flexivity Defense) for the third derivability condition of the Hilbert-Bernays-Löb generalization of Godel's Second Incompleteness Theorem (G2). We argue that (i) in certain settings (rouglily, those where the representing theory of an arithmetization is allowed to be a proper subtheory of the represented theory), use of the Reflexivity Defense to justify the tliird condition induces a fourth condition, and that (ii) the justification of this fourth condition faces serious obstacles. We (...)
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  • Register Computations on Ordinals.Peter Koepke & Ryan Siders - 2008 - Archive for Mathematical Logic 47 (6):529-548.
    We generalize ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. Ordinal register machines are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the Zermelo-Fraenkel axioms ZFC. This allows the following characterization of computable sets: a set of ordinals is ordinal register (...)
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  • Logic with the Quantifier "There Exist Uncountably Many".H. Jerome Keisler - 1970 - Annals of Mathematical Logic 1 (1):1.
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  • Consistency Results About Ordinal Definability.Kenneth McAloon - 1971 - Annals of Mathematical Logic 2 (4):449.
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  • Stationary Logic.Jon Barwise - 1978 - Annals of Mathematical Logic 13 (2):171.
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  • On Embedding Models of Arithmetic Into Reduced Powers.Juliette Kennedy - 2003 - Matematica Contemporanea 24 (1):91--115.
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  • Thorn-Forking as Local Forking.Hans Adler - 2009 - Journal of Mathematical Logic 9 (1):21-38.
    A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be (...)
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  • Purity of Methods.Michael Detlefsen & Andrew Arana - 2011 - Philosophers' Imprint 11.
    Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. This preference for “purity” (which German writers commonly referred to as “methoden Reinheit”) has taken various forms. It has also been persistent. This notwithstanding, it has not been analyzed at even a basic philosophical level. In this paper we give a basic analysis of one conception of purity—what we call topical purity—and (...)
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  • Brouwer's Cambridge Lectures on Intuitionism.R. J. Grayson - 1983 - Journal of Symbolic Logic 48 (1):214-215.
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  • Foundations of Constructive Analysis.John Myhill - 1972 - Journal of Symbolic Logic 37 (4):744-747.
  • On the Relationship Between Plane and Solid Geometry.Andrew Arana & Paolo Mancosu - 2012 - Review of Symbolic Logic 5 (2):294-353.
    Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.
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  • Axioms for Abstract Model Theory.K. J. Barwise - 1974 - Annals of Mathematical Logic 7 (2/3):221.
  • Philosophy of Logic.Willard V. O. Quine - 1986 - Philosophy 17 (3):392-393.
    With his customary incisiveness, W. V. Quine presents logic as the product of two factors, truth and grammar-but argues against the doctrine that the logical truths are true because of grammar or language. Rather, in presenting a general theory of grammar and discussing the boundaries and possible extensions of logic, Quine argues that logic is not a mere matter of words.
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  • Probabilities on Finite Models.Ronald Fagin - 1976 - Journal of Symbolic Logic 41 (1):50-58.
  • Poincaré Vs. Russell on the Rôle of Logic in Mathematicst.Michael Detlefsen - 1993 - Philosophia Mathematica 1 (1):24-49.
    In the early years of this century, Poincaré and Russell engaged in a debate concerning the nature of mathematical reasoning. Siding with Kant, Poincaré argued that mathematical reasoning is characteristically non-logical in character. Russell urged the contrary view, maintaining that (i) the plausibility originally enjoyed by Kant's view was due primarily to the underdeveloped state of logic in his (i.e., Kant's) time, and that (ii) with the aid of recent developments in logic, it is possible to demonstrate its falsity. This (...)
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  • Which Number Theoretic Problems Can Be Solved in Recursive Progressions on Π1 1-Paths Through O?G. Kreisel - 1972 - Journal of Symbolic Logic 37 (2):311-334.
  • Constructing Strongly Equivalent Nonisomorphic Models for Unstable Theories.Tapani Hyttinen & Heikki Tuuri - 1991 - Annals of Pure and Applied Logic 52 (3):203-248.
    If T is an unstable theory of cardinality <λ or countable stable theory with OTOP or countable superstable theory with DOP, λω λω1 in the superstable with DOP case) is regular and λ<λ=λ, then we construct for T strongly equivalent nonisomorphic models of cardinality λ. This can be viewed as a strong nonstructure theorem for such theories. We also consider the case when T is unsuperstable and develop further a result of Shelah about the existence of L∞,λ-equivalent nonisomorphic models for (...)
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  • Absolute Logics and L∞Ω.K. Jon Barwise - 1972 - Annals of Mathematical Logic 4 (3):309-340.
  • Axioms for Abstract Model Theory.K. Jon Barwise - 1974 - Annals of Mathematical Logic 7 (2-3):221-265.
  • On the Role of Ramsey Quantifiers in First Order Arithmetic.James H. Schmerl & Stephen G. Simpson - 1982 - Journal of Symbolic Logic 47 (2):423-435.
  • Set Theory with a Filter Quantifier.Matt Kaufmann - 1983 - Journal of Symbolic Logic 48 (2):263-287.
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  • Intuitionism, an Introduction.A. Heyting - 1958 - Studia Logica 7:277-278.
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  • The Varieties of Rigorous Experience.Juliet Floyd - 2013 - In Michael Beaney (ed.), The Oxford Handbook of the History of Analytic Philosophy. Oxford University Press.
  • On the Metamathematics of Algebra.Abraham Robinson - 1952 - Journal of Symbolic Logic 17 (3):205-207.
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  • The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
     
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