147 found
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  1.  54
    Solomon Feferman (2012). And so On...: Reasoning with Infinite Diagrams. Synthese 186 (1):371 - 386.
    This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a "pre" form of this thesis that every proof can be presented in everyday statements-only form.
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  2.  1
    Solomon Feferman (2012). And so On... : Reasoning with Infinite Diagrams. Synthese 186 (1):371-386.
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  3. Solomon Feferman, Harvey M. Friedman, Penelope Maddy & John R. Steel (2000). Does Mathematics Need New Axioms? Bulletin of Symbolic Logic 6 (4):401-446.
    Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...)
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  4.  78
    Solomon Feferman (1991). Reflecting on Incompleteness. Journal of Symbolic Logic 56 (1):1-49.
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  5.  7
    Solomon Feferman (1998). In the Light of Logic. Oxford University Press.
    In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom (...)
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  6. Anita Burdman Feferman & Solomon Feferman (2005). Alfred Tarski, Life and Logic. Bulletin of Symbolic Logic 11 (4):535-540.
     
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  7.  18
    Solomon Feferman (1984). Toward Useful Type-Free Theories. I. Journal of Symbolic Logic 49 (1):75-111.
  8. Solomon Feferman, J. N. Crossley, Maurice Boffa, Dirk van Dalen & Kenneth Mcaloon (1984). A Language and Axioms for Explicit Mathematics. Journal of Symbolic Logic 49 (1):308-311.
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  9.  23
    Solomon Feferman (2013). Foundations of Unlimited Category Theory: What Remains to Be Done. Review of Symbolic Logic 6 (1):6-15.
    Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for or category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met (...)
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  10. Solomon Feferman (2000). Mathematical Intuition Vs. Mathematical Monsters. Synthese 125 (3):317-332.
    Geometrical and physical intuition, both untutored andcultivated, is ubiquitous in the research, teaching,and development of mathematics. A number ofmathematical ``monsters'', or pathological objects, havebeen produced which – according to somemathematicians – seriously challenge the reliability ofintuition. We examine several famous geometrical,topological and set-theoretical examples of suchmonsters in order to see to what extent, if at all,intuition is undermined in its everyday roles.
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  11.  51
    Solomon Feferman (2010). Set-Theoretical Invariance Criteria for Logicality. Notre Dame Journal of Formal Logic 51 (1):3-20.
    This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language L∞,∞. It continues with a review of critiques of the Tarski-Sher thesis, and a proposal in response to one of those critiques via homomorphism invariance. That has quite divergent characterization results depending (...)
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  12.  1
    Solomon Feferman & Gerhard Jäger (1996). Systems of Explicit Mathematics with Non-Constructive Μ-Operator. Part II. Annals of Pure and Applied Logic 79 (1):37-52.
    This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).
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  13. Jon Barwise & Solomon Feferman (1985). Model-Theoretic Logics.
     
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  14.  60
    Solomon Feferman (2008). My Route to Arithmetization. Theoria 63 (3):168-181.
    I had the pleasure of renewing my acquaintance with Per Lindström at the meeting of the Seventh Scandinavian Logic Symposium, held in Uppsala in August 1996. There at lunch one day, Per said he had long been curious about the development of some of the ideas in my paper [1960] on the arithmetization of metamathematics. In particular, I had used the construction of a non-standard definition !* of the set of axioms of P (Peano Arithmetic) to show that P + (...)
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  15.  23
    Solomon Feferman (1988). Hilbert's Program Relativized: Proof-Theoretical and Foundational Reductions. Journal of Symbolic Logic 53 (2):364-384.
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  16.  1
    Solomon Feferman & Gerhard Jäger (1993). Systems of Explicit Mathematics with Non-Constructive Μ-Operator. Part I. Annals of Pure and Applied Logic 65 (3):243-263.
    Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: BON plus set induction (...)
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  17. Solomon Feferman, Is the Continuum Hypothesis a Definite Mathematical Problem?
    The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.
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  18.  47
    Solomon Feferman (1962). Transfinite Recursive Progressions of Axiomatic Theories. Journal of Symbolic Logic 27 (3):259-316.
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  19.  37
    Solomon Feferman, Godel's Program for New Axioms: Why, Where, How and What?
    From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be (...)
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  20. Solomon Feferman (ed.) (1995). Kurt Gödel, Collected Works. Oxford University Press.
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  21. Solomon Feferman (1968). Systems of Predicative Analysis, II: Representations of Ordinals. Journal of Symbolic Logic 33 (2):193-220.
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  22.  44
    Solomon Feferman (1999). Logic, Logics, and Logicism. Notre Dame Journal of Formal Logic 40 (1):31-54.
    The paper starts with an examination and critique of Tarski’s wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L∞,∞. Also characterized similarly is a natural generalization of Tarski’s thesis, due to Sher, in terms of bijections between domains. My main (...)
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  23. Solomon Feferman, Which Quantifiers Are Logical?
    ✤ It is the characterization of those forms of reasoning that lead invariably from true sentences to true sentences, independently of the subject matter.
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  24. Jon Barwise, Solomon Feferman & David Israel (1986). Meeting of the Association for Symbolic Logic: Stanford, California, 1985. Journal of Symbolic Logic 51 (3):832-862.
  25.  6
    Solomon Feferman & Thomas Strahm (2000). The Unfolding of Non-Finitist Arithmetic. Annals of Pure and Applied Logic 104 (1-3):75-96.
    The unfolding of schematic formal systems is a novel concept which was initiated in Feferman , Gödel ’96, Lecture Notes in Logic, Springer, Berlin, 1996, pp. 3–22). This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-finitist arithmetic . In particular, we examine two restricted unfoldings and , as well as a full unfolding, . The principal results then state: is equivalent to ; is equivalent to ; is equivalent to . Thus is proof-theoretically equivalent (...)
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  26.  58
    Solomon Feferman (2008). Axioms for Determinateness and Truth. Review of Symbolic Logic 1 (2):204-217.
    elaboration of the last part of my Tarski Lecture, “Truth unbound”, UC Berkeley, 3 April 2006, and of the lecture, “A nicer formal theory of non-hierarchical truth”, Workshop on Mathematical Methods in Philosophy, Banff , 18-23 Feb. 2007.
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  27.  47
    Solomon Feferman, Relationships Between Constructive, Predicative and Classical Systems of Analysis.
    Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative" de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives to (...)
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  28.  27
    Solomon Feferman (1964). Systems of Predicative Analysis. Journal of Symbolic Logic 29 (1):1-30.
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  29.  81
    Solomon Feferman, Foundations of Category Theory: What Remains to Be Done.
    • Session on CF&FCT proposed by E. Landry; participants: G. Hellman, E. Landry, J.-P. Marquis and C. McLarty..
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  30.  14
    Solomon Feferman & Thomas Strahm (2010). Unfolding Finitist Arithmetic. Review of Symbolic Logic 3 (4):665-689.
    The concept of the (full) unfolding of a schematic system is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted ? The program to determine for various systems of foundational significance was previously carried out for a system of nonfinitist arithmetic, ; it was shown that is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system (...)
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  31. Solomon Feferman, Conceptions of the Continuum.
    Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.
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  32.  65
    Solomon Feferman & Geoffrey Hellman (1995). Predicative Foundations of Arithmetic. Journal of Philosophical Logic 24 (1):1 - 17.
  33.  45
    Solomon Feferman, About and Around Computing Over the Reals.
    1. One theory or many? In 2004 a very interesting and readable article by Lenore Blum, entitled “Computing over the reals: Where Turing meets Newton,” appeared in the Notices of the American Mathematical Society. It explained a basic model of computation over the reals due to Blum, Michael Shub and Steve Smale (1989), subsequently exposited at length in their influential book, Complexity and Real Computation (1997), coauthored with Felipe Cucker. The ‘Turing’ in the title of Blum’s article refers of course (...)
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  34.  30
    Solomon Feferman, Predicativity.
    What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and (...)
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  35. Solomon Feferman (2009). Gödel, Nagel, Minds, and Machines. Journal of Philosophy 106 (4):201-219.
    Ernest Nagel Lecture, Columbia University, Sept. 27, 2007.
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  36.  27
    Solomon Feferman (1995). Definedness. Erkenntnis 43 (3):295 - 320.
    Questions of definedness are ubiquitous in mathematics. Informally, these involve reasoning about expressions which may or may not have a value. This paper surveys work on logics in which such reasoning can be carried out directly, especially in computational contexts. It begins with a general logic of partial terms, continues with partial combinatory and lambda calculi, and concludes with an expressively rich theory of partial functions and polymorphic types, where termination of functional programs can be established in a natural way.
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  37.  17
    Solomon Feferman, Finitary Inductively Presented Logics.
    A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems)met in practice. A f.i.p. theory FS0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS0 is a conservative extension of PRA. The aims of this work are (i)conceptual, (ii)pedagogical and (iii)practical. The system FS0 serves under (i)and (ii)as a theoretical framework for the formalization of metamathematics. The general approach (...)
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  38.  69
    Solomon Feferman (2008). Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on Finitism, Constructivity and Hilbert's Program. Dialectica 62 (2: Table of Contents"/> Select):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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  39.  58
    Solomon Feferman, The Development of Programs for the Foundations of Mathematics in the First Third of the 20th Century.
    The most prominent “schools” or programs for the foundations of mathematics that took shape in the first third of the 20th century emerged directly from, or in response to, developments in mathematics and logic in the latter part of the 19th century. The first of these programs, so-called logicism, had as its aim the reduction of mathematics to purely logical principles. In order to understand properly its achievements and resulting problems, it is necessary to review the background from that previous (...)
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  40. Solomon Feferman, Conceptual Structuralism and the Continuum.
    • This comes from my general view of the nature of mathematics, that it is humanly based and that it deals with more or less clear conceptions of mathematical structures; for want of a better word, I call that view conceptual structuralism.
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  41.  22
    Solomon Feferman, Tarski's Influence on Computer Science.
    The following is the text of an invited lecture for the LICS 2005 meeting held in Chicago June 26-29, 2005.1 Except for the addition of references, footnotes, corrections of a few points and stylistic changes, the text is essentially as delivered. Subsequent to the lecture I received interesting comments from several colleagues that would have led me to expand on some of the topics as well as the list of references, had I had the time to do so.
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  42.  28
    Solomon Feferman (2004). Comments on “Predicativity as a philosophical position” by G. Hellman. Revue Internationale de Philosophie 3:313-323.
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  43.  38
    Solomon Feferman (1992). Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992:442 - 455.
    Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize (...)
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  44.  19
    Solomon Feferman (2000). Does Reductive Proof Theory Have a Viable Rationale? Erkenntnis 53 (1-2):63-96.
    The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms which aim to reduce all ofmathematics to one supposedly ``universal'' system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational reduction rationale. However, (...)
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  45.  49
    Solomon Feferman (1989). Infinity in Mathematics: Is Cantor Necessary? Philosophical Topics 17 (2):23-45.
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  46.  38
    Solomon Feferman (1985). Intensionality in Mathematics. Journal of Philosophical Logic 14 (1):41 - 55.
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  47.  37
    Solomon Feferman (2008). Harmonious Logic: Craig's Interpolation Theorem and Its Descendants. Synthese 164 (3):341 - 357.
    Though deceptively simple and plausible on the face of it, Craig's interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, (...)
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  48. Solomon Feferman (1992). "Turing's\ Oracle": From Absolute to Relative Computability and Back. In Javier Echeverria, Andoni Ibarra & Thomas Mormann (eds.), The Space of Mathematics. De Gruyter 314--348.
  49.  30
    Solomon Feferman, Challenges to Predicative Foundations of Arithmetic.
    This is a sequel to our article “Predicative foundations of arithmetic” (1995), referred to in the following as [PFA]; here we review and clarify what was accomplished in [PFA], present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by [PFA] was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons (1992). Another critique is due to Daniel Isaacson (1987). Most recently, Alexander George and Daniel (...)
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  50.  26
    Solomon Feferman (1978). The Logic of Mathematical Discovery Vs. The Logical Structure of Mathematics. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1978:309 - 327.
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