Search results for 'Jeremy D. Avigad' (try it on Scholar)

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  1. Jeremy D. Avigad (2002). Review: Sergei N. Artemov, Explicit Provability and Constructive Semantics. [REVIEW] Bulletin of Symbolic Logic 8 (3):432-433.score: 870.0
  2. Lev D. Beklemishev, Stephen Cook, Olivier Lessmann, Simon Thomas, Jeremy Avigad, Arnold Beckmann, Tim Carlson, Robert L. Constable & Kosta Došen (2003). 2002 European Summer Meeting of the Association for Symbolic Logic Logic Colloquium'02. Bulletin of Symbolic Logic 9 (1):71.score: 810.0
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  3. Jeremy Avigad, Notes on II-Conservativity, W-Submodels, and the Collection Schema.score: 480.0
    Jeremy Avigad . Notes on II-conservativity, w-submodels, and the Collection Schema.
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  4. Jesse Hughes, Steve Awodey, Dana Scott, Jeremy Avigad & Lawrence Moss, A Study of Categorres of Algebras and Coalgebras.score: 450.0
    This thesis is intended t0 help develop the theory 0f coalgebras by, Hrst, taking classic theorems in the theory 0f universal algebras amd dualizing them and, second, developing an interna] 10gic for categories 0f coalgebras. We begin with an introduction t0 the categorical approach t0 algebras and the dual 110tion 0f coalgebras. Following this, we discuss (c0)a,lg€bra.s for 2. (c0)monad and develop 2. theory 0f regular subcoalgebras which will be used in the interna] logic. We also prove that categories 0f (...)
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  5. Jeremy Avigad, Computability and Convergence.score: 450.0
    For most of its history, mathematics was fairly constructive: • Euclidean geometry was based on geometric construction. • Algebra sought explicit solutions to equations. Analysis, probability, etc. were focused on calculations. Nineteenth century developments in analysis challenged this view. A sequence (an) in a metric space is said Cauchy if for every ε > 0, there is an m such that for every n, n ≥ m, d (a n , a n ) < ε.
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  6. E. Dean J. Avigad & J. Mumma (2009). Sur Quelques Points d'Algebre Homologique. Review of Symbolic Logic 2 (4):700-768.score: 360.0
     
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  7. Jeremy Avigad, Edward Dean & John Mumma (2009). A Formal System for Euclid's Elements. Review of Symbolic Logic 2 (4):700--768.score: 240.0
    We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.
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  8. Jeremy Avigad, Philosophy of Mathematics.score: 240.0
    The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in (...)
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  9. Jeremy Avigad (2010). Understanding, Formal Verification, and the Philosophy of Mathematics. Journal of the Indian Council of Philosophical Research 27:161-197.score: 240.0
  10. Jeremy Avigad (2003). Number Theory and Elementary Arithmetic. Philosophia Mathematica 11 (3):257-284.score: 240.0
    is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.
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  11. Jeremy Avigad, “Clarifying the Nature of the Infinite”: The Development of Metamathematics and Proof Theory.score: 240.0
    We discuss the development of metamathematics in the Hilbert school, and Hilbert’s proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
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  12. Jeremy Avigad, Philosophy of Mathematics: 5 Questions.score: 240.0
    In 1977, when I was nine years old, Doubleday released Asimov on Numbers, a collection of essays that had first appeared in Isaac Asimov’s Science Fiction and Fantasy column. My mother, recognizing my penchant for science fiction and mathematics, bought me a copy as soon as it hit the bookstores. The essays covered topics such as number systems, combinatorial curiosities, imaginary numbers, and π. I was especially taken, however, by an essay titled “Varieties of the infinite,” which included a photograph (...)
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  13. Jeremy Avigad (2009). Marcus Giaquinto. Visual Thinking in Mathematics: An Epistemological Study. Philosophia Mathematica 17 (1):95-108.score: 240.0
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  14. Jeremy Avigad (2004). Forcing in Proof Theory. Bulletin of Symbolic Logic 10 (3):305-333.score: 240.0
    Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...)
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  15. Jeremy Avigad, Notes on a Formalization of the Prime Number Theorem.score: 240.0
    On September 6, 2004, using the Isabelle proof assistant, I verified the following statement: (%x. pi x * ln (real x) / (real x)) ----> 1 The system thereby confirmed that the prime number theorem is a consequence of the axioms of higher-order logic together with an axiom asserting the existence of an infinite set. All told, our number theory session, including the proof of the prime number theorem and supporting libraries, constitutes 673 pages of proof scripts, or roughly 30,000 (...)
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  16. Jeremy Avigad & Jeffrey Helzner (2002). Transfer Principles in Nonstandard Intuitionistic Arithmetic. Archive for Mathematical Logic 41 (6):581-602.score: 240.0
    Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of these rules destroy (...)
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  17. Jeremy Avigad, Proof Theory.score: 240.0
    At the turn of the nineteenth century, mathematics exhibited a style of argumentation that was more explicitly computational than is common today. Over the course of the century, the introduction of abstract algebraic methods helped unify developments in analysis, number theory, geometry, and the theory of equations; and work by mathematicians like Dedekind, Cantor, and Hilbert towards the end of the century introduced set-theoretic language and infinitary methods that served to downplay or suppress computational content. This shift in emphasis away (...)
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  18. Jeremy Avigad, Steven Kieffer & Harvey Friedman, A Language for Mathematical Knowledge Management.score: 240.0
    We argue that the language of Zermelo Fraenkel set theory with definitions and partial functions provides the most promising bedrock semantics for communicating and sharing mathematical knowledge. We then describe a syntactic sugaring of that language that provides a way of writing remarkably readable assertions without straying far from the set-theoretic semantics. We illustrate with some examples of formalized textbook definitions from elementary set theory and point-set topology. We also present statistics concerning the complexity of these definitions, under various complexity (...)
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  19. Jeremy Avigad (2006). Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals. In Jose Ferreiros Jeremy Gray (ed.), The architecture of modern mathematics.score: 240.0
    Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel (...)
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  20. Marcus Giaquinto & Jeremy Avigad, By Marcus Giaquinto.score: 240.0
    Published in 1891, Edmund Husserl’s first book, Philosophie der Arithmetik, aimed to “prepare the scientific foundations for a future construction of that discipline.” His goals should seem reasonable to contemporary philosophers of mathematics: . . . through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, separating the correct from the erroneous, in order, thus informed, to set in their place new ones which are, if possible, more adequately secured. [7, p. 5]2 But the (...)
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  21. Jeremy Avigad, Computers in Mathematical Inquiry.score: 240.0
    In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, although they do not fall squarely under a traditional philosophical purview. The goal of this article is to try to articulate some of these questions more clearly, and assess the philosophical methods that may be brought to bear. In Section 3, I note that most of the issues can be classified under two headings: some (...)
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  22. Jeremy Avigad, Eliminating Definitions and Skolem Functions in First-Order Logic.score: 240.0
    From proofs in any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions in polynomial time. From proofs in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions in polynomial time.
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  23. Jeremy Avigad (2006). Mathematical Method and Proof. Synthese 153 (1):105 - 159.score: 240.0
    On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...)
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  24. Jeremy Avigad, A Decision Procedure for Linear “Big o” Equations.score: 240.0
    Let F be the set of functions from an infinite set, S, to an ordered ring, R.
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  25. Jeremy Avigad & Richard Zach, The Epsilon Calculus. Stanford Encyclopedia of Philosophy.score: 240.0
    The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term..
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  26. Jeremy Avigad & Harvey Friedman, Combining Decision Procedures for the Reals.score: 240.0
    We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which “local'’ decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let $Tadd[QQ]$ be the first-order theory of the real numbers in the language with symbols $0, 1, +, -.
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  27. Jeremy Avigad, Understanding Proofs.score: 240.0
    “Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a spring-carriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug their ship and only coast along her sides.” (Herman Melville, Moby Dick, Chapter 94).
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  28. Andrea Asperti & Jeremy Avigad, Zen and the Art of Formalization.score: 240.0
    N. G. de Bruijn, now professor emeritus of the Eindhoven University of Technology, was a pioneer in the field of interactive theorem proving. From 1967 to the end of the 1970’s, his work on the Automath system introduced the architecture that is common to most of today’s proof assistants, and much of the basic technology. But de Bruijn was a mathematician first and foremost, as evidenced by the many mathematical notions and results that bear his name, among them de Bruijn (...)
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  29. Jeremy Avigad (2002). An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations. Journal of Mathematical Logic 2 (01):91-112.score: 240.0
    The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal analysis.
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  30. Jeremy Avigad (1996). On the Relationships Between ATR0 and $\Widehat{ID}_{. Journal of Symbolic Logic 61 (3):768 - 779.score: 240.0
    We show that the theory ATR 0 is equivalent to a second-order generalization of the theory $\widehat{ID}_{ . As a result, ATR 0 is conservative over $\widehat{ID}_{ for arithmetic sentences, though proofs in ATR 0 can be much shorter than their $\widehat{ID}_{ counterparts.
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  31. Jeremy Avigad (2009). The Metamathematics of Ergodic Theory. Annals of Pure and Applied Logic 157 (2):64-76.score: 240.0
    The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns (...)
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  32. Jeremy Avigad, Algebraic Proofs of Cut Elimination.score: 240.0
    Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the double-negation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negation-normal form of θ. The translation is used to show that cut-elimination theorems for classical logic can be viewed as special (...)
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  33. Jeremy Avigad & Henry Towsner, Functional Interpretation and Inductive Definitions.score: 240.0
    Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.
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  34. Jeremy Avigad, Formalizing O Notation in Isabelle/Hol.score: 240.0
    We describe a formalization of asymptotic O notation using the Isabelle/HOL proof assistant.
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  35. Jeremy Avigad & Solomon Feferman, Gödel's Functional ("Dialectica") Interpretation.score: 240.0
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  36. Jeremy Avigad, Kevin Donnelly, David Gray & Paul Raff, A Formally Verified Proof of the Prime Number Theorem.score: 240.0
    The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.
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  37. Jeremy Avigad, Eliminating Definitions and Skolem Functions in First-Order Logic.score: 240.0
    When working with a first-order theory, it is often convenient to use definitions. That is, if ϕ(x) is a first-order formula with the free variables shown, one can introduce a new relation symbol R to abbreviate ϕ, with defining axiom ∀x (R(x) ↔ ϕ(x)). Of course, this definition can later be eliminated from a proof, simply by replacing every instance of R by ϕ. But suppose the proof involves nested definitions, with a sequence of relation symbols R0, . . . (...)
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  38. Jeremy Avigad (2006). Fundamental Notions of Analysis in Subsystems of Second-Order Arithmetic. Annals of Pure and Applied Logic 139 (1):138-184.score: 240.0
    We develop fundamental aspects of the theory of metric, Hilbert, and Banach spaces in the context of subsystems of second-order arithmetic. In particular, we explore issues having to do with distances, closed subsets and subspaces, closures, bases, norms, and projections. We pay close attention to variations that arise when formalizing definitions and theorems, and study the relationships between them.
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  39. Jeremy Avigad, Kevin Donnelly, David Gray & Adam Kramer, Number Theory.score: 240.0
    1.1 Some examples of rule induction on permutations . . . . . . . 6 1.2 Ways of making new permutations . . . . . . . . . . . . . . . 7 1.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Removing elements . . . . . . . . . . (...)
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  40. Jeremy Avigad, By Calixto Badesa.score: 240.0
    From ancient times to the beginning of the nineteenth century, mathematics was commonly viewed as the general science of quantity, with two main branches: geometry, which deals with continuous quantities, and arithmetic, which deals with quantities that are discrete. Mathematical logic does not fit neatly into this taxonomy. In 1847, George Boole [1] offered an alternative characterization of the subject in order to make room for this new discipline: mathematics should be understood to include the use of any symbolic calculus (...)
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  41. Jeremy Avigad, Ordinal Analysis Without Proofs.score: 240.0
    An approach to ordinal analysis is presented which is finitary, but highlights the semantic content of the theories under consideration, rather than the syntactic structure of their proofs. In this paper the methods are applied to the analysis of theories extending Peano arithmetic with transfinite induction and transfinite arithmetic hierarchies.
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  42. Jeremy Avigad, Weak Theories of Nonstandard Arithmetic and Analysis.score: 240.0
    A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched.
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  43. Jeremy Avigad & Richard Sommer (1997). A Model-Theoretic Approach to Ordinal Analysis. Bulletin of Symbolic Logic 3 (1):17-52.score: 240.0
    We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.
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  44. Jeremy Avigad, Dedekind's 1871 Version of the Theory of Ideals.score: 240.0
    By the middle of the nineteenth century, it had become clear to mathematicians that the study of finite field extensions of the rational numbers is indispensable to number theory, even if one’s ultimate goal is to understand properties of diophantine expressions and equations in the ordinary integers. It can happen, however, that the “integers” in such extensions fail to satisfy unique factorization, a property that is central to reasoning about the ordinary integers. In 1844, Ernst Kummer observed that unique factorization (...)
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  45. Jeremy Avigad (1996). Formalizing Forcing Arguments in Subsystems of Second-Order Arithmetic. Annals of Pure and Applied Logic 82 (2):165-191.score: 240.0
    We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen the conservation theorems of Harrington and Brown-Simpson, giving an effective proof that WKL+0 is conservative over RCA0 with no significant increase in the lengths of proofs.
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  46. Jeremy Avigad, Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals.score: 240.0
    Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation.
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  47. Jeremy Avigad (forthcoming). On the Relationships Between ATR 0 And ID_<Ω. Journal of Symbolic Logic.score: 240.0
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  48. Jeremy Avigad, Proof Mining.score: 240.0
    Hilbert’s program: • Formalize abstract, infinitary, nonconstructive mathematics. • Prove consistency using only finitary methods.
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  49. Sergei Artemov, Peter Koellner, Michael Rabin, Jeremy Avigad, Wilfried Sieg, William Tait & Haim Gaifman (2006). Of the Association for Symbolic Logic. Bulletin of Symbolic Logic 12 (3-4):503.score: 240.0
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  50. Jeremy Avigad & Richard Sommer (1999). The Model-Theoretic Ordinal Analysis of Theories of Predicative Strength. Journal of Symbolic Logic 64 (1):327-349.score: 240.0
    We use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first- and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ0.
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