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38Extracting Algorithms from Intuitionistic ProofsMathematical Logic Quarterly 44 (2): 143-160. 1998.This paper presents a new method - which does not rely on the cut-elimination theorem - for characterizing the provably total functions of certain intuitionistic subsystems of arithmetic. The new method hinges on a realizability argument within an infinitary language. We illustrate the method for the intuitionistic counterpart of Buss's theory Smath image, and we briefly sketch it for the other levels of bounded arithmetic and for the theory IΣ1.
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8Feasible Operations and Applicative Theories Based on ληBulletin of Symbolic Logic 8 (4): 534. 2002.
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26Benton, RA, 527 Blackburn, P., 281 Braüner, T., 359 Brink, C., 543Journal of Philosophical Logic 31 (615). 2002.
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Liu, Y., B21 Massey, C., B75 Mattingley, JB, 53 Melinger, A., B11 Meseguer, E., B1Cognition 98 309. 2006.
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22The abstract type of the real numbersArchive for Mathematical Logic 60 (7): 1005-1017. 2021.In finite type arithmetic, the real numbers are represented by rapidly converging Cauchy sequences of rational numbers. Ulrich Kohlenbach introduced abstract types for certain structures such as metric spaces, normed spaces, Hilbert spaces, etc. With these types, the elements of the spaces are given directly, not through the mediation of a representation. However, these abstract spaces presuppose the real numbers. In this paper, we show how to set up an abstract type for the real numbers. The ap…Read more
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15Bounds for indexes of nilpotency in commutative ring theory: A proof mining approachBulletin of Symbolic Logic 26 (3-4): 257-267. 2020.It is well-known that an element of a commutative ring with identity is nilpotent if, and only if, it lies in every prime ideal of the ring. A modification of this fact is amenable to a very simple proof mining analysis. We formulate a quantitative version of this modification and obtain an explicit bound. We present an application. This proof mining analysis is the leitmotif for some comments and observations on the methodology of computational extraction. In particular, we emphasize that the f…Read more
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23The FAN principle and weak König's lemma in herbrandized second-order arithmeticAnnals of Pure and Applied Logic 171 (9): 102843. 2020.We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are …Read more
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14Elementary Proof of Strong Normalization for Atomic FBulletin of the Section of Logic 45 (1): 1-15. 2016.We give an elementary proof of the strong normalization of the atomic polymorphic calculus Fat.
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20Arithmetic, proof theory, and computational complexity, edited by Peter Clote and Krajíček Jan, Oxford logic guides, no. 23, Clarendon Press, Oxford University Press, Oxford and New York1993, xiii + 428 pp (review)Journal of Symbolic Logic 60 (3): 1014-1017. 1995.
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31Mitsuru Tada and Makoto Tatsuta. The function ⌊a/m⌋ in sharply bounded arithmetic. Archive for mathematical logic, vol. 37 no. 1 , pp. 51–57 (review)Bulletin of Symbolic Logic 7 (3): 391-391. 2001.
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40Thomas Strahm. Polynomial time operations in explicit mathematics. The journal of symbolic logic, vol. 62 , pp. 575–594. - Andrea Cantini. Feasible operations and applicative theories based on λη. Mathematical logic quarterly, vol. 46 , pp. 291–312 (review)Bulletin of Symbolic Logic 8 (4): 534-535. 2002.
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6th Conference on Computability in Europe "Programs, Proofs, Processes"Bulletin of Symbolic Logic 17 (3): 478-479. 2011.
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11Zigzag and Fregean ArithmeticIn Hassan Tahiri (ed.), The Philosophers and Mathematics: Festschrift for Roshdi Rashed, Springer Verlag. pp. 81-100. 2018.In Frege’s logicism, numbers are logical objects in the sense that they are extensions of certain concepts. Frege’s logical system is inconsistent, but Richard Heck showed that its restriction to predicative quantification is consistent. This predicative fragment is, nevertheless, too weak to develop arithmetic. In this paper, I will consider an extension of Heck’s system with impredicative quantifiers. In this extended system, both predicative and impredicative quantifiers co-exist but it is on…Read more
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15Categoricity and Mathematical KnowledgeRevista Portuguesa de Filosofia 73 (3-4): 1423-1436. 2017.We argue that the basic notions of mathematics can only be properly formulated in an informal way. Mathematical notions transcend formalizations and their study involves the consideration of other mathematical notions. We explain the fundamental role of categoricity theorems in making these studies possible. We arrive at the conclusion that the enterprise of mathematics is not infallible and that it ultimately relies on degrees of evidence.
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19A herbrandized functional interpretation of classical first-order logicArchive for Mathematical Logic 56 (5-6): 523-539. 2017.We introduce a new typed combinatory calculus with a type constructor that, to each type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, associates the star type σ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{a…Read more
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12A Most Artistic Package of a Jumble of IdeasDialectica 62 (2): 205-222. 2008.In the course of ten short sections, we comment on Gödel's seminal dialectica paper of fifty years ago and its aftermath. We start by suggesting that Gödel's use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics, in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel's recasting of intuitionistic arithmetic via the dialectica interpretation, dis…Read more
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12Programs, proofs, processes: 6th Conference on Computability in Europe, CiE, 2010, Ponta Delgada, Azores, Portugal, June 30-July 4, 2010 ; proceedings (edited book, review)Springer. 2010.The LNCS series reports state-of-the-art results in computer science research, development, and education, at a high level and in both printed and electronic form.
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142Amending Frege’s Grundgesetze der ArithmetikSynthese 147 (1): 3-19. 2005.Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary P…Read more
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14Review: Peter Clote, Jan Krajicek, Arithmetic, proof theory, and computational complexity (review)Journal of Symbolic Logic 60 (3): 1014-1017. 1995.
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31Nonstandardness and the bounded functional interpretationAnnals of Pure and Applied Logic 166 (6): 701-712. 2015.
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31Bounded functional interpretation and feasible analysisAnnals of Pure and Applied Logic 145 (2): 115-129. 2007.In this article we study applications of the bounded functional interpretation to theories of feasible arithmetic and analysis. The main results show that the novel interpretation is sound for considerable generalizations of weak König’s Lemma, even in the presence of very weak induction. Moreover, when this is combined with Cook and Urquhart’s variant of the functional interpretation, one obtains effective versions of conservation results regarding weak König’s Lemma which have been so far only…Read more
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78A note on finiteness in the predicative foundations of arithmeticJournal of Philosophical Logic 28 (2): 165-174. 1999.Recently, Feferman and Hellman (and Aczel) showed how to establish the existence and categoricity of a natural number system by predicative means given the primitive notion of a finite set of individuals and given also a suitable pairing function operating on individuals. This short paper shows that this existence and categoricity result does not rely (even indirectly) on finite-set induction, thereby sustaining Feferman and Hellman's point in favor of the view that natural number induction can …Read more
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18Two General Results on Intuitionistic Bounded TheoriesMathematical Logic Quarterly 45 (3): 399-407. 1999.We study, within the framework of intuitionistic logic, two well-known general results of bounded arithmetic. Firstly, Parikh's theorem on the existence of bounding terms for the provably total functions. Secondly, the result which states that adding the scheme of bounded collection to bounded theories does not yield new II2 consequences
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32The Finitistic Consistency of Heck’s Predicative Fregean SystemNotre Dame Journal of Formal Logic 56 (1): 61-79. 2015.Frege’s theory is inconsistent. However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak. We also prove the finitistic consistency of the extension of Heck’s theory to $\Delta^{1}_{1}$-comprehension and of Heck’s ramified predicative second-order system
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20Interpretability in Robinson's QBulletin of Symbolic Logic 19 (3): 289-317. 2013.Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is animpassable barrierin the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted inbut also what…Read more