Results for 'Real arithmetic'

995 found
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  1.  10
    Domains for computation in mathematics, physics and exact real arithmetic.Abbas Edalat - 1997 - Bulletin of Symbolic Logic 3 (4):401-452.
    We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence (...)
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  2.  29
    The Arithmetical Hierarchy of Real Numbers.Xizhong Zheng & Klaus Weihrauch - 2001 - Mathematical Logic Quarterly 47 (1):51-66.
    A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left computable iff it is the supremum of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable sequences of rational numbers we introduce a non-collapsing hierarchy {Σn, Πn, Δn : n ∈ ℕ} of real numbers. We characterize the classes Σ2, Π2 and Δ2 in various ways (...)
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  3.  15
    Real closures of models of weak arithmetic.Emil Jeřábek & Leszek Aleksander Kołodziejczyk - 2013 - Archive for Mathematical Logic 52 (1):143-157.
    D’Aquino et al. (J Symb Log 75(1):1–11, 2010) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by IΔ0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss’ bounded arithmetic: PV or $${\Sigma^b_1-IND^{|x|_k}}$$. It also holds for IΔ0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the (...)
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  4.  94
    Real Numbers and Set theory – Extending the Neo-Fregean Programme Beyond Arithmetic.Bob Hale - 2005 - Synthese 147 (1):21-41.
    It is known that Hume’s Principle, adjoined to a suitable formulation of second-order logic, gives a theory which is almost certainly consistent4 and suffices for arithmetic in the sense that it yields the Dedekind-Peano axioms as theorems. While Hume’s Principle cannot be taken as a definition in any strict sense requiring that it provide for the eliminative paraphrase of its definiendum in every admissible type of occurrence, we hold that it can be viewed as an implicit definition of a (...)
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  5.  58
    Random reals, the rainbow Ramsey theorem, and arithmetic conservation.Chris J. Conidis & Theodore A. Slaman - 2013 - Journal of Symbolic Logic 78 (1):195-206.
    We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey (...)
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  6.  47
    Real closed fields and models of Peano arithmetic.P. D'Aquino, J. F. Knight & S. Starchenko - 2010 - Journal of Symbolic Logic 75 (1):1-11.
    Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), is (...)
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  7.  4
    Encoding true second‐order arithmetic in the real‐algebraic structure of models of intuitionistic elementary analysis.Miklós Erdélyi-Szabó - 2021 - Mathematical Logic Quarterly 67 (3):329-341.
    Based on the paper [4] we show that true second‐order arithmetic is interpretable over the real‐algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras.
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  8. Real closed fields and models of arithmetic (vol 75, pg 1, 2010).P. D'Aquino, J. F. Knight & S. Starchenko - 2012 - Journal of Symbolic Logic 77 (2).
  9.  49
    Conceptual Integration of Arithmetic Operations With Real‐World Knowledge: Evidence From Event‐Related Potentials.Amy M. Guthormsen, Kristie J. Fisher, Miriam Bassok, Lee Osterhout, Melissa DeWolf & Keith J. Holyoak - 2016 - Cognitive Science 40 (3):723-757.
    Research on language processing has shown that the disruption of conceptual integration gives rise to specific patterns of event-related brain potentials —N400 and P600 effects. Here, we report similar ERP effects when adults performed cross-domain conceptual integration of analogous semantic and mathematical relations. In a problem-solving task, when participants generated labeled answers to semantically aligned and misaligned arithmetic problems, the second object label in misaligned problems yielded an N400 effect for addition problems. In a verification task, when participants judged (...)
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  10.  6
    Finite automata, real time processes and counting problems in bounded arithmetics.Mirosław Kutyłowski - 1988 - Journal of Symbolic Logic 53 (1):243-258.
    In this paper we present a negative solution of counting problems for some classes slightly different from bounded arithmetic (▵ 0 sets). To get the results we study properties of chains of finite automata.
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  11.  54
    The strong soundness theorem for real closed fields and Hilbert’s Nullstellensatz in second order arithmetic.Nobuyuki Sakamoto & Kazuyuki Tanaka - 2004 - Archive for Mathematical Logic 43 (3):337-349.
    By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA (...)
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  12.  11
    The arithmetic of Z-numbers: theory and applications.Rafik A. Aliev - 2015 - Chennai: World Scientific. Edited by Oleg H. Huseynov, Rashad R. Aliyev & Akif A. Alizadeh.
    Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature. This book (...)
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  13.  18
    Basic Laws of Arithmetic.Gottlob Frege - 1893 - Oxford, U.K.: Oxford University Press. Edited by Philip A. Ebert, Marcus Rossberg & Crispin Wright.
    The first complete English translation of a groundbreaking work. An ambitious account of the relation of mathematics to logic. Includes a foreword by Crispin Wright, translators' Introduction, and an appendix on Frege's logic by Roy T. Cook. The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It (...)
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  14.  26
    Corrigendum to: “Real closed fields and models of arithmetic”.P. D'Aquino, J. F. Knight & S. Starchenko - 2012 - Journal of Symbolic Logic 77 (2):726-726.
  15.  46
    On the available partial respects in which an axiomatization for real valued arithmetic can recognize its consistency.Dan E. Willard - 2006 - Journal of Symbolic Logic 71 (4):1189-1199.
    Gödel’s Second Incompleteness Theorem states axiom systems of sufficient strength are unable to verify their own consistency. We will show that axiomatizations for a computer’s floating point arithmetic can recognize their cut-free consistency in a stronger respect than is feasible under integer arithmetics. This paper will include both new generalizations of the Second Incompleteness Theorem and techniques for evading it.
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  16.  17
    Peano arithmetic may not be interpretable in the monadic theory of linear orders.Shmuel Lifsches & Saharon Shelah - 1997 - Journal of Symbolic Logic 62 (3):848-872.
    Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We will show here that it is consistent that the monadic second-order theory of no chain interprets Peano Arithmetic.
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  17.  25
    Reals by Abstraction.Bob Hale - 2000 - The Proceedings of the Twentieth World Congress of Philosophy 6:197-207.
    While Frege’s own attempt to provide a purely logical foundation for arithmetic failed, Hume’s principle suffices as a foundation for elementary arithmetic. It is known that the resulting system is consistent—or at least if second-order arithmetic is. Some philosophers deny that HP can be regarded as either a truth of logic or as analytic in any reasonable sense. Others—like Crispin Wright and I—take the opposed view. Rather than defend our claim that HP is a conceptual truth about (...)
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  18.  7
    Every Countable Model of Arithmetic or Set Theory has a Pointwise-Definable End Extension.Joel David Hamkins - forthcoming - Kriterion – Journal of Philosophy.
    According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set (...)
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  19.  17
    Arithmetic as Propaedeutic to Theology: The Brethren of Purity.François Beets - 2015 - Balkan Journal of Philosophy 7 (1):71-76.
    In the 10th century, the Brethren of Purity conceived a henological arithmetic which they believed could explain the mathematical structure of the cosmos (as a macroanthropos), and could lead the student to the discovery of the real substance of his own soul (as a micro-cosmos), a discovery which is the first step towards knowledge of metaphysical and theological truth.
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  20.  10
    Analysis and Necessity in Arithmetic in Light of Maimon’s Concept of Number as Ratio.Idit Chikurel - 2023 - Kant Studien 114 (1):33-67.
    The article examines how Salomon Maimon’s concept of number as ratio can be used to demonstrate that arithmetical judgments are analytical. Based on his critique of Kant’s synthetic a priori judgments, I show how this notion of number fulfills Maimon’s requirements for apodictic knowledge. Moreover, I suggest that Maimon was influenced by mathematicians who previously defined number as a ratio, such as Wallis and Newton. Following an analysis of the real definition of this concept, I conclude that within the (...)
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  21. The absolute arithmetic continuum and the unification of all numbers great and small.Philip Ehrlich - 2012 - Bulletin of Symbolic Logic 18 (1):1-45.
    In his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including $-\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega-\pi$ to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said (...)
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  22.  25
    A Real Number Structure that is Effectively Categorical.Peter Hertling - 1999 - Mathematical Logic Quarterly 45 (2):147-182.
    On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in (...)
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  23. Numbers and Arithmetic: Neither Hardwired Nor Out There.Rafael Núñez - 2009 - Biological Theory 4 (1):68-83.
    What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical (...)
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  24.  25
    Reals by Abstraction.Bob Hale - 2000 - Philosophia Mathematica 8 (2):100--123.
    On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek (...)
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  25.  62
    Angus Macintyre, Kenneth McKenna, and Lou van den Dries. Elimination of quantifiers in algebraic structures. Advances in mathematics, vol. 47 , pp. 74–87. - L. P. D. van den Dries. A linearly ordered ring whose theory admits elimination of quantifiers is a real closed field. Proceedings of the American Mathematical Society, vol. 79 , pp. 97–100. - Bruce I. Rose. Rings which admit elimination of quantifiers. The journal of symbolic logic, vol. 43 , pp. 92–112; Corrigendum, vol. 44 , pp. 109–110. - Chantal Berline. Rings which admit elimination of quantifiers. The journal of symbolic logic, vol. 43 , vol. 46 , pp. 56–58. - M. Boffa, A. Macintyre, and F. Point. The quantifier elimination problem for rings without nilpotent elements and for semi-simple rings. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture. [REVIEW]Gregory L. Cherlin - 1985 - Journal of Symbolic Logic 50 (4):1079-1080.
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  26.  25
    Stål Anderaa (Oslo), A Traktenbrot inseparability theorem for groups. Peter Dybjer (G öteborg), Normalization by Yoneda embedding (joint work with D. Cubric and PJ Scott). Abbas Edalat (Imperial College), Dynamical systems, measures, fractals, and exact real number arithmetic via domain theory. [REVIEW]Anita Feferman, Solomon Feferman, Robert Goldblatt, Yuri Gurevich, Klaus Grue, Sven Ove Hansson, Lauri Hella, Robert K. Meyer & Petri Mäenpää - 1997 - Bulletin of Symbolic Logic 3 (4).
  27.  33
    Logicism as Making Arithmetic Explicit.Vojtěch Kolman - 2015 - Erkenntnis 80 (3):487-503.
    This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so (...)
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  28.  16
    Numerosity, number, arithmetization, measurement and psychology.Thomas M. Nelson & S. Howard Bartley - 1961 - Philosophy of Science 28 (2):178-203.
    The paper aims to put certain basic mathematical elements and operations into an empirical perspective, evaluate the empirical status of various analytic operations widely used within psychology and suggest alternatives to procedures criticized as inadequate. Experimentation shows the "manyness" of items to be a perceptual quality for both young children and animals and that natural operations are performed by naive children analogous to those performed by persons tutored in arithmetic. Number, counting, arithmetic operations therefore can make distinctions that (...)
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  29. WEIHRAUCH, K. and KREITZ, C., Representations of the real numbers and of the open subsets of the set of real numbers WILKIE, AJ and PARIS, JB, On the scheme of induction for bounded arithmetic formulas. [REVIEW]Las Kirby & R. Diaconescu - 1987 - Annals of Pure and Applied Logic 35:303.
  30.  3
    Weak theories of nonstandard arithmetic and analysis.Jeremy Avigad - manuscript
    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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  31.  1
    The Absolute Arithmetic Continuum and Its Geometric Counterpart.Philip Ehrlich - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 1677-1718.
    In a number of works, we have suggested that whereas the ordered field R of real numbers should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), the ordered field No of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo NBG). In the present chapter, as part of a more general exposition of the absolute arithmetic continuum, we will outline some of the properties of the system of surreal (...)
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  32.  72
    Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint.Crispin Wright - 2000 - Notre Dame Journal of Formal Logic 41 (4):317--334.
    We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term Frege's (...)
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  33.  17
    Undecidability of the Real-Algebraic Structure of Scott's Model.Miklós Erdélyi-Szabó - 1998 - Mathematical Logic Quarterly 44 (3):344-348.
    We show that true first-order arithmetic of the positive integers is interpretable over the real-algebraic structure of Scott's topological model for intuitionistic analysis. From this the undecidability of the structure follows.
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  34.  31
    Undecidability of the Real-Algebraic Structure of Models of Intuitionistic Elementary Analysis.Miklós Erdélyi-Szabó - 2000 - Journal of Symbolic Logic 65 (3):1014-1030.
    We show that true first-order arithmetic is interpretable over the real-algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras. From this the undecidability of the structures follows. We also show that Scott's model is equivalent to true second-order arithmetic. In the appendix we argue that undecidability on the language of ordered rings follows from intuitionistically plausible properties of the real numbers.
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  35.  22
    Second order arithmetic as the model companion of set theory.Giorgio Venturi & Matteo Viale - 2023 - Archive for Mathematical Logic 62 (1):29-53.
    This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is forcible (...)
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  36.  43
    Semantic alignment across whole-number arithmetic and rational numbers: evidence from a Russian perspective.Yulia A. Tyumeneva, Galina Larina, Ekaterina Alexandrova, Melissa DeWolf, Miriam Bassok & Keith J. Holyoak - 2018 - Thinking and Reasoning 24 (2):198-220.
    Solutions to word problems are moderated by the semantic alignment of real-world relations with mathematical operations. Categorical relations between entities are aligned with addition, whereas certain functional relations between entities are aligned with division. Similarly, discreteness vs. continuity of quantities is aligned with different formats for rational numbers. These alignments have been found both in textbooks and in the performance of college students in the USA and in South Korea. The current study examined evidence for alignments in Russia. Textbook (...)
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  37.  35
    Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics.Marek Czachor - 2020 - Foundations of Science 26 (1):75-95.
    Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition β1⊕β2=tanh+tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\oplus \beta _2=\tanh \big +\tanh ^{-1}\big )$$\end{document}, although multiplication β1⊙β2=tanh·tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} (...)
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  38.  21
    Borel quasi-orderings in subsystems of second-order arithmetic.Alberto Marcone - 1991 - Annals of Pure and Applied Logic 54 (3):265-291.
    We study the provability in subsystems of second-order arithmetic of two theorems of Harrington and Shelah [6] about Borel quasi-orderings of the reals. These theorems turn out to be provable in ATR0, thus giving further evidence to the observation that ATR0 is the minimal subsystem of second-order arithmetic in which significant portion of descriptive set theory can be developed. As in [6] considering the lightface versions of the theorems will be instrumental in their proof and the main techniques (...)
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  39.  26
    Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap (review).John MacFarlane - 2001 - Journal of the History of Philosophy 39 (3):454-456.
    In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 454-456 [Access article in PDF] Michael Potter. Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap.New York: Oxford University Press, 2000. Pp. x + 305. Cloth, $45.00. This book tells the story of a remarkable series of answers to two related questions:(1) How can arithmetic be necessary and knowable a priori? [End Page 454](2) What accounts for the (...)
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  40. Frege meets dedekind: A neologicist treatment of real analysis.Stewart Shapiro - 2000 - Notre Dame Journal of Formal Logic 41 (4):335--364.
    This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a (...)
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  41.  12
    Empiricism in arithmetic and analysis.E. B. Davies - 2003 - Philosophia Mathematica 11 (1):53-66.
    We discuss the philosophical status of the statement that (9n – 1) is divisible by 8 for various sizes of the number n. We argue that even this simple problem reveals deep tensions between truth and verification. Using Gillies's empiricist classification of theories into levels, we propose that statements in arithmetic should be classified into three different levels depending on the sizes of the numbers involved. We conclude by discussing the relationship between the real number system and the (...)
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  42.  18
    Differential calculus and nilpotent real numbers.Anders Kock - 2003 - Bulletin of Symbolic Logic 9 (2):225-230.
    Do there exist real numbers d with d2 = 0? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid and (...)
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  43.  38
    Periodic points and subsystems of second-order arithmetic.Harvey Friedman, Stephen G. Simpson & Xiaokang Yu - 1993 - Annals of Pure and Applied Logic 62 (1):51-64.
    We study the formalization within sybsystems of second-order arithmetic of theorems concerning periodic points in dynamical systems on the real line. We show that Sharkovsky's theorem is provable in WKL0. We show that, with an additional assumption, Sharkovsky's theorem is provable in RCA0. We show that the existence for all n of n-fold iterates of continuous mappings of the closed unit interval into itself is equivalent to the disjunction of Σ02 induction and weak König's lemma.
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  44.  28
    Proper and piecewise proper families of reals.Victoria Gitman - 2009 - Mathematical Logic Quarterly 55 (5):542-550.
    I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ (...)
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  45.  16
    The Recursively Mahlo Property in Second Order Arithmetic.Michael Rathjen - 1996 - Mathematical Logic Quarterly 42 (1):59-66.
    The paper characterizes the second order arithmetic theorems of a set theory that features a recursively Mahlo universe; thereby complementing prior proof-theoretic investigations on this notion. It is shown that the property of being recursively Mahlo corresponds to a certain kind of β-model reflection in second order arithmetic. Further, this leads to a characterization of the reals recursively computable in the superjump functional.
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  46.  22
    A theorem on partial conservativity in arithmetic.Per Lindström - 2011 - Journal of Symbolic Logic 76 (1):341 - 347.
    Improving on a result of Arana, we construct an effective family (φ r | r ∈ ℚ ⋂ [0, 1]) of Σ n -conservative Π n sentences, increasing in strength as r decreases, with the property that ¬φ p is Π n -conservative over PA + φ q whenever p < q. We also construct a family of Σ n sentences with properties as above except that the roles of Σ n and Π n are reversed. The latter result allows (...)
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  47.  22
    The abstract type of the real numbers.Fernando Ferreira - 2021 - Archive for Mathematical Logic 60 (7):1005-1017.
    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 (...)
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  48.  21
    A variant to Hilbert's theory of the foundations of arithmetic.G. Kreisel - 1953 - British Journal for the Philosophy of Science 4 (14):107-129.
    IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist) criticisms of classical logic, which Hilbert's theory was intended to meet, never even alluded to inconsistencies (in classical arithmetic), and since the investigations of Hilbert's school have always established much more than mere consistency, it is natural to formulate another general problem in the foundations of mathematics: to translate statements (...)
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  49.  66
    Review: Potter, Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap.John MacFarlane - 2001 - Journal of the History of Philosophy 39 (3):454-456.
    In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 454-456 [Access article in PDF] Michael Potter. Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap.New York: Oxford University Press, 2000. Pp. x + 305. Cloth, $45.00. This book tells the story of a remarkable series of answers to two related questions:(1) How can arithmetic be necessary and knowable a priori? [End Page 454](2) What accounts for the (...)
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  50.  11
    Will the real philosopher behind the last logicist please stand up?Philip Robbins - 1998 - Southern Journal of Philosophy 36 (2):265-287.
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