Search results for 'natural numbers' (try it on Scholar)

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  1. Ronald Numbers (2003). Science Without God: Natural Laws and Christian Beliefs. In David C. Lindberg & Ronald L. Numbers (eds.), When Science and Christianity Meet. University of Chicago Press 266.
     
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  2. S. W. P. Steen (1972). Mathematical Logic with Special Reference to the Natural Numbers. Cambridge [Eng.]University Press.
    This book presents a comprehensive treatment of basic mathematical logic. The author's aim is to make exact the vague, intuitive notions of natural number, preciseness, and correctness, and to invent a method whereby these notions can be communicated to others and stored in the memory. He adopts a symbolic language in which ideas about natural numbers can be stated precisely and meaningfully, and then investigates the properties and limitations of this language. The treatment of mathematical concepts in (...)
     
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  3. Edward N. Zalta (1999). Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege"s Grundgesetze in Object Theory. [REVIEW] Journal of Philosophical Logic 28 (6):619-660.
    In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's Grundgesetze. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (...)
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  4.  31
    Wojciech Krysztofiak (2012). Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW] Axiomathes 22 (4):433-456.
    The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in (...)
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  5.  38
    Shunsuke Yatabe (2007). Distinguishing Non-Standard Natural Numbers in a Set Theory Within Łukasiewicz Logic. Archive for Mathematical Logic 46 (3-4):281-287.
    In ${\mathbf{H}}$ , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of ${\mathbf{H}}$ , and we prove an analogy of Hájek’s theorem with a very simple procedure.
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  6.  76
    Øystein Linnebo (2009). The Individuation of the Natural Numbers. In Otavio Bueno & Øystein Linnebo (eds.), New Waves in Philosophy of Mathematics. Palgrave
    It is sometimes suggested that criteria of identity should play a central role in an account of our most fundamental ways of referring to objects. The view is nicely illustrated by an example due to (Quine, 1950). Suppose you are standing at the bank of a river, watching the water that floats by. What is required for you to refer to the river, as opposed to a particular segment of it, or the totality of its water, or the current temporal (...)
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  7.  1
    Michael Rathjen (2008). The Natural Numbers in Constructive Set Theory. Mathematical Logic Quarterly 54 (1):83-97.
    Constructive set theory started with Myhill's seminal 1975 article [8]. This paper will be concerned with axiomatizations of the natural numbers in constructive set theory discerned in [3], clarifying the deductive relationships between these axiomatizations and the strength of various weak constructive set theories.
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  8.  22
    Eric Margolis & Stephen Laurence (2008). How to Learn the Natural Numbers: Inductive Inference and the Acquisition of Number Concepts. Cognition 106 (2):924-939.
    Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A. (2006). Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51–B60.] argue that such an inductive inference is consistent with (...)
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  9. Paolo Mancosu (2009). Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor's Theory of Infinite Number Inevitable? Review of Symbolic Logic 2 (4):612-646.
    Cantorsizesizesizewhole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the partdel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, (...)
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  10.  2
    Dimiter Skordev (2002). Computability of Real Numbers by Using a Given Class of Functions in the Set of the Natural Numbers. Mathematical Logic Quarterly 48 (S1):91-106.
    Given a class ℱ oft otal functions in the set oft he natural numbers, one could study the real numbers that have arbitrarily close rational approximations explicitly expressible by means of functions from ℱ. We do this for classes ℱsatisfying certain closedness conditions. The conditions in question are satisfied for example by the class of all recursive functions, by the class of the primitive recursive ones, by any of the Grzegorczyk classes ℰnwith n ≥ 2, by the (...)
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  11. Øystein Linnebo (2009). Frege's Context Principle and Reference to Natural Numbers. In Sten Lindström (ed.), Logicism, Intuitionism, and Formalism: What Has Become of Them. Springer
    Frege proposed that his Context Principle—which says that a word has meaning only in the context of a proposition—can be used to explain reference, both in general and to mathematical objects in particular. I develop a version of this proposal and outline answers to some important challenges that the resulting account of reference faces. Then I show how this account can be applied to arithmetic to yield an explanation of our reference to the natural numbers and of their (...)
     
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  12.  17
    Roi Cohen Kadosh & Vincent Walsh (2008). From Magnitude to Natural Numbers: A Developmental Neurocognitive Perspective. Behavioral and Brain Sciences 31 (6):647-648.
    In their target article, Rips et al. have presented the view that there is no necessary dependency between natural numbers and internal magnitude. However, they do not give enough weight to neuroimaging and neuropsychological studies. We provide evidence demonstrating that the acquisition of natural numbers depends on magnitude representation and that natural numbers develop from a general magnitude mechanism in the parietal lobes.
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  13.  59
    Jeffrey F. Sicha (1970). Counting and the Natural Numbers. Philosophy of Science 37 (3):405-416.
    Early sections of the paper develop a view of the natural numbers and a view of counting which are suggested by the remarks of several modern philosophers. Further investigation of these views leads to one of the main theses of the paper: a special kind of quantifier, the "numerical quantifier" is essential to counting. The remainder of the paper suggests the rudiments of a new view of the natural numbers, a view which maintains that numerical quantifiers (...)
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  14.  29
    Robert Paré & Leopoldo Román (1989). Monoidal Categories with Natural Numbers Object. Studia Logica 48 (3):361 - 376.
    The notion of a natural numbers object in a monoidal category is defined and it is shown that the theory of primitive recursive functions can be developed. This is done by considering the category of cocommutative comonoids which is cartesian, and where the theory of natural numbers objects is well developed. A number of examples illustrate the usefulness of the concept.
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  15.  4
    Kamila Bendová (2001). On Ordering and Multiplication of Natural Numbers. Archive for Mathematical Logic 40 (1):19-23.
    Even if the ordering of all natural number is (known to be) not definable from multiplication of natural numbers and ordering of primes, there is a simple axiom system in the language $(\times,<,1)$ such that the multiplicative structure of positive integers has a unique expansion by a linear order coinciding with the standard order for primes and satisfying the axioms – namely the standard one.
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  16.  19
    Erich Reck, Frege's Natural Numbers: Motivations and Modifications.
    Frege's main contributions to logic and the philosophy of mathematics are, on the one hand, his introduction of modern relational and quantificational logic and, on the other, his analysis of the concept of number. My focus in this paper will be on the latter, although the two are closely related, of course, in ways that will also play a role. More specifically, I will discuss Frege's logicist reconceptualization of the natural numbers with the goal of clarifying two aspects: (...)
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  17.  18
    C. Barry Jay (1989). A Note on Natural Numbers Objects in Monoidal Categories. Studia Logica 48 (3):389 - 393.
    The internal language of a monoidal category yields simple proofs of results about a natural numbers object therein.
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  18.  3
    Annika M. Wille (2004). The Variety of Lattice-Ordered Monoids Generated by the Natural Numbers. Studia Logica 76 (2):275 - 290.
    We study the variety Var() of lattice-ordered monoids generated by the natural numbers. In particular, we show that it contains all 2-generated positively ordered lattice-ordered monoids satisfying appropriate distributive laws. Moreover, we establish that the cancellative totally ordered members of Var() are submonoids of ultrapowers of and can be embedded into ordered fields. In addition, the structure of ultrapowers relevant to the finitely generated case is analyzed. Finally, we provide a complete isomorphy invariant in the two-generated case.
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  19.  6
    Erich Reck (2003). Frege, Natural Numbers, and Arithmetic's Umbilical Cord. Manuscrito 26 (2):427-70.
    A central part of Frege's logicism is his reconstruction of the natural numbers as equivalence classes of equinumerous concepts or classes. In this paper, I examine the relationship of this reconstruction both to earlier views, from Mill all the way back to Plato, and to later formalist and structuralist views; I thus situate Frege within what may be called the “rise of pure mathematics” in the nineteenth century. Doing so allows us to acknowledge continuities between Frege's and other (...)
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  20.  13
    Dwight Read (2008). Learning Natural Numbers is Conceptually Different Than Learning Counting Numbers. Behavioral and Brain Sciences 31 (6):667-668.
    How children learn number concepts reflects the conceptual and logical distinction between counting numbers, based on a same-size concept for collections of objects, and natural numbers, constructed as an algebra defined by the Peano axioms for arithmetic. Cross-cultural research illustrates the cultural specificity of counting number systems, and hence the cultural context must be taken into account.
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  21.  11
    Marie-Pascale Noël, Jacques Grégoire, Gaëlle Meert & Xavier Seron (2008). The Innate Schema of Natural Numbers Does Not Explain Historical, Cultural, and Developmental Differences. Behavioral and Brain Sciences 31 (6):664-665.
    Rips et al.'s proposition cannot account for the facts that (1) a historical look at the word number systems suggests that the concept of natural numbers has been progressively elaborated; (2) people from cultures without an elaborate counting system do not master the concept of natural numbers; (3) children take time to master natural numbers; and (4) the competing advantage of the postulated math schema in the natural selection process is not obvious.
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  22.  8
    Steven C. Leth (1988). Sequences in Countable Nonstandard Models of the Natural Numbers. Studia Logica 47 (3):243 - 263.
    Two different equivalence relations on countable nonstandard models of the natural numbers are considered. Properties of a standard sequence A are correlated with topological properties of the equivalence classes of the transfer of A. This provides a method for translating results from analysis into theorems about sequences of natural numbers.
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  23.  4
    Daniel Dzierzgowski (1996). Finite Sets and Natural Numbers in Intuitionistic TT. Notre Dame Journal of Formal Logic 37 (4):585-601.
    We show how to interpret Heyting's arithmetic in an intuitionistic version of TT, Russell's Simple Theory of Types. We also exhibit properties of finite sets in this theory and compare them with the corresponding properties in classical TT. Finally, we prove that arithmetic can be interpreted in intuitionistic TT, the subsystem of intuitionistic TT involving only three types. The definitions of intuitionistic TT and its finite sets and natural numbers are obtained in a straightforward way from the classical (...)
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  24.  6
    Wen-Chi Chiang (2008). What is Still Needed? On Nativist Proposals for Acquiring Concepts of Natural Numbers. Behavioral and Brain Sciences 31 (6):646-647.
    Rips et al.'s analyses have boosted the plausibility of proposals that the human mind embodies some critical properties of natural numbers. I suggest that such proposals can be further evaluated by infant studies, neuropsychological data, and evolution-based considerations, and additionally, that Rips et al.'s model may need to be modified in order to more completely reflect infants' quantitative abilities.
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  25.  6
    Mathieu Le Corre (2008). Why Cardinalities Are the “NaturalNatural Numbers. Behavioral and Brain Sciences 31 (6):659-659.
    According to Rips et al., numerical cognition develops out of two independent sets of cognitive primitives – one that supports enumeration, and one that supports arithmetic and the concepts of natural numbers. I argue against this proposal because it incorrectly predicts that natural number concepts could develop without prior knowledge of enumeration.
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  26.  13
    Carlo Proietti (2008). Natural Numbers and Infinitesimals: A Discussion Between Benno Kerry and Georg Cantor. History and Philosophy of Logic 29 (4):343-359.
    During the first months of 1887, while completing the drafts of his Mitteilungen zur Lehre vom Transfiniten, Georg Cantor maintained a continuous correspondence with Benno Kerry. Their exchange essentially concerned two main topics in the philosophy of mathematics, namely, (a) the concept of natural number and (b) the infinitesimals. Cantor's and Kerry's positions turned out to be irreconcilable, mostly because of Kerry's irremediably psychologistic outlook, according to Cantor at least. In this study, I will examine and reconstruct the main (...)
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  27.  73
    Felix Mühlhölzer (2010). Mathematical Intuition and Natural Numbers: A Critical Discussion. [REVIEW] Erkenntnis 73 (2):265–292.
    Charles Parsons’ book “Mathematical Thought and Its Objects” of 2008 (Cambridge University Press, New York) is critically discussed by concentrating on one of Parsons’ main themes: the role of intuition in our understanding of arithmetic (“intuition” in the specific sense of Kant and Hilbert). Parsons argues for a version of structuralism which is restricted by the condition that some paradigmatic structure should be presented that makes clear the actual existence of structures of the necessary sort. Parsons’ paradigmatic structure is the (...)
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  28.  6
    Stella F. Lourenco & Susan C. Levine (2008). Early Numerical Representations and the Natural Numbers: Is There Really a Complete Disconnect? Behavioral and Brain Sciences 31 (6):660-660.
    The proposal of Rips et al. is motivated by discontinuity and input claims. The discontinuity claim is that no continuity exists between early (nonverbal) numerical representations and natural number. The input claim is that particular experiences (e.g., cardinality-related talk and object-based activities) do not aid in natural number construction. We discuss reasons to doubt both claims in their strongest forms.
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  29.  0
    Christian Michaux & Roger Villemaire (1996). Presburger Arithmetic and Recognizability of Sets of Natural Numbers by Automata: New Proofs of Cobham's and Semenov's Theorems. Annals of Pure and Applied Logic 77 (3):251-277.
    Let be the set of nonnegative integers. We show the two following facts about Presburger's arithmetic:1. 1. Let . If L is not definable in , + then there is an definable in , such that there is no bound on the distance between two consecutive elements of L′. and2. 2. is definable in , + if and only if every subset of which is definable in is definable in , +. These two Theorems are of independent interest but we (...)
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  30. Friederike Moltmann (2013). Reference to Numbers in Natural Language. Philosophical Studies 162 (3):499 - 536.
    A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this (...)
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  31.  48
    Friederike Moltmann (2013). Reference to Numbers in Natural Language. Philosophical Studies 162 (3):499 - 536.
    A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this (...)
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  32.  1
    Piergiorgio Odifreddi (1989). Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers. Sole Distributors for the Usa and Canada, Elsevier Science Pub. Co..
    Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity classes, ranging from small (...)
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  33.  5
    Lance J. Rips, Jennifer Asmuth & Amber Bloomfield (2006). Giving the Boot to the Bootstrap: How Not to Learn the Natural Numbers. Cognition 101 (3):B51-B60.
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  34.  5
    Ekaterina B. Fokina & Sy‐David Friedman (2012). On Σ11 Equivalence Relations Over the Natural Numbers. Mathematical Logic Quarterly 58 (1‐2):113-124.
    We study the structure of Σ11 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly equation imagei.e., Σ11 but not equation image equivalence classes. We also show the existence of incomparable Σ11 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ11 (...)
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  35.  3
    Julia F. Knight (1973). Complete Types and the Natural Numbers. Journal of Symbolic Logic 38 (3):413-415.
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  36.  39
    Harold T. Hodes (1990). Where Do the Natural Numbers Come From? Synthese 84 (3):347-407.
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  37.  4
    B. Banaschewski (1991). Fixpoints Without the Natural Numbers. Mathematical Logic Quarterly 37 (8):125-128.
  38.  3
    Lou van den Dries & Yiannis N. Moschovakis (2004). The Euclidean Algorithm on the Natural Numbers Æ= 0, 1,... Can Be Specified Succinctly by the Recursive Program. Bulletin of Symbolic Logic 10 (3).
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  39.  35
    Gregory Landini (1996). The Definability of the Set of Natural Numbers in the 1925 Principia Mathematica. Journal of Philosophical Logic 25 (6):597 - 615.
    In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Godel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot (...)
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  40.  5
    E. J. Lowe (1993). Are the Natural Numbers Individuals or Sorts? Analysis 53 (3):142 - 146.
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  41.  3
    Philip Mirowski (1992). Looking for Those Natural Numbers: Dimensionless Constants and the Idea of Natural Measurement. Science in Context 5 (1).
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  42.  8
    Antonin Sochor (1975). Contribution to the Theory of Semisets VI: (Non‐Existence of the Class of All Absolute Natural Numbers). Mathematical Logic Quarterly 21 (1):439-442.
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  43.  9
    T. J. Smiley (1988). Frege's `Series of Natural Numbers'. Mind 97 (388):583-584.
  44.  6
    Felix Muhlholzer (2010). Mathematical Intuition and Natural Numbers: A Critical Discussion: Charles Parsons, Mathematical Thought and Its Objects, Cambridge University Press, New York, 2008, Xx+ 378 Pp. [REVIEW] Erkenntnis 73 (2):265-292.
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  45.  1
    Thomjas Bedürftig (1989). Another Characterization of the Natural Numbers. Mathematical Logic Quarterly 35 (2):185-186.
  46.  1
    Alonzo Church (1950). Review: Baruch Germansky, An Alternative Proof of a Theorem of Equivalence Concerning Axioms of Natural Numbers. [REVIEW] Journal of Symbolic Logic 15 (4):282-282.
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  47.  9
    Renling Jin (2001). Existence of Some Sparse Sets of Nonstandard Natural Numbers. Journal of Symbolic Logic 66 (2):959-973.
    Answers are given to two questions concerning the existence of some sparse subsets of $\mathscr{H} = \{0, 1,..., H - 1\} \subseteq * \mathbb{N}$ , where H is a hyperfinite integer. In § 1, we answer a question of Kanovei by showing that for a given cut U in H, there exists a countably determined set $X \subseteq \mathscr{H}$ which contains exactly one element in each U-monad, if and only if U = a · N for some $a \in \mathscr{H} (...)
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  48. John Myhill (1974). The Undefinability of the Set of Natural Numbers in the Ramified Principia. In George Nakhnikian (ed.), Bertrand Russell's Philosophy. [London]Duckworth 19--27.
     
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  49.  8
    Harold T. Hodes (1990). Where Do the Natural Numbers Come From? In Memory of Geoffrey Joseph. Synthese 84 (3):347 - 407.
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  50.  3
    Michael D. Resnik (1982). Review: David Bostock, Logic and Arithmetic. Volume 1. Natural Numbers; David Bostock, Logic and Arithmetic. Volume 2. Rational and Irrational Numbers. [REVIEW] Journal of Symbolic Logic 47 (3):708-713.
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