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

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  1. S. W. P. Steen (1972). Mathematical Logic with Special Reference to the Natural Numbers. Cambridge [Eng.]University Press.score: 168.0
    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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  2. 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.score: 160.0
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  3. Wojciech Krysztofiak (2012). Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW] Axiomathes 22 (4):433-456.score: 156.0
    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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  4. 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.score: 156.0
    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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  5. Øystein Linnebo (2009). The Individuation of the Natural Numbers. In Otavio Bueno & Øystein Linnebo (eds.), New Waves in Philosophy of Mathematics. Palgrave.score: 150.0
    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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  6. Shunsuke Yatabe (2007). Distinguishing Non-Standard Natural Numbers in a Set Theory Within Łukasiewicz Logic. Archive for Mathematical Logic 46 (3-4):281-287.score: 150.0
    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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  7. Kristen Pilner Blair, Miriam Rosenberg-Lee, Jessica M. Tsang, Daniel L. Schwartz & Vinod Menon (2012). Beyond Natural Numbers: Negative Number Representation in Parietal Cortex. Frontiers in Human Neuroscience 6.score: 132.0
    Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related fMRI design, we examined brain (...)
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  8. 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.score: 120.0
    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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  9. 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.score: 120.0
    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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  10. Robert Paré & Leopoldo Román (1989). Monoidal Categories with Natural Numbers Object. Studia Logica 48 (3):361 - 376.score: 120.0
    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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  11. Roi Cohen Kadosh & Vincent Walsh (2008). From Magnitude to Natural Numbers: A Developmental Neurocognitive Perspective. Behavioral and Brain Sciences 31 (6):647-648.score: 120.0
    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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  12. Eric Margolis & Stephen Laurence (2008). How to Learn the Natural Numbers: Inductive Inference and the Acquisition of Number Concepts. Cognition 106 (2):924-939.score: 120.0
    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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  13. Erich Reck, Frege's Natural Numbers: Motivations and Modifications.score: 120.0
    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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  14. Jeffrey F. Sicha (1970). Counting and the Natural Numbers. Philosophy of Science 37 (3):405-416.score: 120.0
    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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  15. C. Barry Jay (1989). A Note on Natural Numbers Objects in Monoidal Categories. Studia Logica 48 (3):389 - 393.score: 120.0
    The internal language of a monoidal category yields simple proofs of results about a natural numbers object therein.
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  16. Dwight Read (2008). Learning Natural Numbers is Conceptually Different Than Learning Counting Numbers. Behavioral and Brain Sciences 31 (6):667-668.score: 120.0
    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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  17. 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.score: 120.0
    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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  18. Steven C. Leth (1988). Sequences in Countable Nonstandard Models of the Natural Numbers. Studia Logica 47 (3):243 - 263.score: 120.0
    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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  19. 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.score: 120.0
    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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  20. Mathieu Le Corre (2008). Why Cardinalities Are the “NaturalNatural Numbers. Behavioral and Brain Sciences 31 (6):659-659.score: 120.0
    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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  21. Daniel Dzierzgowski (1996). Finite Sets and Natural Numbers in Intuitionistic TT. Notre Dame Journal of Formal Logic 37 (4):585-601.score: 120.0
    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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  22. Kamila Bendová (2001). On Ordering and Multiplication of Natural Numbers. Archive for Mathematical Logic 40 (1):19-23.score: 120.0
    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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  23. Annika M. Wille (2004). The Variety of Lattice-Ordered Monoids Generated by the Natural Numbers. Studia Logica 76 (2):275 - 290.score: 120.0
    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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  24. Michael Rathjen (2008). The Natural Numbers in Constructive Set Theory. Mathematical Logic Quarterly 54 (1):83-97.score: 118.0
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  25. Friederike Moltmann (2013). Reference to Numbers in Natural Language. Philosophical Studies 162 (3):499 - 536.score: 114.0
    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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  26. Felix Mühlhölzer (2010). Mathematical Intuition and Natural Numbers: A Critical Discussion. [REVIEW] Erkenntnis 73 (2):265–292.score: 104.0
    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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  27. Carlo Proietti (2008). Natural Numbers and Infinitesimals: A Discussion Between Benno Kerry and Georg Cantor. History and Philosophy of Logic 29 (4):343-359.score: 104.0
    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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  28. 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.score: 104.0
    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. Lance J. Rips, Amber Bloomfield & Jennifer Asmuth (2008). From Numerical Concepts to Concepts of Number. Behavioral and Brain Sciences 31 (6):623-642.score: 96.0
    Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the (...) number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)
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  30. 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.score: 94.0
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  31. 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..score: 92.0
    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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  32. Harold T. Hodes (1990). Where Do the Natural Numbers Come From? Synthese 84 (3):347-407.score: 90.0
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  33. Gregory Landini (1996). The Definability of the Set of Natural Numbers in the 1925 Principia Mathematica. Journal of Philosophical Logic 25 (6):597 - 615.score: 90.0
    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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  34. Charles Sayward (2010). Dialogues Concerning Natural Numbers. Peter Lang.score: 90.0
    Two philosophical theories, mathematical Platonism and nominalism, are the background of six dialogues in this book. There are five characters in these dialogues: three are nominalists; the fourth is a Platonist; the main character is somewhat skeptical on most issues in the philosophy of mathematics, and is particularly skeptical regarding the two background theories.
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  35. Christopher Barney (2003). Ultrafilters on the Natural Numbers. Journal of Symbolic Logic 68 (3):764-784.score: 90.0
    We study the problem of existence and generic existence of ultrafilters on ω. We prove a conjecture of $J\ddot{o}rg$ Brendle's showing that there is an ultrafilter that is countably closed but is not an ordinal ultrafilter under CH. We also show that Canjar's previous partial characterization of the generic existence of Q-points is the best that can be done. More simply put, there is no normal cardinal invariant equality that fully characterizes the generic existence of Q-points. We then sharpen results (...)
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  36. Harold T. Hodes (1990). Where Do the Natural Numbers Come From? In Memory of Geoffrey Joseph. Synthese 84 (3):347 - 407.score: 90.0
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  37. Renling Jin (2001). Existence of Some Sparse Sets of Nonstandard Natural Numbers. Journal of Symbolic Logic 66 (2):959-973.score: 90.0
    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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  38. Shianghaw Wang (1943). A System of Completely Independent Axioms for the Sequence of Natural Numbers. Journal of Symbolic Logic 8 (1):41-44.score: 90.0
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  39. Mariana Córdoba (2013). ¿Relatividad ontológica o radicalidad ontológica? La respuesta estructuralista de Shapiro al problema de la identificación y la obstinación por el realismo. Revista de Filosofía (Madrid) 38 (1):7-28.score: 90.0
    In this paper I will analyze some philosophically relevant aspects involved in the dissolution of Benacerraf’s problem of fixing the identity of natural numbers by Shapiro’s structuralism. My fundamental aim is to present three criticisms to Shapiro’s position –to his conception of language, to his characterization of structures as ante rem, and to his dramatic conception of ontology. Some of these criticisms will also be directed to Benacerraf’s identification problem.
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  40. G. D. Duthie (1959). Notes on the Logic of Natural Numbers. Philosophical Quarterly 9 (36):217-230.score: 90.0
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  41. W. D. Hart (1991). Natural Numbers. Crítica 23 (69):61 - 81.score: 90.0
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  42. Geoffrey Hunter (1979). Dummett's Arguments About the Natural Numbers. Proceedings of the Aristotelian Society 80:115 - 126.score: 90.0
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  43. T. J. Smiley (1988). Frege's `Series of Natural Numbers'. Mind 97 (388):583-584.score: 90.0
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  44. Frederic B. Fitch (1949). On Natural Numbers, Integers, and Rationals. Journal of Symbolic Logic 14 (2):81-84.score: 90.0
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  45. Ekaterina B. Fokina & Sy‐David Friedman (2012). On Σ11 Equivalence Relations Over the Natural Numbers. Mathematical Logic Quarterly 58 (1‐2):113-124.score: 90.0
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  46. E. J. Lowe (1993). Are the Natural Numbers Individuals or Sorts? Analysis 53 (3):142 - 146.score: 90.0
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  47. Daniel Dzierzgowski (1998). Finite Sets and Natural Numbers in Intuitionistic TT Without Extensionality. Studia Logica 61 (3):417-428.score: 90.0
    In this paper, we prove that Heyting's arithmetic can be interpreted in an intuitionistic version of Russell's Simple Theory of Types without extensionality.
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  48. Francis J. Tytus (1967). An Elementary Construction of the Natural Numbers. Notre Dame Journal of Formal Logic 8 (4):297-300.score: 90.0
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  49. Erich Reck (2003). Frege, Natural Numbers, and Arithmetic's Umbilical Cord. Manuscrito 26 (2):427-70.score: 90.0
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