Search results for 'Set theory' (try it on Scholar)

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  1. Mary Tiles (1989/2004). The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. Dover Publications.score: 240.0
    David Hilbert famously remarked, “No one will drive us from the paradise that Cantor has created.” This volume offers a guided tour of modern mathematics’ Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor’s transfinite paradise; axiomatic set theory; logical objects and logical types; independence results and the universe of sets; and the constructs and reality of mathematical structure. Philosophers (...)
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  2. Han Geurdes, Heisenberg Quantum Mechanics, Numeral Set-Theory And.score: 240.0
    In the paper we will employ set theory to study the formal aspects of quantum mechanics without explicitly making use of space-time. It is demonstrated that von Neuman and Zermelo numeral sets, previously efectively used in the explanation of Hardy’s paradox, follow a Heisenberg quantum form. Here monadic union plays the role of time derivative. The logical counterpart of monadic union plays the part of the Hamiltonian in the commutator. The use of numerals and monadic union in the classical (...)
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  3. Jeffrey W. Roland (2010). Concept Grounding and Knowledge of Set Theory. Philosophia 38 (1):179-193.score: 240.0
    C. S. Jenkins has recently proposed an account of arithmetical knowledge designed to be realist, empiricist, and apriorist: realist in that what’s the case in arithmetic doesn’t rely on us being any particular way; empiricist in that arithmetic knowledge crucially depends on the senses; and apriorist in that it accommodates the time-honored judgment that there is something special about arithmetical knowledge, something we have historically labeled with ‘a priori’. I’m here concerned with the prospects for extending Jenkins’s account beyond arithmetic—in (...)
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  4. P. T. Johnstone (1987). Notes on Logic and Set Theory. Cambridge University Press.score: 240.0
    A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included (...)
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  5. Salvatore Florio & Stewart Shapiro (2014). Set Theory, Type Theory, and Absolute Generality. Mind 123 (489):157-174.score: 240.0
    In light of the close connection between the ontological hierarchy of set theory and the ideological hierarchy of type theory, Øystein Linnebo and Agustín Rayo have recently offered an argument in favour of the view that the set-theoretic universe is open-ended. In this paper, we argue that, since the connection between the two hierarchies is indeed tight, any philosophical conclusions cut both ways. One should either hold that both the ontological hierarchy and the ideological hierarchy are open-ended, or (...)
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  6. Richard Pettigrew (2010). The Foundations of Arithmetic in Finite Bounded Zermelo Set Theory. Cahiers du Centre de Logique 17:99-118.score: 240.0
    In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of (...)
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  7. Zach Weber (2010). Extensionality and Restriction in Naive Set Theory. Studia Logica 94 (1):87 - 104.score: 240.0
    The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set (...), leads to triviality. (shrink)
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  8. Richard Pettigrew (2009). On Interpretations of Bounded Arithmetic and Bounded Set Theory. Notre Dame Journal of Formal Logic 50 (2):141-152.score: 240.0
    In 'On interpretations of arithmetic and set theory', Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.

    THEOREM 1 The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.

    In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot (...)
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  9. Michael D. Potter (2004). Set Theory and its Philosophy: A Critical Introduction. Oxford University Press.score: 240.0
    Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes (...)
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  10. Rodrigo A. Freire (2012). On Existence in Set Theory. Notre Dame Journal of Formal Logic 53 (4):525-547.score: 240.0
    The aim of the present paper is to provide a robust classification of valid sentences in set theory by means of existence and related notions and, in this way, to capture similarities and dissimilarities among the axioms of set theory. In order to achieve this, precise definitions for the notions of productive and nonproductive assertions, constructive and nonconstructive productive assertions, and conditional and unconditional productive assertions, among others, will be presented. These definitions constitute the result of a semantical (...)
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  11. Ross T. Brady (2014). The Simple Consistency of Naive Set Theory Using Metavaluations. Journal of Philosophical Logic 43 (2-3):261-281.score: 240.0
    The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of (...)
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  12. Petr Andreev & Karel Hrbacek (2004). Standard Sets in Nonstandard Set Theory. Journal of Symbolic Logic 69 (1):165-182.score: 240.0
    We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.
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  13. William M. Farmer & Joshua D. Guttman (2000). A Set Theory with Support for Partial Functions. Studia Logica 66 (1):59-78.score: 240.0
    Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for (...)
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  14. Johan van Benthem, Giovanna D'Agostino, Angelo Montanari & Alberto Policriti (1998). Modal Deduction in Second-Order Logic and Set Theory - II. Studia Logica 60 (3):387-420.score: 240.0
    In this paper, we generalize the set-theoretic translation method for poly-modal logic introduced in [11] to extended modal logics. Instead of devising an ad-hoc translation for each logic, we develop a general framework within which a number of extended modal logics can be dealt with. We first extend the basic set-theoretic translation method to weak monadic second-order logic through a suitable change in the underlying set theory that connects up in interesting ways with constructibility; then, we show how to (...)
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  15. Étienne Matheron & Miroslav Zelený (2007). Descriptive Set Theory of Families of Small Sets. Bulletin of Symbolic Logic 13 (4):482-537.score: 240.0
    This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.
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  16. Jonas R. Becker Arenhart (2012). Finite Cardinals in Quasi-Set Theory. Studia Logica 100 (3):437-452.score: 240.0
    Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a theory is to deal properly with collections of items like particles in non-relativistic quantum mechanics when these are understood as being non-individuals in the sense that they may be indistinguishable although identity does not apply to (...)
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  17. Peter Verdée (2013). Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics. Foundations of Science 18 (4):655-680.score: 240.0
    In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also (...)
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  18. Jonas Rafael Becker Arenhart (2011). A Discussion on Finite Quasi-Cardinals in Quasi-Set Theory. Foundations of Physics 41 (8):1338-1354.score: 240.0
    Quasi-set theory Q is an alternative set-theory designed to deal mathematically with collections of indistinguishable objects. The intended interpretation for those objects is the indistinguishable particles of non-relativistic quantum mechanics, under one specific interpretation of that theory. The notion of cardinal of a collection in Q is treated by the concept of quasi-cardinal, which in the usual formulations of the theory is introduced as a primitive symbol, since the usual means of cardinal definition fail for collections (...)
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  19. Joel David Hamkins, David Linetsky & Jonas Reitz (2013). Pointwise Definable Models of Set Theory. Journal of Symbolic Logic 78 (1):139-156.score: 240.0
    A pointwise definable model is one in which every object is \loos definable without parameters. In a model of set theory, this property strengthens $V=\HOD$, but is not first-order expressible. Nevertheless, if \ZFC\ is consistent, then there are continuum many pointwise definable models of \ZFC. If there is a transitive model of \ZFC, then there are continuum many pointwise definable transitive models of \ZFC. What is more, every countable model of \ZFC\ has a class forcing extension that is pointwise (...)
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  20. Richard Kaye & Tin Lok Wong (2007). On Interpretations of Arithmetic and Set Theory. Notre Dame Journal of Formal Logic 48 (4):497-510.score: 240.0
    This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. (...)
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  21. George J. Tourlakis (2003). Lectures in Logic and Set Theory. Cambridge University Press.score: 240.0
    This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume II, on formal (ZFC) set theory, incorporates a self-contained 'chapter 0' on proof (...)
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  22. Frode Bjørdal (2011). The Inadequacy of a Proposed Paraconsistent Set Theory. Review of Symbolic Logic 4 (1):106-108.score: 240.0
    We show that a paraconsistent set theory proposed in Weber (2010) is strong enough to provide a quite classical nonprimitive notion of identity, so that the relation is an equivalence relation and also obeys full substitutivity: a = b -> F(b)). With this as background it is shown that the proposed theory also proves the negation of x=x. While not by itself showing that the proposed system is trivial in the sense of proving all statements, it is argued (...)
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  23. Eugenio Omodeo & Alberto Policriti (2010). The Bernays-Schönfinkel-Ramsey Class for Set Theory: Semidecidability. Journal of Symbolic Logic 75 (2):459-480.score: 240.0
    As is well-known, the Bernays-Schönfinkel-Ramsey class of all prenex ∃*∀* -sentences which are valid in classical first-order logic is decidable. This paper paves the way to an analogous result which the authors deem to hold when the only available predicate symbols are ∈ and =, no constants or function symbols are present, and one moves inside a (rather generic) Set Theory whose axioms yield the well-foundedness of membership and the existence of infinite sets. Here semi-decidability of the satisfiability problem (...)
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  24. Roland Hinnion & Thierry Libert (2008). Topological Models for Extensional Partial Set Theory. Notre Dame Journal of Formal Logic 49 (1):39-53.score: 240.0
    We state the consistency problem of extensional partial set theory and prove two complementary results toward a definitive solution. The proof of one of our results makes use of an extension of the topological construction that was originally applied in the paraconsistent case.
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  25. A. R. D. Mathias (2001). Slim Models of Zermelo Set Theory. Journal of Symbolic Logic 66 (2):487-496.score: 240.0
    Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence $\langle A_{\lambda} \mid \lambda \text{a limit ordinal} \rangle$ where for each $\lambda, A_{\lambda} \subseteq ^{\lambda}2$ , there is a supertransitive inner model of Zermelo containing all ordinals in which for every λ (...)
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  26. Maribel Anacona, Luis Carlos Arboleda & F. Javier Pérez-Fernández (2014). On Bourbaki's Axiomatic System for Set Theory. Synthese 191 (17):4069-4098.score: 240.0
    In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to Bourbaki’s (...)
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  27. Ray-Ming Chen & Michael Rathjen (2012). Lifschitz Realizability for Intuitionistic Zermelo–Fraenkel Set Theory. Archive for Mathematical Logic 51 (7-8):789-818.score: 240.0
    A variant of realizability for Heyting arithmetic which validates Church’s thesis with uniqueness condition, but not the general form of Church’s thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101–106, 1979). A Lifschitz counterpart to Kleene’s realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805–821, 1990). In that paper he also extended Lifschitz’ realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo–Fraenkel set theory, IZF. The (...)
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  28. Vladimir Kanovei & Michael Reeken (2000). A Nonstandard Set Theory in the Displaystylein-Language. Archive for Mathematical Logic 39 (6):403-416.score: 240.0
    . We demonstrate that a comprehensive nonstandard set theory can be developed in the standard $\displaystyle{\in}$ -language. As an illustration, a nonstandard ${\sf Law of Large Numbers}$ is obtained.
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  29. Penelope Maddy (2011). Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford University Press.score: 240.0
    Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account (...)
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  30. 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: 240.0
    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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  31. Benjamin D. Miller (2012). The Graph-Theoretic Approach to Descriptive Set Theory. Bulletin of Symbolic Logic 18 (4):554-575.score: 216.0
    We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.
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  32. Alberto Policriti & Eugenio Omodeo (2012). The Bernays—Schönfinkel—Ramsey Class for Set Theory: Decidability. Journal of Symbolic Logic 77 (3):896-918.score: 216.0
    As proved recently, the satisfaction problem for all prenex formulae in the set-theoretic Bernays-Shönfinkel-Ramsey class is semi-decidable over von Neumann's cumulative hierarchy. Here that semi-decidability result is strengthened into a decidability result for the same collection of formulae.
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  33. Øystein Linnebo (2012). Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory. [REVIEW] Philosophy 87 (01):133-137.score: 210.0
  34. Yehoshua Bar-Hillel (ed.) (1970). Mathematical Logic and Foundations of Set Theory. Amsterdam,North-Holland Pub. Co..score: 210.0
    LN , so f lies in the elementary submodel M'. Clearly co 9 M' . It follows that 6 = {f(n): n em} is included in M'. Hence the ordinals of M' form an initial ...
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  35. Petr Hájek (2005). On Arithmetic in the Cantor-Łukasiewicz Fuzzy Set Theory. Archive for Mathematical Logic 44 (6):763-782.score: 210.0
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  36. Giacomo Lenzi (2011). On a Positive Set Theory with Inequality. Mathematical Logic Quarterly 57 (5):474-480.score: 210.0
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  37. Laurence Kirby (2010). Substandard Models of Finite Set Theory. Mathematical Logic Quarterly 56 (6):631-642.score: 210.0
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  38. M. Rathjen (2001). Kripke-Platek Set Theory and the Anti-Foundation Axiom. Mathematical Logic Quarterly 47 (4):435-440.score: 210.0
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  39. W. Degen & J. Johannsen (2000). Cumulative Higher-Order Logic as a Foundation for Set Theory. Mathematical Logic Quarterly 46 (2):147-170.score: 210.0
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  40. Ulrich Felgner (1971). Models of Zf-Set Theory. New York,Springer-Verlag.score: 210.0
     
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  41. Thomas J. Jech (1971). Lectures in Set Theory. New York,Springer-Verlag.score: 210.0
     
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  42. Stefano Baratella & Ruggero Ferro (1995). Non Standard Regular Finite Set Theory. Mathematical Logic Quarterly 41 (2):161-172.score: 210.0
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  43. Maarten Wicher Visser Bunder (1969). Set Theory Based on Combinatory Logic. Groningen, V. R. B. --Offsetdrukkerij (Kleine Der a 3-4).score: 210.0
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  44. Paul J. Cohen (1966). Set Theory and the Continuum Hypothesis. New York, W. A. Benjamin.score: 210.0
  45. Olivier Esser (2003). On the Axiom of Extensionality in the Positive Set Theory. Mathematical Logic Quarterly 49 (1):97-100.score: 210.0
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  46. Vladimir Kanovei & Michael Reeken (1999). Special Model Axiom in Nonstandard Set Theory. Mathematical Logic Quarterly 45 (3):371-384.score: 210.0
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  47. Thierry Libert & Olivier Esser (2005). On Topological Set Theory. Mathematical Logic Quarterly 51 (3):263-273.score: 210.0
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  48. Maria Emilia Maietti & Silvio Valentini (1999). Can You Add Power‐Sets to Martin‐Lof's Intuitionistic Set Theory? Mathematical Logic Quarterly 45 (4):521-532.score: 210.0
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  49. Michael Rathjen (2006). A Note on Bar Induction in Constructive Set Theory. Mathematical Logic Quarterly 52 (3):253-258.score: 210.0
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  50. Michael Rathjen (2008). The Natural Numbers in Constructive Set Theory. Mathematical Logic Quarterly 54 (1):83-97.score: 210.0
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