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  1.  23
    Interpreting the Infinitesimal Mathematics of Leibniz and Euler.Jacques Bair, Piotr Błaszczyk, Robert Ely, Valérie Henry, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, Patrick Reeder, David M. Schaps, David Sherry & Steven Shnider - 2017 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 48 (2):195-238.
    We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...)
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  2.  7
    Leibniz Versus Ishiguro: Closing a Quarter Century of Syncategoremania.Tiziana Bascelli, Piotr Błaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, David M. Schaps & David Sherry - 2016 - Hopos: The Journal of the International Society for the History of Philosophy of Science 6 (1):117-147.
    Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...)
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  3. Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics.Vladimir Kanovei, Mikhail G. Katz & Thomas Mormann - 2013 - Foundations of Science 18 (2):259-296.
    We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...)
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  4.  5
    Toward a History of Mathematics Focused on Procedures.Piotr Błaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze & David Sherry - 2017 - Foundations of Science 22 (4):763-783.
    Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...)
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  5.  3
    A Definable E 0 Class Containing No Definable Elements.Vladimir Kanovei & Vassily Lyubetsky - 2015 - Archive for Mathematical Logic 54 (5-6):711-723.
  6.  11
    Gregory’s Sixth Operation.Tiziana Bascelli, Piotr Błaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Tahl Nowik, David M. Schaps & David Sherry - 2018 - Foundations of Science 23 (1):133-144.
    In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...)
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  7.  14
    A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos.Piotr Błaszczyk, Alexandre Borovik, Vladimir Kanovei, Mikhail G. Katz, Taras Kudryk, Semen S. Kutateladze & David Sherry - 2016 - Logica Universalis 10 (4):393-405.
    We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...)
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  8.  4
    A Groszek-Laver Pair of Undistinguishable E0-Classes.Mohammad Golshani, Vladimir Kanovei & Vassily Lyubetsky - 2017 - Mathematical Logic Quarterly 63 (1-2):19-31.
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  9.  8
    Controversies in the Foundations of Analysis: Comments on Schubring’s Conflicts.Piotr Błaszczyk, Vladimir Kanovei, Mikhail G. Katz & David Sherry - 2017 - Foundations of Science 22 (1):125-140.
    Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...)
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  10.  42
    A Definable Nonstandard Model of the Reals.Vladimir Kanovei & Saharon Shelah - 2004 - Journal of Symbolic Logic 69 (1):159-164.
    We prove, in ZFC,the existence of a definable, countably saturated elementary extension of the reals.
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  11.  15
    Proofs and Retributions, Or: Why Sarah Can’T Take Limits.Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz & Mary Schaps - 2015 - Foundations of Science 20 (1):1-25.
    The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...)
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  12.  95
    What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel.Vladimir Kanovei, K. Katz, M. Katz & Thomas Mormann - 2018 - Journal of Humanistic Mathematics 8 (1):108 - 119.
    Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.
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  13.  7
    DefinableE0classes at Arbitrary Projective Levels.Vladimir Kanovei & Vassily Lyubetsky - 2018 - Annals of Pure and Applied Logic 169 (9):851-871.
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  14.  6
    Minimal Axiomatic Frameworks for Definable Hyperreals with Transfer.Frederik S. Herzberg, Vladimir Kanovei, Mikhail Katz & Vassily Lyubetsky - 2018 - Journal of Symbolic Logic 83 (1):385-391.
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  15.  9
    Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms.Tiziana Bascelli, Piotr Błaszczyk, Alexandre Borovik, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, David M. Schaps & David Sherry - 2018 - Foundations of Science 23 (2):267-296.
    Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.
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  16.  4
    Counterexamples to Countable-sectionΠ21uniformization andΠ31separation.Vladimir Kanovei & Vassily Lyubetsky - 2016 - Annals of Pure and Applied Logic 167 (3):262-283.
  17.  5
    Internal Approach to External Sets and Universes.Vladimir Kanovei & Michael Reeken - 1995 - Studia Logica 55 (2):229-257.
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  18. Is Leibnizian Calculus Embeddable in First Order Logic?Piotr Błaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Taras Kudryk, Thomas Mormann & David Sherry - 2017 - Foundations of Science 22 (4):73 - 88.
    To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...)
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  19.  4
    Internal Approach to External Sets and Universes.Vladimir Kanovei & Michael Reeken - 1995 - Studia Logica 55 (3):347-376.
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  20.  4
    On Coding Uncountable Sets by Reals.Joan Bagaria & Vladimir Kanovei - 2010 - Mathematical Logic Quarterly 56 (4):409-424.
    If A ⊆ ω1, then there exists a cardinal preserving generic extension [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ][x ] of [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ] by a real x such that1) A ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ] and A is Δ1HC in [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ];2) x is minimal over [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ], that is, if a set Y belongs to [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ], then either x ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A, Y ] or Y (...)
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  21.  66
    Internal Approach to External Sets and Universes.Vladimir Kanovei & Michael Reeken - 1995 - Studia Logica 55 (2):347 - 376.
    In this article we show how the universe of BST, bounded set theory can be enlarged by definable subclasses of sets so that Separation and Replacement are true in the enlargement for all formulas, including those in which the standardness predicate may occur. Thus BST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop topics in nonstandard analysis inaccessible in the framework of a purely internal approach, such as Loeb measures.
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  22.  31
    An Ulm-Type Classification Theorem for Equivalence Relations in Solovay Model.Vladimir Kanovei - 1997 - Journal of Symbolic Logic 62 (4):1333-1351.
    We prove that in the Solovay model, every OD equivalence relation, E, over the reals, either admits an OD reduction to the equality relation on the set of all countable (of length $ ) binary sequences, or continuously embeds E 0 , the Vitali equivalence. If E is a Σ 1 1 (resp. Σ 1 2 ) relation then the reduction above can be chosen in the class of all ▵ 1 (resp. ▵ 2 ) functions. The proofs are based (...)
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  23.  29
    On Non-Wellfounded Iterations of the Perfect Set Forcing.Vladimir Kanovei - 1999 - Journal of Symbolic Logic 64 (2):551-574.
    We prove that if I is a partially ordered set in a countable transitive model M of ZFC then M can be extended by a generic sequence of reals a i , i ∈ I, such that ℵ M 1 is preserved and every a i is Sacks generic over $\mathfrak{M}[\langle \mathbf{a}_j: j . The structure of the degrees of M-constructibility of reals in the extension is investigated. As applications of the methods involved, we define a cardinal invariant to distinguish (...)
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  24.  23
    A Nonstandard Set Theory in the $\Displaystyle\in$ -Language.Vladimir Kanovei & Michael Reeken - 2000 - Archive for Mathematical Logic 39 (6):403-416.
    . 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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  25.  2
    Definable Minimal Collapse Functions at Arbitrary Projective Levels.Vladimir Kanovei & Vassily Lyubetsky - 2019 - Journal of Symbolic Logic 84 (1):266-289.
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  26.  6
    A Model of Second-Order Arithmetic Satisfying AC but Not DC.Sy-David Friedman, Victoria Gitman & Vladimir Kanovei - 2019 - Journal of Mathematical Logic 19 (1):1850013.
    We show that there is a β-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a Π21-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of ZFC−. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. G. Kanovei, On descriptive forms of (...)
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  27.  29
    Elementary Extensions of External Classes in a Nonstandard Universe.Vladimir Kanovei & Michael Reeken - 1998 - Studia Logica 60 (2):253-273.
    In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "external" class (...)
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  28.  15
    Ulm Classification of Analytic Equivalence Relations in Generic Universes.Vladimir Kanovei - 1998 - Mathematical Logic Quarterly 44 (3):287-303.
  29.  12
    A Nonstandard Set Theory in the [Mathematical Formula]-Language.Vladimir Kanovei & Michael Reeken - 2000 - Archive for Mathematical Logic 6.
  30.  10
    Isomorphism Property in Nonstandard Extensions of theZFC Universe.Vladimir Kanovei & Michael Reeken - 1997 - Annals of Pure and Applied Logic 88 (1):1-25.
    We study models of HST . This theory admits an adequate formulation of the isomorphism propertyIP, which postulates that any two elementarily equivalent internally presented structures of a well-orderable language are isomorphic. We prove that IP is independent of HST and consistent with HST.
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  31.  7
    Countable OD Sets of Reals Belong to the Ground Model.Vladimir Kanovei & Vassily Lyubetsky - 2018 - Archive for Mathematical Logic 57 (3-4):285-298.
    It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \ elements.
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  32.  5
    Linearization of Definable Order Relations.Vladimir Kanovei - 2000 - Annals of Pure and Applied Logic 102 (1-2):69-100.
    We prove that if ≼ is an analytic partial order then either ≼ can be extended to a Δ 2 1 linear order similar to an antichain in 2 ω 1 , ordered lexicographically, or a certain Borel partial order ⩽ 0 embeds in ≼. Similar linearization results are presented, for κ -bi-Souslin partial orders and real-ordinal definable orders in the Solovay model. A corollary for analytic equivalence relations says that any Σ 1 1 equivalence relation E , such that (...)
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  33.  24
    Extending Standard Models of ZFC to Models of Nonstandard Set Theories.Vladimir Kanovei & Michael Reeken - 2000 - Studia Logica 64 (1):37-59.
    We study those models of ZFCwhich are embeddable, as the class of all standard sets, in a model of internal set theory >ISTor models of some other nonstandard set theories.
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  34.  22
    A Version of the Jensen–Johnsbråten Coding at Arbitrary Level N≥ 3.Vladimir Kanovei - 2001 - Archive for Mathematical Logic 40 (8):615-628.
    We generalize, on higher projective levels, a construction of “incompatible” generic Δ1 3 real singletons given by Jensen and Johnsbråten.
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  35.  13
    On Baire Measurable Homomorphisms of Quotients of the Additive Group of the Reals.Vladimir Kanovei & Michael Reeken - 2000 - Mathematical Logic Quarterly 46 (3):377-384.
    The quotient ℝ/G of the additive group of the reals modulo a countable subgroup G does not admit nontrivial Baire measurable automorphisms.
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  36.  11
    Special Model Axiom in Nonstandard Set Theory.Vladimir Kanovei & Michael Reeken - 1999 - Mathematical Logic Quarterly 45 (3):371-384.
    We demonstrate that the special model axiom SMA of Ross admits a natural formalization in Kawai's nonstandard set theory KST but is independent of KST. As an application of our methods to classical model theory, we present a short proof of the consistency of the existence of a k+ like k-saturated model of PA for a given cardinal k.
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  37.  29
    On External Scott Algebras in Nonstandard Models of Peano Arithmetic.Vladimir Kanovei - 1996 - Journal of Symbolic Logic 61 (2):586-607.
    We prove that a necessary and sufficient condition for a countable set L of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of ω by a formula of the PA language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: L is closed under arithmetical definability and contains 0 (ω) , the set of all (Gödel numbers of) true arithmetical sentences. Some results (...)
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  38.  13
    On Effective Σ‐Boundedness and Σ‐Compactness.Vladimir Kanovei & Vassily Lyubetsky - 2013 - Mathematical Logic Quarterly 59 (3):147-166.
  39.  11
    On a Spector Ultrapower for the Solovay Model.Vladimir Kanovei & Michiel van Lambalgen - 1997 - Mathematical Logic Quarterly 43 (3):389-395.
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  40.  9
    Loeb Measure From the Point of View of a Coin Flipping Game.Vladimir Kanovei & Michael Reeken - 1996 - Mathematical Logic Quarterly 42 (1):19-26.
    A hyperfinitely long coin flipping game between the Gambler and the Casino, associated with a given set A, is considered. It turns out that the Gambler has a winning strategy if and only if A has Loeb measure 0. The Casino has a winning strategy if and only if A contains an internal subset of positive Loeb measure.
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  41. On Coding Uncountable Sets by Reals.Joan Bagaria I. Pigrau & Vladimir Kanovei - 2010 - Mathematical Logic Quarterly 56 (4):409-424.
     
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  42.  56
    Internal Approach to External Sets and Universes: Part 3: Partially Saturated Universes.Vladimir Kanovei & Michael Reeken - 1996 - Studia Logica 56 (3):293-322.
    In this article ‡ we show how the universe of HST, Hrbaček set theory admits a system of subuniverses which keep the Replacement, model Power set and Choice, and also keep as much of Saturation as it is necessary. This gives sufficient tools to develop the most complicated topics in nonstandard analysis, such as Loeb measures.
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