•  827
    Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics
    with Mikhail G. Katz and Thomas Mormann
    Foundations of Science 18 (2): 259-296. 2013.
    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 conclus…Read more
  •  310
    What Makes a Theory of Infinitesimals Useful? A View by Klein and Fraenkel
    with K. Katz, M. Katz, and Thomas Mormann
    Journal of Humanistic Mathematics 8 (1). 2018.
    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.
  •  281
    Is Leibnizian calculus embeddable in first order logic?
    with Piotr Błaszczyk, Karin U. Katz, Mikhail G. Katz, Taras Kudryk, Thomas Mormann, and David Sherry
    Foundations of Science 22 (4). 2017.
    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…Read more
  •  140
    Internal approach to external sets and universes
    with Michael Reeken
    Studia Logica 55 (2). 1995.
    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
  •  91
    A nonstandard set theory in the $\displaystyle\in$ -language
    with Michael Reeken
    Archive for Mathematical Logic 39 (6): 403-416. 2000.
    . 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
  •  81
    Internal Approach to External Sets and Universes: Part 3: Partially Saturated Universes
    with Michael Reeken
    Studia Logica 56 (3): 293-322. 1996.
    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.
  •  80
    A version of the Jensen–Johnsbråten coding at arbitrary level n≥ 3
    Archive for Mathematical Logic 40 (8): 615-628. 2001.
    We generalize, on higher projective levels, a construction of “incompatible” generic Δ1 3 real singletons given by Jensen and Johnsbråten
  •  48
    A definable nonstandard model of the reals
    with Saharon Shelah
    Journal of Symbolic Logic 69 (1): 159-164. 2004.
    We prove, in ZFC,the existence of a definable, countably saturated elementary extension of the reals
  •  44
    Elementary extensions of external classes in a nonstandard universe
    with Michael Reeken
    Studia Logica 60 (2): 253-273. 1998.
    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 "e…Read more
  •  44
    On non-wellfounded iterations of the perfect set forcing
    Journal of Symbolic Logic 64 (2): 551-574. 1999.
    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 product and iterated Sacks extensions, and give a short proo…Read more
  •  40
    We prove that in IST, Nelson's internal set theory, the Uniqueness and Collection principles, hold for all (including external) formulas. A corollary of the Collection theorem shows that in IST there are no definable mappings of a set X onto a set Y of greater (not equal) cardinality unless both sets are finite and #(Y) ≤ n #(X) for some standard n. Proofs are based on a rather general technique which may be applied to other nonstandard structures. In particular we prove that in a nonstandard mo…Read more
  •  33
    An Ulm-type classification theorem for equivalence relations in Solovay model
    Journal of Symbolic Logic 62 (4): 1333-1351. 1997.
    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 on a topology generated by OD sets
  •  30
    On external Scott algebras in nonstandard models of peano arithmetic
    Journal of Symbolic Logic 61 (2): 586-607. 1996.
    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 related to definability o…Read more
  •  28
    Proofs and Retributions, Or: Why Sarah Can’t Take Limits
    with Karin U. Katz, Mikhail G. Katz, and Mary Schaps
    Foundations of Science 20 (1): 1-25. 2015.
    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 a…Read more
  •  28
    Extending standard models of ZFC to models of nonstandard set theories
    with Michael Reeken
    Studia Logica 64 (1): 37-59. 2000.
    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.
  •  26
    A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos
    with Piotr Błaszczyk, Alexandre Borovik, Mikhail G. Katz, Taras Kudryk, Semen S. Kutateladze, and David Sherry
    Logica Universalis 10 (4): 393-405. 2016.
    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 mathematic…Read more
  •  20
    Isomorphism property in nonstandard extensions of theZFC universe
    with Michael Reeken
    Annals of Pure and Applied Logic 88 (1): 1-25. 1997.
    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
  •  19
    Controversies in the Foundations of Analysis: Comments on Schubring’s Conflicts
    with Piotr Błaszczyk, Mikhail G. Katz, and David Sherry
    Foundations of Science 22 (1): 125-140. 2017.
    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 m…Read more
  •  19
    Linearization of definable order relations
    Annals of Pure and Applied Logic 102 (1-2): 69-100. 2000.
    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 E 0 does not embed in E , is fully determined by i…Read more
  •  18
    A definable E 0 class containing no definable elements
    with Vassily Lyubetsky
    Archive for Mathematical Logic 54 (5-6): 711-723. 2015.
    A generic extension L[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{L}[x]}$$\end{document} by a real x is defined, in which the E0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgree…Read more
  •  17
    Gregory’s Sixth Operation
    with Tiziana Bascelli, Piotr Błaszczyk, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Tahl Nowik, David M. Schaps, and David Sherry
    Foundations of Science 23 (1): 133-144. 2018.
    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 Gregory…Read more
  •  17
    Cauchy’s Infinitesimals, His Sum Theorem, and Foundational Paradigms
    with Tiziana Bascelli, Piotr Błaszczyk, Alexandre Borovik, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, Thomas McGaffey, David M. Schaps, and David Sherry
    Foundations of Science 23 (2): 267-296. 2018.
    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.
  •  15
    Ulm Classification of Analytic Equivalence Relations in Generic Universes
    Mathematical Logic Quarterly 44 (3): 287-303. 1998.
  •  15
    Leibniz versus Ishiguro: Closing a Quarter Century of Syncategoremania
    with Tiziana Bascelli, Piotr Błaszczyk, Karin U. Katz, Mikhail G. Katz, David M. Schaps, and David Sherry
    Hopos: The Journal of the International Society for the History of Philosophy of Science 6 (1): 117-147. 2016.
    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 a…Read more
  •  15
    On Baire Measurable Homomorphisms of Quotients of the Additive Group of the Reals
    with Michael Reeken
    Mathematical Logic Quarterly 46 (3): 377-384. 2000.
    The quotient ℝ/G of the additive group of the reals modulo a countable subgroup G does not admit nontrivial Baire measurable automorphisms
  •  14
    Special Model Axiom in Nonstandard Set Theory
    with Michael Reeken
    Mathematical Logic Quarterly 45 (3): 371-384. 1999.
    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
  •  13
    A nonstandard set theory in the [mathematical formula]-language
    with Michael Reeken
    Archive for Mathematical Logic 39 (6): 403-416. 2000.
  •  13
    Counterexamples to countable-section Π21 uniformization and Π31 separation
    with Vassily Lyubetsky
    Annals of Pure and Applied Logic 167 (3): 262-283. 2016.
  •  13
    On effective σ‐boundedness and σ‐compactness
    with Vassily Lyubetsky
    Mathematical Logic Quarterly 59 (3): 147-166. 2013.
  •  12
    DefinableE0classes at arbitrary projective levels
    with Vassily Lyubetsky
    Annals of Pure and Applied Logic 169 (9): 851-871. 2018.