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  1. John L. Bell, Basic Set Theory.
    to indicate that the object a is an element or member of the class A. We assume that every member of a class is an object. Lower-case letters a, b, c, x, y, z, … will always denote objects, and later, sets. Equality between classes is governed by the Axiom of Extensionality.
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  2. John L. Bell, Cohesiveness.
    ABSTRACT: It is characteristic of a continuum that it be “all of one piece”, in the sense of being inseparable into two (or more) disjoint nonempty parts. By taking “part” to mean open (or closed) subset of the space, one obtains the usual topological concept of connectedness . Thus a space S is defined to be connected if it cannot be partitioned into two disjoint nonempty open (or closed) subsets – or equivalently, given any partition of S into two open (...)
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  3. John L. Bell, Continuity and Infinitesimals.
    The usual meaning of the word continuous is “unbroken” or “uninterrupted”: thus a continuous entity —a continuum—has no “gaps.” We commonly suppose that space and time are continuous, and certain philosophers have maintained that all natural processes occur continuously: witness, for example, Leibniz's famous apothegm natura non facit saltus—“nature makes no jump.” In mathematics the word is used in the same general sense, but has had to be furnished with increasingly precise definitions. So, for instance, in the later 18th century (...)
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  4. John L. Bell, Constructive Context.
    One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor's theorem that the cardinality of any set is strictly less than that of its power set. The other method of proving Cantor's theorem — employed by Cantor himself in showing that the set of real numbers is uncountable — is that of diagonalization. Typically, diagonalization arguments are used to show that function spaces are "large" in a suitable sense. (...)
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  5. John L. Bell, Comparing the Smooth and Dedekind Reals in Smooth Infinitesimal Analysis.
    Axioms for the continuum, or smooth real line R. These include the usual axioms for a commutative ring with unit expressed in terms of two operations + and i , and two distinguished elements 0 ≠ 1. In addition we stipulate that R is a local ring, i.e., the following axiom: ∃y x i y = 1 ∨ ∃y (1 – x) i y = 1. Axioms for the strict order relation < on R. These are: 1. a < b (...)
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  6. John L. Bell, Logical Reflections On the Kochen-Specker Theorem.
    IN THEIR WELL-KNOWN PAPER, Kochen and Specker (1967) introduce the concept of partial Boolean algebra (pBa) and show that certain (finitely generated) partial Boolean algebras arising in quantum theory fail to possess morphisms to any Boolean algebra (we call such pBa's intractable in the sequel). In this note we begin by discussing partial..
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  7. John L. Bell, M L.
    A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that beefing up WST with moderate extensionality principles or quotient sets enables the derivation to go through.
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  8. John L. Bell, Notes On Formal Logic.
    On the contrary, I find nothing in logistic but shackles. It does not help us at all in the direction of conciseness, far from it; and if it requires 27 equations to establish that 1 is a number, how many will it require to demonstrate a real theorem?
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  9. John L. Bell, Two Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets.
    I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with (...)
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  10. John L. Bell, The Development of Categorical Logic.
    5.5. Every topos is linguistic: the equivalence theorem.
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  11. John L. Bell, The Incredible Shrinking Manifold.
    Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic geometry are (...)
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  12. John L. Bell, Types, Sets and Categories.
    This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard to the latter) with any pretensions to completeness within the space of a comparatively short article, (...)
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  13. John L. Bell, Algorithmicity and Consciousness.
    Why should one believe that conscious awareness is solely the result of organizational complexity? What is the connection between consciousness and combinatorics: transformation of quantity into quality? The claim that the former is reducible to the other seems unconvincing—as unlike as chalk and cheese! In his book1 Penrose is at least attempting to compare like with like: the enigma of consciousness with the progress of physics.
     
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  14. John L. Bell, Notes on Toposes and Local Set Theories.
    This book is written for those who are in sympathy with its spirit. This spirit is different from the one which informs the vast stream of European and American civilization in which all of us stand. That spirit expresses itself in an onwards movement, in building ever larger and more complicated structures; the other in striving in clarity and perspicuity in no matter what structure. The first tries to grasp the world by way of its periphery—in its variety; the second (...)
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  15. John L. Bell (2011). Set Theory: Boolean-Valued Models and Independence Proofs. OUP Oxford.
    This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory,. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice. Aimed at graduate students and researchers in mathematics, mathematical logic, philosophy, and computer science, the third edition has been extensively updated with expanded introductory material, new chapters, and (...)
     
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  16. John L. Bell, Hermann Weyl. Stanford Encyclopedia of Philosophy.
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  17. John L. Bell, Infinitary Logic. Stanford Encyclopedia of Philosophy.
    Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...)
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  18. John L. Bell, The Axiom of Choice. Stanford Encyclopedia of Philosophy.
    The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...)
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  19. John L. Bell (2008). Corrigendum to “Incompleteness in a General Setting”. Bulletin of Symbolic Logic 14 (1):122-122.
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  20. A. Kock & John L. Bell (2007). REVIEWS-Synthetic Differential Geometry. Bulletin of Symbolic Logic 13 (2).
     
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  21. John L. Bell (2006). Paul Rusnock. Bolzano's Philosophy and the Emergence of Modern Mathematics. Studien Zur Österreichischen Philosophie [Studies in Austrian Philosophy], Vol. 30. Amsterdam & Atlanta: Editions Rodopi, 2000. Isbn 90-420-1501-2. Pp. 218. [REVIEW] Philosophia Mathematica 14 (3):362-364.
  22. John L. Bell (2005). Divergent Conceptions of the Continuum in 19th and Early 20th Century Mathematics and Philosophy. Axiomathes 15 (1):63-84.
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  23. John L. Bell (2005). Oppositions and Paradoxes in Mathematics and Philosophy. Axiomathes 15 (2):165-180.
    In this paper a number of oppositions which have haunted mathematics and philosophy are described and analyzed. These include the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable.
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  24. John L. Bell (2004). Whole and Part in Mathematics. Axiomathes 14 (4):285-294.
    The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.
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  25. John L. Bell (2001). Observations on Category Theory. Axiomathes 12 (1-2):151-155.
    is a presentation of mathematics in terms of the fundamental concepts of transformation, and composition of transformations. While the importance of these concepts had long been recognized in algebra (for example, by Galois through the idea of a group of permutations) and in geometry (for example, by Klein in his Erlanger Programm), the truly universal role they play in mathematics did not really begin to be appreciated until the rise of abstract algebra in the 1930s. In abstract algebra the idea (...)
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  26. John L. Bell, David DeVidi & Graham Solomon (2001). Logical Options: An Introduction to Classical and Alternative Logics. Broadview Press.
    Logical Options introduces the extensions and alternatives to classical logic which are most discussed in the philosophical literature: many-sorted logic, second-order logic, modal logics, intuitionistic logic, three-valued logic, fuzzy logic, and free logic. Each logic is introduced with a brief description of some aspect of its philosophical significance, and wherever possible semantic and proof methods are employed to facilitate comparison of the various systems. The book is designed to be useful for philosophy students and professional philosophers who have learned some (...)
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  27. John L. Bell (2000). Continuity and the Logic of Perception. Transcendent Philosophy 1 (2):1-7.
    If we imagine a chess-board with alternate blue and red squares, then this is something in which the individual red and blue areas allow themselves to be distinguished from each other in juxtaposition, and something similar holds also if we imagine each of the squares divided into four smaller squares also alternating between these two colours. If, however, we were to continue with such divisions until we had exceeded the boundary of noticeability for the individual small squares which result, then (...)
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  28. John L. Bell (2000). Hermann Weyl on Intuition and the Continuum. Philosophia Mathematica 8 (3):259-273.
    Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.
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  29. John L. Bell (2000). Sets and Classes as Many. Journal of Philosophical Logic 29 (6):585-601.
    In this paper the view is developed that classes should not be understood as individuals, but, rather, as "classes as many" of individuals. To correlate classes with individuals "labelling" and "colabelling" functions are introduced and sets identified with a certain subdomain of the classes on which the labelling and colabelling functions are mutually inverse. A minimal axiomatization of the resulting system is formulated and some of its extensions are related to various systems of set theory, including nonwellfounded set theories.
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  30. John L. Bell (1999). Finite Sets and Frege Structures. Journal of Symbolic Logic 64 (4):1552-1556.
    Call a family F of subsets of a set E inductive if ∅ ∈ F and F is closed under unions with disjoint singletons, that is, if ∀X∈F ∀x∈E–X(X ∪ {x} ∈ F]. A Frege structure is a pair (E.
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  31. John L. Bell (1999). Frege's Theorem in a Constructive Setting. Journal of Symbolic Logic 64 (2):486-488.
    then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., (...)
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  32. John L. Bell (1996). Polymodal Lattices and Polymodal Logic. Mathematical Logic Quarterly 42 (1):219-233.
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  33. John L. Bell & William Demopoulos (1996). Elementary Propositions and Independence. Notre Dame Journal of Formal Logic 37 (1):112-124.
    This paper is concerned with Wittgenstein's early doctrine of the independence of elementary propositions. Using the notion of a free generator for a logical calculus–a concept we claim was anticipated by Wittgenstein–we show precisely why certain difficulties associated with his doctrine cannot be overcome. We then show that Russell's version of logical atomism–with independent particulars instead of elementary propositions–avoids the same difficulties.
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  34. John L. Bell & Silvia Gebellato (1996). Precovers, Modalities and Universal Closure Operators in a Topos. Mathematical Logic Quarterly 42 (1):289-299.
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  35. David L. Kemmerer, Kenneth Aizawa, Donald H. Berman, Stacey L. Edgar, James E. Tomberlin, J. Christopher Maloney, John L. Bell, Stuart C. Shapiro, Georges Rey, Morton L. Schagrin, Robert A. Wilson & Patrick J. Hayes (1995). Book Reviews. [REVIEW] Minds and Machines 5 (3):411-465.
  36. John L. Bell (1994). Fregean Extensions of First‐Order Theories. Mathematical Logic Quarterly 40 (1):27-30.
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  37. John L. Bell (1993). Hilbert's Ε‐Operator in Intuitionistic Type Theories. Mathematical Logic Quarterly 39 (1):323-337.
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