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Predicative Fragments of Frege Arithmetic
, 2003
"... Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply al ..."
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Frege Arithmetic (FA) is the secondorder theory whose sole nonlogical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be onetoone correlated. According to Frege’s Theorem, FA and some natural definitions imply all of secondorder Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying secondorder logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. 1
Natural Logicism via the Logic of Orderly Pairing by
, 2008
"... Schumm, Timothy Smiley and Matthias Wille. Comments by two anonymous referees have also led to significant improvements. The aim here is to describe how to complete the constructive logicist program, in the author’s book AntiRealism and Logic, of deriving all the PeanoDedekind postulates for arith ..."
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Schumm, Timothy Smiley and Matthias Wille. Comments by two anonymous referees have also led to significant improvements. The aim here is to describe how to complete the constructive logicist program, in the author’s book AntiRealism and Logic, of deriving all the PeanoDedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neoFregean done so. These outstanding axioms need to be derived in a way fully in keeping with the spirit and the letter of Frege’s logicism and his doctrine of definition. To that end this study develops a logic, in the GentzenPrawitz style of natural deduction, for the operation of orderly pairing. The logic is an extension of free firstorder logic with identity. Orderly pairing is treated as a primitive. No notion of set is presupposed, nor any settheoretic notion of membership. The formation of ordered pairs, and the two projection operations yielding their left and right coordinates, form a coeval family of logical notions. The challenge is to furnish them with introduction and elimination rules that capture their exact meanings, and no more. Orderly pairing as a logical primitive is then used in order to introduce addition and multiplication in a conceptually satisfying way within a constructive logicist theory of the natural numbers. Because of its reliance, throughout, on senseconstituting rules of natural deduction, the completed account can be described as ‘natural logicism’. 2 1 Introduction: historical
BERNAYS AND SET THEORY
"... Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Göd ..."
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Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the twovolume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of firstorder logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent reevaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higherorder reflection principles, and produced a stream of
CONCATENATION AS A BASIS FOR Q AND THE INTUITIONISTIC VARIANT OF NELSON’S CLASSIC RESULT
, 2008
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Logic and Metaphysics ∗
"... In this article, we canvass a few of the interesting topics that philosophers can pursue as part of the simultaneous study of logic and metaphysics. To keep the discussion to a manageable length, we limit our survey to deductive, as opposed to inductive, logic. Though most of this article ..."
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In this article, we canvass a few of the interesting topics that philosophers can pursue as part of the simultaneous study of logic and metaphysics. To keep the discussion to a manageable length, we limit our survey to deductive, as opposed to inductive, logic. Though most of this article
THE NATURE OF CONTEMPORARY CORE MATHEMATICS
, 2010
"... Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments ..."
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Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. A particular concern is the significance for mathematics education: elementary education remains modeled on the mathematics of the nineteenth century and before, and use of modern methodologies might give advantages similar to those seen in mathematics. This draft is about 90 % complete, and comments are welcome. 1.
From Hilbert’s Program to a Logic Tool Box
"... www.cs.technion.ac.il/∼janos Abstract. In this paper I discuss what, according to my long experience, every computer scientists should know from logic. We concentrate on issues of modeling, interpretability and levels of abstraction. We discuss what the minimal toolbox of logic tools should look lik ..."
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www.cs.technion.ac.il/∼janos Abstract. In this paper I discuss what, according to my long experience, every computer scientists should know from logic. We concentrate on issues of modeling, interpretability and levels of abstraction. We discuss what the minimal toolbox of logic tools should look like for a computer scientist who is involved in designing and analyzing reliable systems. We shall conclude that many classical topics dear to logicians are less important than usually presented, and that less known ideas from logic may be more useful for the working computer scientist. For Witek Marek, first mentor, then colleague and true friend, on the occasion of his 65th birthday.
On the notion of object A logical genealogy∗
"... Abstract. We argue that logic is not a uniform terrain where all truths lie on a par. We analyze the apparatus of firstorder classical logic with identity into two main ingredients: a deeper and wider component and, on top of it, a narrower component which consists of principles that articulate our ..."
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Abstract. We argue that logic is not a uniform terrain where all truths lie on a par. We analyze the apparatus of firstorder classical logic with identity into two main ingredients: a deeper and wider component and, on top of it, a narrower component which consists of principles that articulate our modern notion of object.