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The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
No Syllogisms for the Numerical Syllogistic
"... Abstract. The numerical syllogistic is the extension of the traditional syllogistic with numerical quantifiers of the forms at least C and at most C. It is known that, for the traditional syllogistic, a finite collection of rules, similar in spirit to the classical syllogisms, constitutes a sound an ..."
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Abstract. The numerical syllogistic is the extension of the traditional syllogistic with numerical quantifiers of the forms at least C and at most C. It is known that, for the traditional syllogistic, a finite collection of rules, similar in spirit to the classical syllogisms, constitutes a sound and complete proofsystem. The question arises as to whether such a proof system exists for the numerical syllogistic. This paper answers that question in the negative: no finite collection of syllogismlike rules, broadly conceived, is sound and complete for the numerical syllogistic. 1
Which arithmeticisation for which logicism? Russell on relations and quantities in The Principles of Mathematics. History and Philosophy of Logic
"... This article aims first at showing that Russell’s general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles ’ does not require a preliminary reduction of all mathematics ..."
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This article aims first at showing that Russell’s general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles ’ does not require a preliminary reduction of all mathematics
Ludwig Boltzmann over the Poincaré Recurrence
"... Ernst Zermelo is familiar to mathematicians as the creator of the controversial Axiom of Choice in 1904 and the theorem, based on the Axiom of Choice, that every set can be well ordered. Many will be aware that in 1908 he axiomatized set theory—in a form later modified by Abraham Fraenkel (1922) and ..."
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Ernst Zermelo is familiar to mathematicians as the creator of the controversial Axiom of Choice in 1904 and the theorem, based on the Axiom of Choice, that every set can be well ordered. Many will be aware that in 1908 he axiomatized set theory—in a form later modified by Abraham Fraenkel (1922) and then by Zermelo himself (1930). Some will know of Zermelo’s conflict with
Gödel’s Incompleteness Theorems
"... In 1931, when he was only 25 years of age, the great Austrian logician Kurt Gödel (1906– 1978) published an epochmaking paper [16] (for an English translation see [8, pp. 5–38]), in which he proved that an effectively definable consistent mathematical theory which is strong enough to prove Peano’s ..."
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In 1931, when he was only 25 years of age, the great Austrian logician Kurt Gödel (1906– 1978) published an epochmaking paper [16] (for an English translation see [8, pp. 5–38]), in which he proved that an effectively definable consistent mathematical theory which is strong enough to prove Peano’s postulates of elementary arithmetic cannot prove its own
SET THEORY FROM CANTOR TO COHEN
"... Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun int ..."
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Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The
The Mathematical Origins of 19th Century Algebra of Logic ∗
, 2003
"... Most 19th century scholars would have agreed to the opinion that philosophers are responsible for research on logic. On the other hand, the history ..."
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Most 19th century scholars would have agreed to the opinion that philosophers are responsible for research on logic. On the other hand, the history
c○University of Notre Dame BOOK REVIEW Geraldine Brady. From Peirce to Skolem: A Neglected Chapter in
"... The thesis of this book is that Löwenheim’s and Skolem’s work on what is now known as the downward Löwenheim–Skolem theorem developed directly from Schröder’s Algebra der Logik, which was itself an avowed elaboration of the ..."
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The thesis of this book is that Löwenheim’s and Skolem’s work on what is now known as the downward Löwenheim–Skolem theorem developed directly from Schröder’s Algebra der Logik, which was itself an avowed elaboration of the