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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
Algebra of Logic, Quantification Theory, and the Square of Opposition ∗
"... 1967 was one of the most important years for the historiography of modern logic. In that year Jean van Heijenoort published his From Frege to Gödel, according to its subtitle “A Source Book in Mathematical Logic, 1879–1931”. The seminal character of this collection is due to the fact that for the fi ..."
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1967 was one of the most important years for the historiography of modern logic. In that year Jean van Heijenoort published his From Frege to Gödel, according to its subtitle “A Source Book in Mathematical Logic, 1879–1931”. The seminal character of this collection is due to the fact that for the first
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