From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931

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Harvard University Press, 15 ene 2002 - 680 páginas
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The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded by Leibniz, may be said to have been initiated by Boole, De Morgan, and Jevons, but it was the publication in 1879 of Gottlob Frege's Begriffsschrift that opened a great epoch in the history of logic by presenting, in full-fledged form, the propositional calculus and quantification theory.

Frege's book, translated in its entirety, begins the present volume. The emergence of two new fields, set theory and foundations of mathematics, on the borders of logic, mathematics, and philosophy, is depicted by the texts that follow. Peano and Dedekind illustrate the trend that led to Principia Mathematica. Burali-Forti, Cantor, Russell, Richard, and König mark the appearance of the modern paradoxes. Hilbert, Russell, and Zermelo show various ways of overcoming these paradoxes and initiate, respectively, proof theory, the theory of types, and axiomatic set theory. Skolem generalizes Löwenheim's theorem, and heand Fraenkel amend Zermelo's axiomatization of set theory, while von Neumann offers a somewhat different system. The controversy between Hubert and Brouwer during the twenties is presented in papers of theirs and in others by Weyl, Bernays, Ackermann, and Kolmogorov. The volume concludes with papers by Herbrand and by Gödel, including the latter's famous incompleteness paper.

Of the forty-five contributions here collected all but five are presented in extenso. Those not originally written in English have been translated with exemplary care and exactness; the translators are themselves mathematical logicians as well as skilled interpreters of sometimes obscure texts. Each paper is introduced by a note that sets it in perspective, explains its importance, and points out difficulties in interpretation. Editorial comments and footnotes are interpolated where needed, and an extensive bibliography is included.

 

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Índice

Frege 1879 Begriffsschrift a formula language modeled upon that of arithmetic for pure thought
Peano 1889 The principles of arithmetic presented by a new method
Hilbert 1927 The foundations of mathematics
Dedekind 1890a Letter to Keferstein BuraliForti 1897 and 1897a A question on transfinite numbers and On wellordered classes Cantor 1899 Letter t...
Padoa 1900 Logical introduction to any deductive theory
Russell 1902 Letter to Frege
Frege 1902 Letter to Russell
Hilbert 1904 On the foundations of logic and arithmetic
Fraenkel 1922b The notion definite and the independence of the axiom of choice Skolem 1922 Some remarks on axiomatized set theory
Skolem 1923 The foundations of elementary arithmetic established by means of the recursive mode of thought without the use of apparent variables ...
von Neumann 1923 On the introduction of transfinite numbers Schönfinkel 1924 On the building blocks of mathematical logic
Hilbert 1925 On the infinite
von Neumann 1925 An axiomatization of set theory Kolmogorov 1925 On the principle of excluded middle
Finsler 1926 Formal proofs and undecidability
Brouwer 1927 On the domains of definition of functions
Weyl 1927 Comments on Hilberts second lecture on the foundations of mathematics

Zermelo 1904 Proof that every set can be wellordered
Richard 1905 The principles of mathematics and the problem of sets
König 1905a On the foundations of set theory and the continuum problem Russell 1908a Mathematical logic as based on the theory of types Zermel...
Zermelo 1908a Investigations in the foundations of set theory I
Löwenheim 1915 On possibilities in the calculus of relatives
A simplified proof of a theorem by L Löwenheim and generalizations of the theorem
Post 1921 Introduction to a general theory of elementary propositions
Skolem 1928 On mathematical logic
The properties of true propositions
Gödel 1930a The completeness of the axioms of the functional calculus of logic
Herbrand 1931b On the consistency of arithmetic
Index
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