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基本説明
New in paperback. Hardcover is out of print. A complete translation of Gottlob Frege's Begriffsschrift begins the volume. The volume concludes with papers by Herbrand and by Gödel, including the latter's famous incompleteness paper.
Full Description
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 he and 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.
Contents
1. Frege (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought 2. Peano (1889). The principles of arithmetic, presented by a new method 3.Dedekind (1890a). Letter to Keferstein Burali-Forti (1897 and 1897a). A question on transfinite numbers and On well-ordered classes 4.Cantor (1899). Letter to Dedekind 5.Padoa (1900). Logical introduction to any deductive theory 6,Russell (1902). Letter to Frege 7.Frege (1902). Letter to Russell 8.Hilbert (1904). On the foundations of logic and arithmetic 9.Zermelo (1904). Proof that every set can be well-ordered 10.Richard (1905). The principles of mathematics and the problem of sets 11.Konig (1905a). On the foundations of set theory and the continuum problem 12.Russell (1908a). Mathematical logic as based on the theory of types 13.Zermelo (1908). A new proof of the possibility of a well-ordering 14.Zermelo (l908a). Investigations in the foundations of set theory I Whitehead and Russell (1910). Incomplete symbols: Descriptions 15.Wiener (1914). A simplification of the logic of relations 16.Lowenheim (1915). On possibilities in the calculus of relatives 17.Skolem (1920). Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions: A simplified proof of a theorem by L. Lowenheim and generalizations of the 18.theorem 19.Post (1921). Introduction to a general theory of elementary propositions 20.Fraenkel (1922b). The notion "definite" and the independence of the axiom of choice 21.Skolem (1922). Some remarks on axiomatized set theory 22.Skolem (1923). The foundations of elementary arithmetic established by means of the recursive mode of thought, without the use of apparent variables ranging over infinite domains 23.Brouwer (1923b, 1954, and 1954a). On the significance of the principle of excluded middle in mathematics, especially in function theory, Addenda and corrigenda, and Further addenda and corrigenda von Neumann (1923). On the introduction of transfinite numbers Schonfinkel (1924). On the building blocks of mathematical logic filbert (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 filbert (1927). The foundations of mathematics Weyl (1927). Comments on Hilbert's second lecture on the foundations of mathematics Bernays (1927). Appendix to Hilbert's lecture "The foundations of mathematics" Brouwer (1927a). Intuitionistic reflections on formalism Ackermann (1928). On filbert's construction of the real numbers Skolem (1928). On mathematical logic Herbrand (1930). Investigations in proof theory: The properties of true propositions Godel (l930a). The completeness of the axioms of the functional calculus of logic Godel (1930b, 1931, and l931a). Some metamathematical results on completeness and consistency, On formally undecidable propositions of Principia mathematica and related systems I, and On completeness and consistency Herbrand (1931b). On the consistency of arithmetic References Index
