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

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Harvard University Press, 1967 - Mathematics - 664 pages
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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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Contents

DEFINITION OF THE SYMBOLS
10
Inference The Aristotelian modes of inference
17
Peano 1889 The principles of arithmetic presented by a new method
83
Dedekind 1890a Letter to Keferstein
98
BuraliForti 1897 and 1897a A question on transfinite numbers
104
Cantor 1899 Letter to Dedekind
113
Russell 1902 Letter to Frege
124
Zermelo 1904 Proof that every set can be wellordered
139
Skolem 1922 Some remarks on axiomatized set theory
290
Skolem 1923 The foundations of elementary arithmetic established
302
Brouwer 1923b 1954 and 1954a On the significance of the principle
334
Schonfinkel 1924 On the building blocks of mathematical logic
355
von Neumann 1925 An axiomatization of set theory
393
Kolmogorov 1925 On the principle of excluded middle
414
Finsler 1926 Formal proofs and undecidability
438
Hilbert 1927 The foundations of mathematics
464

Konig 1905a On the foundations of set theory and the continuum
145
Zermelo 1908 A new proof of the possibility of a wellordering
183
Zermelo 1908a Investigations in the foundations of set theory I
199
Descriptions
216
Wiener 1914 A simplification of the logic of relations
224
Skolem 1920 Logicocombinatorial investigations in the satisfiability
252
Post 1921 Introduction to a general theory of elementary
264
Fraenkel 1922b The notion definite and the independence of
284
Weyl 1927 Comments on Hilberts second lecture on the foundations
480
Ackermann 1928 On Hilberts construction of the real numbers
493
Skolem 1928 On mathematical logic
508
The properties
525
Godel 1930a The completeness of the axioms of the functional
582
Herbrand 193Ib On the consistency of arithmetic
618
Index
657
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