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

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Overview

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.

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Editorial Reviews

Paolo Mancosu
The outstanding quality of the translations and introductions still make this source book the most important reference for the history of mathematical logic.
Warren Goldfarb
Meticulously edited, with excellent translations and helpful introductory notes, From Frege to Gödel is an indispensable volume for anyone interested in the development of modern logic and its philosophical impact.
Synthese - Andrzej Mostowski
There can be no doubt that the book is a valuable contribution to the logical literature and that it will certainly spread the knowledge of mathematical logic and its history in the nineteenth and twentieth centuries.
Hilary Putnam
If there is one book that every philosopher interested in the history of logic should own, not to mention all the philosophers who pretend they know something about the history of logic, From Frege to Gödel is that book.
W. D. Hart
From Frege to Gödel lays out before our eyes the turbulent panorama in which modern logic came to be.
Michael Friedman
From Frege to Gödel is the single most important collection of original papers from the development of mathematical logic-an invaluable source for all students of the subject.
Juliet Floyd
A Bible for historians of logic and computer science, this invaluable collection will profit anyone interested in the interplay between mathematics and philosophy in the early decades of the twentieth century. It provides a unique and comprehensive way to appreciate how modern mathematical logic unfolded in the hands of its greatest founding practitioners.
Solomon Feferman
Year in, year out, I recommend this book enthusiastically to students and colleagues for sources in the history and philosophy of modern logic and the foundations of mathematics; I use my own copy so much, it is falling apart.
Michael Detlefson
For more than three decades this outstanding collection has been the authoritative source of basic texts in mathematical logic in the English language; it remains without peer to this day.
James Conant
Jean van Heijenoort's Source Book in Mathematical Logic offers a judicious selection of articles, lectures and correspondence on mathematical logic and the foundations of mathematics, covering the whole of the single most fertile period in the history of logic, namely from 1879 (the year of Frege's epochmaking discovery/invention of modern mathematical logic) to 1931 (the year of Gödel's epoch-ending incompleteness theorem). All the translations are impeccable. Each piece is introduced by an expository article and additionally furnished with a battery of supplementary technical, historical, and philosophical comments in the form of additional footnotes. The collection as a whole allows one to relive each of the crucial steps in this formative period in the history of logic, from Frege's introduction of the Begriffsschrift, to the discovery of Russell's paradox (including Frege's heroic and heart-breaking letter of congratulation to Russell(, the development of axiomatic set theory, the program of Russell and Whitehead's Principia Mathematica, Brouwer's intuitionism, Hilbert's proof theory, to the limitative theorems of Skolem and Gödel, to mention only a few of the highlights. Anyone with a serious interest in the history or philosophy of logic will want to own this volume.
Review of Metaphysics
It is difficult to describe this book without praising it...[From Frege to Gödel] is, in effect, the record of an important chapter in the history of thought. No serious student of logic or foundations of mathematics will want to be without it.
Synthese

There can be no doubt that the book is a valuable contribution to the logical literature and that it will certainly spread the knowledge of mathematical logic and its history in the nineteenth and twentieth centuries.
— Andrzej Mostowski

Synthese
There can be no doubt that the book is a valuable contribution to the logical literature and that it will certainly spread the knowledge of mathematical logic and its history in the nineteenth and twentieth centuries.
— Andrzej Mostowski
Andrzej Mostowski
There can be no doubt that the book is a valuable contribution to the logical literature and that it will certainly spread the knowledge of mathematical logic and its history in the nineteenth and twentieth centuries.
Synthese
Review of Metaphysics
It is difficult to describe this book without praising it...[From Frege to Gödel] is, in effect, the record of an important chapter in the history of thought. No serious student of logic or foundations of mathematics will want to be without it.
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Product Details

Meet the Author

Jean van Heijenoort, well known in the fields of mathematical logic and foundations of mathematics, is Professor of Philosophy at Brandeis University and has taught at New York and Columbia Universities.
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Table of 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.König (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.Löwenheim (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. Löwenheim 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

Schönfinkel (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

Gödel (l930a). The completeness of the axioms of the functional calculus of logic

Gödel (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

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