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

Gathered together in this book are the fundamental texts of the great classical period in modern logic. A complete translation of Gottlob Frege's Begriffsschrift--which opened a great epoch in the history of logic by fully presenting propositional calculus and quantification theory--begins the volume. The texts that follow depict the emergence of set theory and foundations of mathematics, two new fields on the borders of logic, mathematics, and philosophy. Essays trace the trends that led to Principia mathematica, the appearance of modern paradoxes, and topics including proof theory, the theory of types, axiomatic set theory, and Löwenheim's theorem. 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.

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