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 Konig 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 Lowenheim'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 Godel, 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 arethemselves 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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