From Frege to Gödel: a source book in mathematical logic, 1879-1931
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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Letters and other signs
Inference The Aristotelian modes of inference
Identity of content 8 Need for a sign for identity of content introduction of such a sign
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From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931
Jean Van Heijenoort
Limited preview - 1967
according apparent variable applied arbitrary argument arithmetic assertion assume assumption axiom of choice axiom of separation axioms Begriffsschrift belongs Brouwer calculus called Cantor cardinal consider consistency consistency proof constructed contains continuum contradiction corresponding Dedekind defined definition denote denumerably infinite descriptive functions disjoint element equation equivalent example exists expression f-sequence fact false finite number first-order proposition follows footnote formal formula Frege given Godel hence Herbrand Hilbert identity ideography II-object induction inference infinite domain integer judgment Lemma letter logic logical value Lowenheim mapping mathematical induction mathematics means natural numbers negation notation notion objects obtain occur order types ordinal number paper paradox Peano possible predicate prenex prenex form primitive proof propositional function provable prove quantification theory quantifiers recursive relation replace result Russell satisfied set theory Skolem subscripts subset substitution Theorem tion transfinite translation truth table truth values well-ordered set well-ordering theorem Zermelo