Outline and key results
The main results established are Gödel's first and second incompleteness theorems, which have had an enormous impact on the field ofPublished English translations
During his lifetime three English translations of Gödel's paper were printed, but the process was not without difficulty. The first English translation was byFortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthology ''The Undecidable''; but it too was not brought to Gödel's attention until almost the last minute, and the new translation was still not wholly to his liking ... when informed that there was not time enough to consider substituting another text, he declared that Mendelson's translation was 'on the whole very good' and agreed to its publication.3 3 Afterward he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints.">sup>3 Afterward he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints./blockquote> The translation by Elliott Mendelson appears in the collection ''The Undecidable'' (Davis 1965:5ff). This translation also received a harsh review by Bauer-Mengelberg (1966), who in addition to giving a detailed list of the typographical errors also described what he believed to be serious errors in the translation. A translation by Jean van Heijenoort appears in the collection ''From Frege to Gödel: A Source Book in Mathematical Logic'' (van Heijenoort 1967). A review byAlonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scien ...(1972) described this as "the most careful translation that has been made" but also gave some specific criticisms of it. Dawson (1997:216) notes:The translation Gödel favored was that by Jean van Heijenoort ... In the preface to the volume van Heijenoort noted that Gödel was one of four authors who had personally read and approved the translations of his works.This approval process was laborious. Gödel introduced changes to his text of 1931, and negotiations between the men were "protracted": "Privately van Heijenoort declared that Gödel was the most doggedly fastidious individual he had ever known." Between them they "exchanged a total of seventy letters and met twice in Gödel's office in order to resolve questions concerning subtleties in the meanings and usage of German and English words." (Dawson 1997:216-217). Although not a translation of the original paper, a very useful 4th version exists that "cover ground quite similar to that covered by Godel's original 1931 paper on undecidability" (Davis 1952:39), as well as Gödel's own extensions of and commentary on the topic. This appears as ''On Undecidable Propositions of Formal Mathematical Systems'' (Davis 1965:39ff) and represents the lectures as transcribed by Stephen Kleene and J. Barkley Rosser while Gödel delivered them at the Institute for Advanced Study in Princeton, New Jersey in 1934. Two pages of errata and additional corrections by Gödel were added by Davis to this version. This version is also notable because in it Gödel first describes the Herbrand suggestion that gave rise to the (general, i.e. Herbrand-Gödel) form ofrecursion Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ....
References
* Stefan Bauer-Mengelberg (1966). Review of ''The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable problems and Computable Functions.'' ''The Journal of Symbolic Logic'', Vol. 31, No. 3. (September 1966), pp. 484–494. *Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scien ...(1972). Review of ''A Source Book in Mathematical Logic 1879–1931.'' ''The Journal of Symbolic Logic'', Vol. 37, No. 2. (June 1972), p. 405. * Martin Davis, ed. (1965). ''The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions'', Raven, New York. Reprint, Dover, 2004. . * Martin Davis, (2000). ''Engines of Logic: Mathematics and the Origin of the Computer'', W. W. Norton & Company, New York. pbk. *Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...(1931), "Über formal unentscheidbare Sätze der ''Principia Mathematica'' und verwandter Systeme I". ''Monatshefte für Mathematik und Physik 38'': 173–198. . Available online via SpringerLink. *Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...(1958). "Über eine bisher noch nicht benüzte Erweiterung des finiten Standpunktes". ''Dialectica'' v. 12, pp. 280–287. Reprinted in English translation in Gödel's ''Collected Works'', vol II, Soloman Feferman et al., eds. Oxford University Press, 1990. * Jean van Heijenoort, ed. (1967). ''From Frege to Gödel: A Source Book on Mathematical Logic 1879–1931''. Harvard University Press. *Bernard Meltzer Bernard C. Meltzer (May 2, 1916 – March 25, 1998) was a United States radio host for several decades. His advice call-in show, "What's Your Problem?," aired from 1967 until the mid-1990s on stations WCAU-AM and WPEN-AM in Philadelphia, ...(1962). ''On Formally Undecidable Propositions of Principia Mathematica and Related Systems.'' Translation of the German original by Kurt Gödel, 1931. Basic Books, 1962. Reprinted, Dover, 1992. . *Raymond Smullyan Raymond Merrill Smullyan (; May 25, 1919 – February 6, 2017) was an American mathematician, magician, concert pianist, logician, Taoist, and philosopher. Born in Far Rockaway, New York, his first career was stage magic. He earned a BSc from ...(1966). Review of ''On Formally Undecidable Propositions of Principia Mathematica and Related Systems.'' ''The American Mathematical Monthly'', Vol. 73, No. 3. (March 1966), pp. 319–322. * John W. Dawson, (1997). ''Logical Dilemmas: The Life and Work of Kurt Gödel'', A. K. Peters, Wellesley, Massachusetts. {{isbn, 1-56881-256-6.
External links
"On formally undecidable propositions of Principia Mathematica and related systems I"
Translated by Martin Hirzel, November 27, 2000.
"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I"
on Wilhelm K. Essler's (Prof. em. of Logic, Goethe-Universität Frankfurt am Main) webpage Mathematical logic Mathematics papers 1931 in science 1931 documents Works originally published in German magazines Works originally published in science and technology magazines Logic literature Works by Kurt Gödel