Published 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 by Bernard Meltzer; it was published in 1963 as a standalone work by Basic Books and has since been reprinted by Dover and reprinted by Hawking (God Created the Integers, Running Press, 2005:1097ff). The Meltzer version was adversely reviewed by Raymond Smullyan (1966). According to Dawson's biography of Gödel (Dawson 1997:216),
"Fortunately, 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
The translation by Elliott Mendelson appears in the collection The Undecidable (Davis 1965:5ff). This translation also received a harsh review by Bauer-Medelburg (1965), 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 by Alonzo Church (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 N.J. 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 of recursion.
Read more about this topic: On Formally Undecidable Propositions Of Principia Mathematica And Related Systems
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