Does anyone know how news of Gödel's incompleteness theorem spread? Did it do so little by little, or was it shouted in dramatic headlines throughout mathematical literature? If anyone can point me to the history of how people reacted to Gödel's result, I would be grateful.


I am asking this question because I have recently been reading a work that comes from a field very far from logic, namely Lie theory, written in 1940 (ten years after Über formal unentscheidbare...") wherein the author seems to be highly mindful of the different mathematical philosophies - he is at least paying lip service to intuitionism and how his work might make sense in that philosophy (see below). Now I should have thought that Gödel's incompleteness theorem would take much of the heat out of debates as to who had the "best" philosophy. Or did it put wind in the sails of the intuitionists, after Gödel had seemingly demolished the formalists, although I believe Gödel would not have seen his incompleteness theorem validating the intuitionists either, being as he was a strong Platonist. Anyhow, here is the quote: it is "Hauptsatz 1" in the paper and I was fascinated to read these words in the far-removed-from-logic field of Lie theory:


From H. Freudenthal "Die Topologie der Lieschen Gruppen Als Algebraisches Ph\"anomen" Annals of Mathematics vol 42. # 5 (1941) wherein he makes the following statement: 


"Main Theorem 1: An isomorphism between two Lie groups, of which one is simple and of the second kind, is needfully continuous. Otherwise put: in the theory of Lie groups, the topology of simple groups of the second kind is a wholly algebraic phenomenon"


Lest you should think that the rewording "otherwise put ..." cannot be construed as a precise statement of a theorem (it does on the surface seem rather vague), Freudenthal goes on to explain: 


"In the latter formulation the main theorem also makes sense for someone who outright refuses [the existence of] discontinuous mappings, such as [someone with] intuitionist leanings"



How quickly did Gödel's Incompleteness Theorem become known and heeded throughout mathematics
I believe the answer to the title of the question is "Quickly". The Wikipedia article on "Gödel's incompleteness theorems" has a nice discussion of the developments in the 1930's. The title of Gödel's paper on the incompleteness theorem is "Über formal unentscheidbare Sätze... I", and apparently he never needed to write part II. 


It seems that the correctness of Gödel's paper was quickly realized, but rather than taking the heat out of any philosophical debate, it provided new and interesting fuel.


