# How quickly did Gödel's Incompleteness Theorem become known and heeded throughout mathematics

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"

• Godel was hardly the last word on the philosophy of mathematics. Also, I think you are reading far too much into a single sentence in Freudenthal's paper. Commented May 20, 2011 at 5:35
• @ Andy - certainly I agree with your statement about Goedel being the last word on philosophy, which is part of the point of my question - how did the different philosophies react to Godel, Turing, Post, Tarski etc? What I find interesting is the mention of different philophies in a paper far removed from logic. I know Freudenthal's Hilbert number was 2 (through van der Waerden) and he would have had much contact with the the latter, being a good friend. It's almost as though the 1920's debate between Brouwer and Hilbert is still to the fore in Freudenthal's mind when he makes the statement. Commented May 20, 2011 at 5:49
• I don't see why this has to do with Goedel per se, still less the incompleteness theorems. Model theory, perhaps. Commented May 20, 2011 at 5:53
• @ Yemon I am only guessing (which is why I would like to {\it know} the history) that the incompleteness theorems might have taken some of the heat out of claims of various philosophies to be superior. I can't believe that adherents to various standpoints would have carried on heedless of such results as Goedel's, Tarski's, Turing's etc Commented May 20, 2011 at 6:00
• Mathias's article "The ignorance of Bourbaki" (dpmms.cam.ac.uk/~ardm/bourbaki.pdf) seems to be quite relevant to your question. Commented May 20, 2011 at 9:12