We reprint the translation of the now famous letter he wrote fifty years ago to von Neumann which was rediscovered in 1988. This letter describes something close to the P versus NP problem years before the field Computational Complexity even had its name. SIGACT and EATCS jointly sponsor a prize named after Gödel because of the letter.
Princeton, 20 March 1956
Dear Mr. von Neumann:
With the greatest sorrow I have learned of your illness. The news came to me as quite unexpected. Morgenstern already last summer told me of a bout of weakness you once had, but at that time he thought that this was not of any greater significance. As I hear, in the last months you have undergone a radical treatment and I am happy that this treatment was successful as desired, and that you are now doing better. I hope and wish for you that your condition will soon improve even more and that the newest medical discoveries, if possible, will lead to a complete recovery.
Since you now, as I hear, are feeling stronger, I would like to allow myself to write you about a mathematical problem, of which your opinion would very much interest me: One can obviously easily construct a Turing machine, which for every formula F in first order predicate logic and every natural number n, allows one to decide if there is a proof of F of length n (length = number of symbols). Let ψ(F,n) be the number of steps the machine requires for this and let φ(n) = maxF ψ(F,n). The question is how fast φ(n) grows for an optimal machine. One can show that φ(n) ≥ k ⋅ n. If there really were a machine with φ(n) ∼ k ⋅ n (or even ∼ k ⋅ n2), this would have consequences of the greatest importance. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem, the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine. After all, one would simply have to choose the natural number n so large that when the machine does not deliver a result, it makes no sense to think more about the problem. Now it seems to me, however, to be completely within the realm of possibility that φ(n) grows that slowly. Since
- it seems that φ(n) ≥ k ⋅ n is the only estimation which one can obtain by a generalization of the proof of the undecidability of the Entscheidungsproblem and
- after all φ(n) ∼ k ⋅ n (or ∼ k ⋅ n2) only means that the number of steps as opposed to trial and error can be reduced from N to log N (or (log N)2).
I do not know if you have heard that "Post's problem", whether there are degrees of unsolvability among problems of the form (∃ y) φ(y,x), where φ is recursive, has been solved in the positive sense by a very young man by the name of Richard Friedberg. The solution is very elegant. Unfortunately, Friedberg does not intend to study mathematics, but rather medicine (apparently under the influence of his father). By the way, what do you think of the attempts to build the foundations of analysis on ramified type theory, which have recently gained momentum? You are probably aware that Paul Lorenzen has pushed ahead with this approach to the theory of Lebesgue measure. However, I believe that in important parts of analysis non-eliminable impredicative proof methods do appear.
I would be very happy to hear something from you personally. Please let me know if there is something that I can do for you. With my best greetings and wishes, as well to your wife,
P.S. I heartily congratulate you on the award that the American government has given to you.
[The text is taken from this page where you can also find the original German text and acknowledgments. John von Neumann, who received the Presidential Medal of Freedom in 1956, had cancer at the time of the letter and passed away in 1957.]