Kurt G¨
odel’s Letter to John von Neumann - 1956
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. Morgen-
stern 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 mathe-
matical 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) = max
F
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 · n
2
), 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 · n
2
) 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
). However, such strong reductions appear in other finite problems, for example in the
computation of the quadratic residue symbol using repeated application of the law of reciprocity. It would
be interesting to know, for instance, the situation concerning the determination of primality of a number and
how strongly in general the number of steps in finite combinatorial problems can be reduced with respect to
simple exhaustive search.
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,
Sincerely yours,
Kurt G¨
odel
P.S. I heartily congratulate you on the award that the American government has given to you.