The Incompleteness Theorem Martin Davis n September 1930 in Königsberg, on the third represented by a string of symbols, and a proof, day of a symposium devoted to the founda- by a finite sequence of such strings. Since these sys- tions of mathematics, the young Kurt Gödel tems were simple combinatorial objects, it seemed launched his bombshell announcing his in- quite possible to apply mathematical methods to Icompleteness theorem. At that time, there study their properties. Hilbert’s program aimed to were three recognized “schools” on the foundations prove, by utterly unimpeachable methods, that of mathematics: the logicism based on the work of these systems were consistent and complete: that Frege, Russell, and Whitehead that saw mathe- they were safe from the catastrophic inconsistency, matics as simply part of logic, Brouwer’s radical in- due to Russell’s paradox, that had struck Frege’s tuitionism, and Hilbert’s proof theory (also called ambitious attempt to bridge the gap between the “formalism”). In fact two days earlier, lectures rep- elements of formal logic and mathematics proper, resenting these schools had been delivered by Car- and that with respect to some specified class of nap, Heyting, and von Neumann respectively. Von statements, each statement of the class could be Neumann may have been the only person in the either proved or refuted within the system. Gödel’s room to have grasped the significance of what incompleteness theorem did away with the sec- Gödel had done. He saw that the goals of Hilbert’s ond of these goals, and shortly thereafter Gödel was proof theory had been shown to be simply unat- able to show that the first was likewise unachiev- tainable. Logicism had also been dealt a death blow, able. Gödel’s theorem had made it clear that no sin- but Carnap, who had known about Gödel’s in- gle formal system could be devised that would en- completeness theorem for over a week when he able all mathematical truths, even those expressible gave his address, seemed not to realize its signif- in terms of basic operations on the natural num- icance. bers, to be provided with a formal proof. Formalization of Mathematics Gödel’s Proof It was Gottlob Frege in his Begriffsschrift of 1879 Gödel proceeded to define a code by means of who had shown how the logical reasoning used in which each expression of a formal system would mathematical proofs can be reduced to the com- have its own natural number, what has come to be binatorial manipulation of symbols. By the 1920s called its Gödel number, associated with it. Thus, foundational work had made it clear that the full expressions of the system that represent proposi- expanse of classical mathematics could be encap- tions about the natural numbers might be seen by sulated in such formal combinatorial systems. In someone privy to the code as also making asser- these systems, a proposition of mathematics was tions, incidentally as it were, about the system it- Martin Davis is professor emeritus of mathematics and self. Working with a particular formal system computer science, New York University, and is a Visiting loosely based on that of Whitehead and Russell and Scholar in mathematics at the University of California, exploiting this idea, Gödel showed how to con- Berkeley. His email address is [email protected]. struct a remarkable expression of the system we 414 NOTICES OF THE AMS VOLUME 53, NUMBER 4 may designate as U. To someone who didn’t know Computability theory provides a perspective the code, U would be seen as expressing a com- from which it can be seen that incompleteness is plicated and peculiar statement about the natural a pervasive fundamental property not dependent numbers. But to someone who could decipher it, on a trifling trick. From this point of view the for- U would be seen as also asserting that some state- mal systems studied by logicians are simply com- ment expressible in the system is unprovable. Look- putable functions that spew out theorems (more ing more closely, it would be found that the state- precisely, Gödel numbers of theorems). Such sys- ment asserted to be unprovable is U itself. Thus we tems are usually given in terms of a set of axioms may say: and rules of inference. One can then imagine an al- U asserts that it is unprovable. gorithm that begins with the axioms and proceeds Thus, if U were false, it would be provable, and by iteratively applying the rules of inference. hence, presumably, true. This contradiction shows To obtain a form of the incompleteness theorem that U is true, and hence, given what it asserts, un- let us begin with the set K whose existence is given provable. There are true statements unprovable in by the theorem above, and consider propositions the given system. of the form n ∈ K where n is a fixed natural num- Of course, this heuristic outline would have ber. We can suppose that, in a particular formal sys- hardly been convincing. But Gödel carefully worked tem these propositions are each represented by a out the details leaving no doubt about the cor- rectness of his conclusions.1 Nevertheless, a whiff corresponding string of symbols we may write as of paradox hung over the matter; it seemed hard Pn . We need only assume that there is an algo- 4 to believe that a trick so close to puzzles usually rithm for obtaining Pn given n. Let us use the sym- offered for amusement could really be used to bol F for some formal system, and write F Pn to demonstrate something profound about mathe- mean that Pn is provable in F. We will say that F matics. is sound if Whenever F P for a particular n, Computability Theory Makes a n it will also be the case that n ∈ K. Contribution Since Pn is intended to stand for the proposition We write N = {0, 1, 2,...} for the set of natural n ∈ K, soundness simply means that the provable numbers. A function f : N → N is called computable statements are true. if there is an algorithm that given an x ∈ N will com- pute f (x). Here the notion of algorithm is assumed Incompleteness Theorem. Let F be a sound formal to involve no restriction as to the amount of time system. Then there is a number n0 such that n0 ∈ K, or space required to complete a computation.2 Fi- but it is not the case that F Pn . nally a set S ⊆ N is called computable if its char- 0 acteristic function Again, we have a true sentence that is not prov- 1ifx ∈ S able. Note that we only succeed in changing the CS (x)= value of the particular number n as we attempt 0 otherwise 0 to create stronger and stronger formal systems that can prove more and more. is computable. Proof of the Incompleteness Theorem. Sup- The following is fundamental: pose that there is no such n0. Then we would have: Theorem. There is a computable function f whose F P for a particular n, if and only if n ∈ K. range n K = {f (0),f(1),f(2),...} Recall that K is the range of the computable func- tion f. Then the following would be an algorithm 3 is not computable. for computing CK(n) for a given value of n, con- tradicting the fact that K is not computable: Begin 1Detailed proofs can be found in a number of textbooks, F for example [3]. In addition Gödel’s clear and meticulous generating the theorems of and at the same time original exposition [8] still repays study. begin computing the successive values 2 Computability theory has provided a number of precise f (0),f(1),f(2),.... If n ∈ K, then n will eventually characterizations to replace this heuristic explanation, show up in the list of values of f so CK(n)=1. Oth- and they have all been proved equivalent to one another. erwise, Pn will eventually show up in the theorem 3 See for example [1]. Computability theory is also known list of F so that CK(n)=0. as recursion theory and used to also be called recursive 4 function theory. Computable functions are also called re- In a traditional formal system, for a given number n, Pn cursive. Sets that are the range of a computable function will be obtained by replacing, in a certain specific formula, as well as the empty set are called recursively enumerable, a symbol for a variable by a “numeral” representing the or more recently, computably enumerable, or listable. number n. APRIL 2006 NOTICES OF THE AMS 415 A Diophantine Perspective and for the successor, sum, and product functions The following result, known variously as MRDP on the natural numbers. The axioms are the familiar and as Matiyasevich’s Theorem, enables it to be seen Peano postulates together with equations serving that the truths unprovable in specified formal sys- to implicitly define sum and product. The induc- tems can have a straightforward mathematical tion postulate, whose informal statement is that a form. set of natural numbers containing 0 and closed under successor must consist of all natural num- Theorem. If S is the range of a computable func- bers, appears in a restricted form: it is stated only tion, then there is a polynomial p(a, x1,...,xm) with for sets definable in terms of the vocabulary.6 integer coefficients such that the equation PA formalizes the elementary number theory of p(a, x1,...,xm)=0has a solution in natural num- the textbooks as well as (via clever coding) sub- bers x1,...,xm for a given value of a if and only if stantial parts of elementary analysis.