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The Incompleteness Theorem Martin Davis

n September 1930 in Königsberg, on the third represented by a string of symbols, and a , day of a symposium devoted to the founda- by a finite sequence of such strings. Since these sys- tions of , 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 , Brouwer’s radical in- due to Russell’s paradox, that had struck Frege’s tuitionism, and Hilbert’s proof (also called ambitious attempt to bridge the gap between the “”). In 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 of nap, Heyting, and von Neumann respectively. Von statements, each 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 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 could be devised that would en- theorem for over a week when he able all mathematical , 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 .

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 , 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 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 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 of may say: and rules of . One can then imagine an al- U asserts that it is unprovable. gorithm that begins with the axioms and proceeds Thus, if U were , it would be provable, and by iteratively applying the rules of inference. hence, presumably, true. This 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 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, simply means that the provable numbers. A 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 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 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 are called recursively enumerable, a for a variable by a “numeral” representing the or more recently, , 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 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. In contrast ∈ a S (see [9, 2]). ZFC formalizes the full scope of modern set- Applying this result to the case S = K, let us theoretic mathematics including such things as call the corresponding polynomial p0 . Now we can and transfinite arithmetic. The vo- think of the expressions Pn as standing for the cabulary can be extremely parsimonious consist- ∈ proposition that p0(n, x1,...,xm)=0has no solu- ing only of the symbol for set membership. For tions in natural numbers, and say that F is Dio- our purposes it will be useful to be slightly less fru- ∅ phantine-sound if F Pn implies that the equation gal, allowing as well symbols (the empty set), { } p0(n, x1,...,xm)=0does indeed fail to have solu- ... (the set consisting of a single ), and tions. Then, the incompleteness theorem of the ∪ (binary union). The axioms are those of Zermelo- previous section takes the form: Fraenkel together with the of choice, and the resulting system is powerful enough to encapsu- F Diophantine Incompleteness Theorem. Let be late the full scope of classical mathematics, and in- Diophantine-sound. Then there is a number n0 such deed, much more (see for example [4]). that the equation p0(n0,x1,...,xm)=0 has no so- We write PA and ZFC for provability in PA lutions in natural numbers although it is not the case and ZFC, respectively. We will use F as a subscript that F P . n0 to refer ambiguously to either of these systems. It is worth remarking that the proof of MRDP is Also we write  F to express non-provability in the entirely constructive so the polynomial p0 could corresponding systems. be produced quite explicitly. In each of PA and ZFC, a simple notation is avail- able for representing the natural numbers by se- Two Formal Systems: PA and ZFC quences of strings we call numerals. We will write What Frege showed is that the ordinary reasoning n for the numeral representing the natural num- in proofs of mathematical theorems amounts to for- ber n. In PA this may be defined as follows using mal manipulations of the propositional connec- the letter s for successor and letting 0 be repre- tives ¬→∨∧together with the quantifiers ∀∃. sented by its usual symbol: Manipulations of the propositional connectives amounts to carrying out the operations of Boolean 0=0; n +1=sn algebra. The quantifiers get in the way of this, and Following von Neumann, the numerals in ZFC can careful rules are needed to justify removing and re- be defined as follows: instating them. Once these rules are specified ∅ ∪{ } (which can be done in a number of equivalent 0= ; n +1=n n ways), the way is open to set up formal systems en- Now, for F standing for either PA or ZFC, associ- capsulating greater or lesser portions of mathe- ated with the polynomial p0 of the previous sec- matics. This involves supplying a vocabulary of tion, there is a formula πF (x0,x1,...,xm) such symbols representing various constants, functions, that for arbitrary natural numbers a, a1,...,am: and relations appropriate to the part of mathe- matics being formalized. Finally a list of axioms if p0(a, a1,...,am) = 0 then F π(a, a1,...,am) must be given: these are written using this vocab- if p0(a, a1,...,am) = 0 then F ¬π(a, a1,...,am) ulary together with the symbols listed above cor- responding to the operations of logical inference. For PA, this is almost a because symbols A symbol for should also be available.5 for addition and multiplication are part of its vo- For the system PA (for “Peano Arithmetic”), the cabulary, and the axioms justify ordinary calcula- vocabulary consists of symbols for the number 0, tions. For ZFC, some circumlocution is needed, but

5Equality can be thought of as the most “advanced” part 6A fuller account of PA will be found in Feferman’s arti- of the underlying logic, or as the most fundamental math- cle [5] in this issue of the Notices. Full details will be found ematical relation. in textbooks such as [3].

416 NOTICES OF THE AMS VOLUME 53, NUMBER 4 the ordinary about addition and multiplica- property just exhibited that their undecidability in tion of natural numbers can still be replicated. a reasonable formal system implies their . It follows from the MRDP theorem that statements Incompleteness Theorem for PA and ZFC. If F is asserting that some computable property holds consistent, then there is a natural number n such 0 for all natural numbers are provably equivalent that (for example in PA) to a ∀-statement. Many fa-  F (∀x1) ...(∀xm)¬π(n0,x1,...,xm), mous problems are thus seen to belong to this and class, in particular, Fermat’s last theorem, the Gold- bach , the four-color theorem, and the  F ¬(∀x1) ...(∀xm)¬π(n0,x1,...,xm). Riemann (see [2]). Thus the sentence (∀x1) ...(∀xm)¬π(n0,x1, ...,xm) is undecidable in F: neither it nor its Beyond ZFC is provable. However, what that sentence At the same time that the ZFC axioms provide a asserts, namely that the equation p0(n0,x1,...,xm) foundation for mathematics, they also can be re- =0 has no solutions in natural numbers, is true. garded as defining a class of mathematical struc- Moreover that truth is a consequence of its unde- tures. From this point of view they can be seen as cidability. For if p0(n0,a1,...,am)=0 we would providing “closure” under such operations as form- have F π(n0, a1,...,am) and using elementary ing the set of all of a given set or the union logic we would obtain of all of its elements. In a normal situation of this kind it would be natural to find the least set closed F (∃x1) ...(∃xm)π(n0,x1,...,xm) under all of the operations called for by the axioms. from which we readily obtain Remarkably, as natural as such an object appears, F ¬(∀x1) ...(∀xm)¬π(n0,x1,...,xm) its existence cannot be proved in ZFC. This is be- cause if such existence could be proved, it would contradicting the claimed undecidability. provide a model for the axioms and hence lead to There has been much confusion about this sit- a proof in ZFC of its own . And this, uation. How is it that we can see that the proposi- Gödel had proved to be impossible. So systems tion is true although a system as powerful as ZFC like ZFC lead in a natural way to extensions, and cannot? The answer is that ZFC can indeed see in each such extension new ∀-propositions be- what we can, namely that if ZFC is consistent then come provable. In fact there will be new values of the proposition is true but undecidable by its n for which the fact that the equation means. In fact, it was precisely by analyzing this 0 p (n ,x ,...,x )=0 has no solutions in natural situation that Gödel could conclude that systems 0 0 1 m numbers becomes provable. What remains unclear like PA and ZFC cannot prove their own consistency, is whether some really mathematically significant thereby shattering Hilbert’s hopes. ∀-propositions, perhaps like some of those men- The fact that ZFC is stronger than PA (in actual fact very much stronger) is exemplified by the fol- tioned at the end of the previous section, require lowing result: means beyond ZFC for their proof. We conclude this article with a recent example announced by Har- Theorem. If PA is consistent, then there is a natural vey Friedman (see [7]) of a ∀-proposition that is un- number n0 such that provable in ZFC, but becomes provable with the aid of a so-called “large cardinal” axiom, an assump-  PA (∀x1) ...(∀xm)¬π(n0,x1,...,xm), tion of the existence of an of a size larger and than any whose existence can be proved in ZFC.8  PA ¬(∀x1) ...(∀xm)¬π(n0,x1,...,xm), Friedman’s example concerns finite directed but graphs (no multiple edges allowed) whose vertices ZFC (∀x1) ...(∀xm)¬π(n0,x1,...,xm). are finite sequences of integers. What is striking is that what looks like a harmless additional conclu- So the undecidability in PA is decided in ZFC! But sion in a theorem provable in ZFC (and even in much then ZFC has its own undecidability and with the weaker systems) results in a proposition that is un- very same formula π. Only the number n changes. 0 provable in ZFC but becomes provable on the ad- The values of n for either system will be enormous 0 dition of a large cardinal axiom, an assumption of since all the complexity of the algorithms for gen- the existence of a set too large for that existence erating theorems these systems provide must be 9 contained in those numbers. to be provable in ZFC. Some preliminary defini- We will refer to statements to the effect that tions are needed. For a natural number n we write some polynomial equation has no solutions in nat- 8The article by Juliet Floyd and Akihiro Kanamori [6] in ∀ 7 ural numbers as -statements. They all have the this issue of the Notices contains some discussion of large 7 0 cardinal axioms. Logicians call these Π1 statements. As this notation sug- gests, they find their place in a hierarchy. 9Specifically, an axiom of the Mahlo type.

APRIL 2006 NOTICES OF THE AMS 417 nˆ for the set {1, 2,...,n}. So nˆk is the set of all se- [3] HERBERT ENDERTON, A Mathematical Introduction to Logic, quences of these numbers of length k. If x, y ∈ nˆk, Academic Press, New York, 1972. [4] , Elements of , Academic Press, New we write x ∗ y for the element of nˆ2k obtained by ——— York, 1977. concatenating x and y. We will work with directed [5] SOLOMON FEFERMAN, The Impact of the Incompleteness graphs G whose vertex set V(G) consists of ele- Theorems on Mathematics, Notices, April 2006. k ments of nˆ for certain fixed n, k. For x, y ∈ V(G) [6] JULIET FLOYD and AKIHIRO KANAMORI, How Gödel Trans- we write (x, y) for a possible edge proceeding from formed Set Theory, Notices, April 2006. 0 x to y. G is called an upgraph if for every edge (x, y) [7] HARVEY FRIEDMAN, “Π1 Incompleteness: Finite Graph of G , we have max(x) < max(y) . We say that Theory 1.” http://www.math.ohio-state.edu/ u, v ∈ nˆ are order equivalent if for all 1 ≤ i,j ≤ , %7Efriedman/pdf/Pi01013006.pdf. [8] KURT GÖDEL, “Über formal unentscheidbare Sätze der we have ui

References [1] NIGEL CUTLAND, Computability: An Introduction to Re- cursive Function Theory, Cambridge University Press, Cambridge, England, and New York, 1980. [2] MARTIN DAVIS, YURI MATIJASEVIC, and JULIA ROBINSON, “Hilbert’s Tenth Problem: Diophantine Equations: Pos- itive Aspects of a Negative Solution”, Proceedings of Symposia in , vol. 28 (1976), pp. 323–378; reprinted in Feferman, Solomon, ed. The Collected Works of Julia Robinson, Amer. Math. Soc. 1996, pp. 269–378.

418 NOTICES OF THE AMS VOLUME 53, NUMBER 4