The Impact of the Incompleteness Theorems on Mathematics Solomon Feferman n addition to this being the centenary of Kurt ting that aside, my view of Gödel’s incompleteness Gödel’s birth, January marked 75 years since theorems is that their relevance to mathematical the publication (1931) of his stunning in- logic (and its offspring in the theory of computa- completeness theorems. Though widely tion) is paramount; further, their philosophical rel- Iknown in one form or another by practicing evance is significant, but in just what way is far from mathematicians, and generally thought to say some- settled; and finally, their mathematical relevance thing fundamental about the limits and potential- outside of logic is very much unsubstantiated but ities of mathematical knowledge, the actual im- is the object of ongoing, tantalizing efforts. My portance of these results for mathematics is little main purpose here is to elaborate this last assess- understood. Nor is this an isolated example among ment. famous results. For example, not long ago, Philip Davis wrote me about what he calls The Paradox Informal and Formal Axiom Systems of Irrelevance: “There are many math problems One big reason for the expressed disconnect is that have achieved the cachet of tremendous sig- that Gödel’s theorems are about formal axiom sys- nificance, e.g., Fermat, four-color, Kepler’s packing, tems of a kind that play no role in daily mathe- Gödel, etc. Of Fermat, I have read: ‘the most famous matical work. Informal axiom systems for various math problem of all time’. Of Gödel, I have read: kinds of structures are of course ubiquitous in ‘the most mathematically significant achievement practice, viz. axioms for groups, rings, fields, vec- of the 20th century’. … Yet, these problems have tor spaces, topological spaces, Hilbert spaces, etc., engaged the attention of relatively few research etc.; these axioms and their basic consequences are mathematicians—even in pure math.” What ac- so familiar it is rarely necessary to appeal to them counts for this disconnect between fame and rel- explicitly, but they serve to define one’s subject mat- evance? Before going into the question for Gödel’s ter. They are to be contrasted with foundational theorems, it should be distinguished in one re- axiom systems for the “mother” structures—the spect from the other examples mentioned, which natural numbers (Peano) and the real numbers in any case form quite a mixed bag. Namely, each (Dedekind)—on the one hand, and for the general of the Fermat, four-color, and Kepler’s packing concepts of set and function (Zermelo-Fraenkel) problems posed a stand-out challenge following ex- used throughout mathematics, on the other. Math- tended efforts to settle them; meeting the challenge ematicians may make explicit appeal to the prin- in each case required new ideas or approaches and ciple of induction for the natural numbers or the intense work, obviously of different degrees. By con- least upper bound principle for the real numbers trast, Gödel’s theorems were simply unexpected, or the axiom of choice for sets, but reference to and their proofs, though requiring novel tech- foundational axiom systems in practice hardly goes niques, were not difficult on the scale of things. Set- beyond that. One informal statement of the basic Peano ax- Solomon Feferman is professor of mathematics and phi- ioms for the natural numbers is that they concern losophy, emeritus, at Stanford University. His email address a structure (N,0,s) where 0 is in N, the successor is [email protected]. function s is a unary one-one map from N into N 434 NOTICES OF THE AMS VOLUME 53, NUMBER 4 which does not have 0 in its range, and the Induc- Consistency, Completeness, and tion Principle is satisfied in the following form: Incompleteness (IP) for any property P(x), if P(0) holds All such formal details are irrelevant to the work- and if for all x in N, P(x) implies P(s(x)) ing mathematician’s use of arguments by induction then for all x in N,P(x) holds. on the natural numbers, but for the logician, the way a formal system S is specified can make the But this is too indefinite to become the subject difference between night and day. This is the case, of precise logical studies, and for that purpose in particular, concerning the questions whether S one needs to say exactly which properties P are ad- is consistent, i.e., no contradiction is provable from missible in (IP), and to do that one needs to spec- S, and whether S is complete, i.e., every sentence ify a formal language L within which we can sin- A is decided by S in the sense that either S proves gle out a class of well-formed formulas (wffs) A A or S proves ¬A. If neither A nor ¬A is provable which express the admitted properties. And to do in S then A is said to be undecidable by S, and S that we have to prescribe a list of basic symbols is said to be incomplete. and we have to say which finite sequences of basic As an example of how matters of consistency and symbols constitute well-formed terms and which completeness can change dramatically depending constitute wffs. Finally, we have to specify which on the formalization taken, consider the subsys- wffs are axioms (both logical and non-logical), and tem of PA obtained by restricting throughout to which relations between wffs are instances of rules terms and formulas that do not contain the mul- of inference. The wffs without free variables are tiplication symbol ×. That system, sometimes called those that constitute definite statements and are Presburger Arithmetic, was shown to be complete called the closed formulas or sentences of L. All by Moses Presburger in 1928, and his proof of of this is what goes into specifying a formal axiom completeness also gives a finite combinatorial system S. proof of its consistency.1 Gödel’s discovery in 1931 In the case of a formal version of the Peano ax- was that we have a radical change when we move ioms, once its basic symbols are specified, and the to the full axiom system PA. What Gödel showed logical symbols are taken to be ¬ (“not”), ∧ (“and”), was that PA is not complete and that, unlike Pres- ∨ (“or”), =⇒ (“implies”), ∀ (“for all”), and ∃ (“there burger Arithmetic, its consistency cannot be es- exists”), one puts in place of the Induction Princi- tablished by finite combinatorial means, at least not ple an Induction Axiom Scheme: those that can be formalized in PA. Before going (IA) A(0) ∧∀x(A(x)=⇒ A(s(x))) =⇒∀xA(x) , into the mathematical significance of these results, where A is an arbitrary wff of the let us take a closer look at how Gödel formulated language L and A(t) indicates the result of and established them not only for PA, but also for 2 substituting the term t for all free a very wide class of its extensions S. To do this occurrences of the variable x in A. he showed that the language of PA is much more expressively complete than appears on the sur- N.B. (IA) is not a single axiom but an infinite col- face. A primitive recursive (p.r.) function on N (in lection of axioms, each instance of which is given any number of arguments) is a function generated by some wff A of our language. from zero and successor both by explicit definition But what about the basic vocabulary of L? Be- and definition by recursion along N. A p.r. relation sides zero and successor, nothing of number- (which may be unary, i.e., a set) is a relation whose theoretical interest can be derived without ex- characteristic function is p.r. Gödel showed that panding it to include at least addition and multi- every p.r. function is definable in the language of plication. As shown by Dedekind, the existence of PA, and its defining equations can be proved there. those operations as given by their recursive defin- For example, the operations of exponentiation xy, ing conditions can be established using (IP) ap- the factorial x!, and the sequence of prime plied to predicates P involving quantification over functions. But for a formal axiom system PA (“Peano 1Presburger’s work was carried out as an “exercise” in a Arithmetic”) for elementary number theory, in seminar at the University of Warsaw run by Alfred Tarski. which one quantifies only over numbers, one needs His proof applies the method of elimination of quantifiers to posit those operations at the outset. The basic to show that every formula is equivalent to a proposi- tional combination of congruences. At its core it makes use vocabulary of PA is thus taken to consist of the con- of the Chinese Remainder Theorem giving conditions for stant symbol 0 and the operation symbols s,+, the existence of solutions of simultaneous congruences. and × together with the relation symbol =. Then 2Gödel’s initial statement of his results was for extensions the axioms indicated above for zero and successor of a variant P of the system of Principia Mathematica, but are supplemented by axioms giving the recursive a year later he announced his results more generally for characterizations of addition and multiplication, a system like PA in place of P; no new methods of proof namely: x +0=x, x + s(y)=s(x + y),x× 0=0, and were required. Nowadays it is known that much weaker x × s(y)=(x × y)+x. systems than PA suffice for his results.