The Impact of the Incompleteness Theorems on Mathematics, Volume

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.

APRIL 2006 NOTICES OF THE AMS 435 numbers px , each of which is p.r., can all be rep- Automatically, every 1-consistent system is con- resented in this way in PA, facts that are not at all sistent, but the converse is not true: by (1), if S is 3 obvious. Each instance of a p.r. relation is decid- consistent it remains consistent when we add ¬DS able by PA; for example if R is a binary p.r. rela- to it as a new axiom, and the resulting system is tion then for each n, m ∈ N, either PA proves not 1-consistent. The following is Gödel’s first in- R(n, m) or it proves ¬R(n, m). completeness theorem:5

Gödel’s Incompleteness Theorems (2) if S is 1-consistent then DS is To apply these notions to the language and de- undecidable by S; hence S is not complete. ductive structure of PA, Gödel assigned natural It turns out that only the first part, (1), is needed numbers to the basic symbols. Then any finite se- for his second incompleteness theorem. Let C be quence σ of symbols gets coded by a number #σ, the sentence ¬(0 = 0), so S is consistent if and only say, using prime power representation; #σ is nowa- if C is not provable in S; this is expressed by the days called the Gödel number (g.n.) of σ. A relation ∀-sentence ¬ProvS (#C), which is denoted ConS. By R between syntactic objects (terms, formulas, etc.) formalizing the proof of (1) it may be shown that is said to be p.r. if the corresponding relation be- the formal implication ConS =⇒ ProvS (#DS ) is tween g.n.’s is p.r. For example, with a basic finite provable in PA. But since ¬ProvS (#DS )=⇒ DS in vocabulary, the sets of terms and wffs are both p.r. PA by the diagonal construction, we have Finally a formal system S for such a language is said ConS =⇒ DS too. Hence: to be p.r. if its set of axioms and its rules of inference are both p.r. The formal system PA and its sub- (3) if S is consistent then S does not prove ConS. systems defined above are all p.r. That is Gödel’s second incompleteness theo- Throughout the following, S is assumed to be rem. Its impact on Hilbert’s consistency program any p.r. formal system that extends PA. Denote by has been much discussed by logicians and histo- ProofS (x, y) the relation which holds just in case y rians and philosophers of mathematics and will not is the g.n. of a proof in S of a wff with g.n. x. Then be gone into here, except to say that it is generally Proof (x, y) is p.r. Using its definition in PA, the for- S agreed that the program as originally conceived can- mula ∃y Proof (x, y) expresses that x is the g.n. of S not be carried out for PA or any of its extensions. a provable formula; this is denoted Prov (x). Finally, S However, various modified forms of the program for each wff A, Prov (#A) expresses in the language S have been and continue to be vigorously pursued of PA that A is provable. By a diagonal argument, within the area of logic called proof theory, inau- Gödel constructed a closed wff D which is prov- S gurated by Hilbert as the tool to carry out his pro- ably equivalent in PA to ¬Prov (#D ); more loosely, S S gram. I recommend Zach (2003) (readily accessible “D says of itself that it is not provable in S.” And, S online) for an excellent overview and introduction indeed, to the literature on Hilbert’s program. (1) if S is consistent, D is not provable in S. S Gödel’s Theorems and Unsettled Hence, in ordinary informal terms, DS is true, Mathematical Problems so S cannot establish all arithmetical truths. This With this background in place we can now return is one way that Gödel’s first incompleteness theo- to the question of the impact of the incompleteness rem is often stated, but actually (1) is only the first theorems on mathematics; in that respect it is part of the way that he stated it. For that we need mainly the first incompleteness theorem that is of a few more slightly technical notions. A sentence concern, and indeed only the first part of it, namely ∃ A of the language of PA is said to be in -form if, (1). A common complaint about this result is that ∃ up to equivalence, it is of the form yR(y) where it just uses the diagonal method to “cook up” an ∀ R defines a p.r. set, and A is said to be in -form example of an undecidable statement. What one if, up to equivalence, ¬A is in ∃-form, or what would really like to show undecidable by PA or comes to the same, if A can be expressed in the some other formal system is a natural number- form ∀yR (y) with R p.r.4 Thus D is in ∀-form 1 1 S theoretical or combinatorial statement of prior in- and its negation is in ∃-form. S is said to be terest. The situation is analogous to Cantor’s use 1-consistent if every ∃-sentence provable in S is true. of the diagonal method to infer the existence of transcendental numbers from the denumerability of 3The way that is done might interest number theorists; see the set of algebraic numbers; however, that did Franzén (2004), Ch. 4, for an exposition. not provide any natural example. The existence of 4Standard logical terminology for ∃-form and ∀-form is 0 0 transcendentals had previously been established Σ1 -form and Π1 -form, respectively. It should be noted that the formulas in ∃-form are closed under existential by an explicit but artificial example by Liouville. quantification and those in ∀-form under universal quan- 5 tification. That is, like quantifiers can be collapsed to a sin- Gödel used a stronger, purely syntactic, assumption in gle one of that type. place of 1-consistency, that he called ω-consistency.

436 NOTICES OF THE AMS VOLUME 53, NUMBER 4 Neither argument helped to show that e and π, p(x1,...,xn)=0, where p is a suitable polynomial among other reals, are transcendental, but they with integer coefficients. So each ∀-sentence A is did at least show that questions of transcendence equivalent to the non-solvability of a suitable dio- are non-vacuous. Similarly, Gödel’s first incom- phantine equation, in particular, sentences known pleteness theorem shows that the question of to be undecidable in particular systems such as the undecidability of sentences by PA or any one of its Gödel sentences DS. The trouble with this result consistent extensions is non-vacuous. That sug- compared to open questions in the literature about gests looking for natural arithmetical statements the solvability of specific diophantine equations in which have resisted attack so far to try to see whether that is because they are not decided by sys- two or three variables or of low degree is that the tems that formalize a significant part of mathe- best value known for the above representation in matical practice, and in particular to look for such terms of number of variables is n =9, and in terms statements in ∀-form. An obvious candidate for a of degree d with a much larger number of variables long time was the Fermat conjecture; now that we is d =4(cf. Jones 1982). know it is true, it would be interesting to see just what principles are needed to establish it from a log- Combinatorial Independence Results ical point of view. Some logicians have speculated Things look more promising if we consider ∀∃- that it has an elementary proof that can be for- sentences, i.e., those that can be brought to the form malized in PA, but we don’t have any evidence so ∀x∃yR(x, y) with R p.r.6 The statement that there far to settle this one way or the other. Another ob- are infinitely many y’s satisfying a p.r. condition vious candidate is the Goldbach conjecture; in- P(y) is an example of a ∀∃-sentence, since it is ex- deed, Gödel often referred to his independent state- pressed by ∀x∃y(y>x∧ P(y)) . In particular, the ments as being “of Goldbach type”, by which he simply meant that they are both expressible in ∀- twin prime conjecture has this form. Again, no form. A far less obvious candidate is the Riemann problems of prior mathematical interest that are Hypothesis; Georg Kreisel showed that this is equiv- in ∀∃-form have been shown to be undecidable in alent to a ∀-statement (see Davis, Matijaseviˇc, and PA or one of its extensions. However, in 1977, Jeff Robinson (1976), p. 335, for an explicit presenta- Paris and Leo Harrington published a proof of the tion of such a sentence). No example like these has independence from PA of a modified form of the been shown to be independent of PA or any of its Finite Ramsey Theorem. The latter says that for presumably consistent extensions. each n, r and k there is an m such that for every r-colored partition π of the n-element subsets of Unsolvable Diophantine Problems M = {l,m} there is a subset H of M of cardinality Consider any system S containing PA that is known at least k such that H is homogeneous for π, i.e., or assumed to be consistent and suppose that A all n-element subsets of H are assigned the same is a ∀-sentence conjectured to be undecidable by S. It turns out that proving its undecidability would color by π . The Paris-Harrington modification ≥ automatically show A to be true, since ¬A is equiv- consists in adding the condition that card(H) alent to a ∃-sentence ∃yR(y); thus if ¬A were true min(H). It may be seen that this statement, call it there would be an n such that R(n) is true, hence PH, is in ∀∃-form. Their result is that PH is not provable in PA and thence in S. So, finally, ¬A provable in PA. But they also showed that PH is would be provable in S, contradicting the sup- true, since it is a consequence of the Infinite Ram- posed undecidability of A by S. The odd thing sey Theorem. The way that PH was shown to be in- ∀ about this is that if we want to show a -sentence dependent of PA was to prove that it implies Con PA; undecidable by a given S, we better expect it to be in fact, it implies the formal statement of the true. And if we can show it to be true by one means 1-consistency of PA. That work gave rise to a num- or another, who cares (other than the logician who ber of similar independence results for stronger sys- is interested in exactly what depends on what) tems S, in each case yielding a ∀∃-sentence A whether it can or can’t be proved in S? Still, the first S which is a variant of a combinatorial result already incompleteness theorem is tantalizing for its prospects in this direction. The closest that one has in the literature such that AS is true but unprovable come is due to the work of Martin Davis, Hilary from S on the assumption that S is 1-consistent. The Putnam, Julia Robinson, and Yuri Matiyasevich re- proof consists in showing that AS implies (and is sulting, finally, in the latter’s negative resolution in some cases equivalent to) the formal statement of Hilbert’s 10th Problem on the algorithmic solv- of its 1-consistency. However, no example is known ability of diophantine equations (cf. Matiyasevich of an unprovable ∀∃-sentence whose truth has 1993). It follows from their work that every ∃- been a matter of prior conjecture. sentence is equivalent in PA to the existence of 6 0 natural numbers x1,...,xn such that Standard logical terminology for these is Π2 sentences.

APRIL 2006 NOTICES OF THE AMS 437 Set Theory and Incompleteness latter axiom, roughly speaking, means Gödel signaled a move into more speculative ter- nothing else but that the totality of sets ritory in footnote 48a (evidently an afterthought) obtainable by exclusive use of the of his 1931 paper: processes of formation of sets expressed in the other axioms forms again As will be shown in Part II of this paper, a set (and, therefore, a new basis for a the true reason for the incompleteness further application of these processes). inherent in all formal systems of math- Other axioms of infinity have been for- ematics is that the formation of ever mulated by P. Mahlo. … These axioms higher types can be continued into the show clearly, not only that the axiomatic transfinite… For it can be shown that the system of set theory as known today is undecidable propositions constructed incomplete, but also that it can be sup- here become decidable whenever ap- plemented without arbitrariness by new propriate higher types are added… An axioms which are only the natural con- analogous situation prevails for the ax- 7 tinuation of those set up so far. (Gödel iom system of set theory. 1947, as reprinted in 1990, p. 182). The reason for this, roughly speaking, is that for Whether or not one agreed with Gödel about the each system S the notion of truth for the language ontological underpinnings of set theory and in par- of S can be developed in an axiomatic system S ticular about the truth or falsity of CH, in the fol- for the subsets of the domain of interpretation of lowing years it was widely believed to be indepen- S; then in S one can prove by induction that the dent of ZFC; that was finally demonstrated in 1963 statements provable in S are all true, and hence that by Paul Cohen using a new method of building S is consistent. Implicit in the quote is that S models of set theory. And, contrary to Gödel’s ex- ought to be accepted if S is accepted. Later, in his pectations, it has subsequently been shown by an famous article on Cantor’s Continuum Problem expansion of Cohen’s method that CH is undecid- (1947), Gödel pointed to the need for new set- able in every plausible extension of ZFC that has theoretic axioms to settle specifically set-theoretic been considered so far, at least along the lines of problems, such as the Continuum Hypothesis (CH). Gödel’s proposal (cf. Martin 1976 and Kanamori At that point, it was only known as a result of 2003). For the most recent work on CH, see the con- his earlier work that AC (the Axiom of Choice) and clusion of Floyd and Kanamori (this issue). CH are consistent relative to Zermelo-Fraenkel ax- But what about arithmetical problems? For a iomatic set theory ZF.8 In Gödel’s 1947 article he number of years, Harvey Friedman has been work- argued that CH is a definite mathematical problem, ing to produce mathematically perspicuous finite and, in fact, he conjectured that it is false while all combinatorial ∀∃-statements A whose proof re- the axioms of ZFC (= ZF + AC) are true. Hence CH quires the use of many Mahlo cardinals and even must be independent of ZFC; he thus concluded stronger axioms of infinity (such as the so-called that one will need new axioms to determine the car- subtle cardinals) and has come up with a variety of dinal number of the continuum. In particular, he candidates; for a fairly recent report, including work suggested for that purpose the possible use of ax- 9 ioms of infinity not provable in ZFC: in progress, see Friedman (2000). The strategy for establishing that such a statement A needs a sys- The simplest of these … assert the ex- tem S incorporating the strong axioms in question istence of inaccessible numbers… The is like that above: one shows that A is equivalent to (or in certain cases is slightly stronger than) the 7 Part II of Gödel (1931) never appeared. Also promised 1-consistency of S. In my discussion of this in Fe- for it was a full proof of the second incompleteness theo- ferman (2000), p. 407, I wrote: “In my view, it is beg- rem, the idea for which was only indicated in Part I. He later explained that since the second incompleteness the- ging the question to claim this shows we need ax- orem had been readily accepted there was no need to ioms of large cardinals in order to demonstrate the publish a complete proof. Actually, the impact of Gödel’s truth of such A, since this only shows that we work was not so rapid as this suggests; the only one who ‘need’ their 1-consistency. However plausible we immediately grasped the first incompleteness theorem might find the latter for one reason or another, it was John von Neumann, who then went on to see for him- doesn’t follow that we should accept those axioms self that the second incompleteness theorem must hold. Others were much slower to absorb the significance of 9More recently, Friedman has announced the need for such Gödel’s results (cf. Dawson 1997, pp. 72–75.) The first de- large cardinal axioms in order to prove a certain combi- tailed proof of the second incompleteness theorem for a natorial statement A that can be expressed in ∀-form; see system Z equivalent to PA appeared in Hilbert and Bernays the final section of Davis (this issue). Here, A implies ConS (1939). and is itself provable in S for S and S embodying suit- 8See Floyd and Kanamori in this issue of the Notices. able large cardinal axioms.

438 NOTICES OF THE AMS VOLUME 53, NUMBER 4 themselves as first-class mathematical principles.” Mathematics 28 (1976), Amer. Math. Soc., Providence, (Cf. also op. cit. p. 412). RI. [2] M. DAVIS, The incompleteness theorem (this issue of Prospects and Practice the Notices). As things stand today, these explorations of the set- [3] M. DAVIS, Y. MATIJASEVICˇ, and J. ROBINSON, Hilbert’s tenth theoretical stratosphere are clearly irrelevant to problem. Diophantine equations: positive aspects of the concerns of most working mathematicians. A a negative solution, in Browder (1976), 323–378. reason for this was also given by Gödel near the [4] J. W. DAWSON JR., Logical Dilemmas. The Life and Work outset of his 1951 Gibbs lecture (posthumously of Kurt Gödel, A. K. Peters, Ltd., Wellesley, MA, 1997. published in 1995), where he said that the “phe- [5] S. FEFERMAN, In the Light of Logic, Oxford University nomenon of the inexhaustibility of mathematics” Press, New York, 1998. [6] , Why the programs for new axioms need to be follows from the fact that “the very formulation of ——— questioned, Bull. Symbolic Logic 6 (2000), 401–413. the axioms [of set theory over the natural numbers] [7] J. FLOYD and A. KANAMORI, How Gödel transformed set up to a certain stage gives rise to the next axiom. theory (this issue of the Notices). It is true that in the mathematics of today the [8] H. FRIEDMAN, Normal mathematics will need new axioms, higher levels of this hierarchy are practically never Bull. Symbolic Logic 6 (2000), 434–446. used. It is safe to say that 99.9% of present-day [9] T. FRANZÉN, Inexhaustibility, Lecture Notes in Logic 28 mathematics is contained in the first three levels (2004), Assoc. for Symbolic Logic, A.K. Peters, Ltd., of this hierarchy.” In fact, modern logical studies Wellesley (distribs.). have shown that the system corresponding to the [10] K. GÖDEL, Über formal unentscheidbare Sätze der second level of this hierarchy, called second-order Principia Mathematica und verwandter Systeme I, arithmetic or analysis and dealing with the theory Monatshefte für Mathematik und Physik 38 (1931), of sets of natural numbers, already accounts for the 173–198. Reprinted with an English translation in bulk of present-day mathematics. Indeed, much Gödel (1986), 144–195. weaker systems suffice, as is demonstrated in [11] ——— , What is Cantor’s continuum problem?, Amer. Simpson (1999). Even more, I have conjectured and Math. Monthly 54 (1947), 515–525, errata, 55, 151. verified to a considerable extent that all of current Reprinted in Gödel (1990), 176–187. scientifically applicable mathematics can be for- [12] ——— , Some basic theorems on the foundations of malized in a system that is proof-theoretically no mathematics and their implications, in Gödel (1995), stronger than PA (cf. Feferman 1998, Ch. 14). pp. 304–323. [The 1951 Gibbs lecture.] Whether or not the kind of inexhaustibility of [13] ——— , Collected Works, Vol. I. Publications 1929–1936, mathematics discovered by Gödel is relevant to (S. Feferman, et al., eds.), Oxford University Press, New present-day pure and applied mathematics, there York, 1986. is a different kind of inexhaustibility which is [14] ——— , Collected Works, Vol. II. Publications 1938- clearly significant for practice: no matter which 1974, (S. Feferman, et al., eds.), Oxford University axiomatic system S is taken to underlie one’s work Press, New York, 1990. at any given stage in the development of our sub- [15] ——— , Collected Works, Vol. III. Unpublished Essays ject, there is a potential infinity of propositions that and Lectures, (S. Feferman, et al., eds.), Oxford Uni- versity Press, New York, 1995. can be demonstrated in S, and at any moment, [16] D. HILBERT and P. BERNAYS, Grundlagen der Mathe- only a finite number of them have been estab- matik, Vol. II, Springer-Verlag, Berlin, 1939. Second, re- lished. Experience shows that significant progress vised edition, 1968. at each such point depends to an enormous extent [17] J. P. JONES, Universal Diophantine equation, J. Sym- on creative ingenuity in the exploitation of ac- bolic Logic 47 (1982), 549–571. cepted principles rather than essentially new prin- [18] A. KANAMORI, The Higher Infinite, Springer-Verlag, ciples. But Gödel’s theorems will always call us to Berlin, 2nd edition, 2003. try to find out what lies beyond them. [19] D. A. MARTIN, Hilbert’s first problem: The continuum Note to the reader: Gödel’s incompleteness pa- hypothesis, in Browder (1976), 81–92. per (1931) is a classic of its kind; elegantly organized [20] Y. MATIYASEVICH, Hilbert’s Tenth Problem, MIT Press, and clearly presented, it progresses steadily and ef- Cambridge, 1993. ficiently from start to finish, with no wasted energy. [21] J. PARIS and L. HARRINGTON, A mathematical incom- The reader can find it in the German original along pleteness in Peano Arithmetic, in Handbook of Math- with a convenient facing English translation in ematical Logic, (J. Barwise, ed.), 1133–1142, North- Vol. I of his Collected Works (1986). I recommend Holland, Amsterdam, 1977. it highly to all who are interested in this landmark [22] S. G. SIMPSON, Subsystems of Second Order Arithmetic, in the history of our subject. Springer-Verlag, Berlin, 1999. [23] R. ZACH, Hilbert’s program, Stanford Encyclopedia References of Philosophy, (E. N. Zalta, ed.), http://plato. [1] F. E. BROWDER (ed.), Mathematical Developments Aris- stanford.edu/archives/fall2003/entries/ ing from Hilbert Problems, Proc. Symposia in Pure hilbert-program/ (2003).

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