The Godel Incompleteness Theorem from a Length-of-Proof PerspectiveAuthor(s): John DawsonSource: The American Mathematical Monthly, Vol. 86, No. 9 (Nov., 1979), pp. 740-747Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2322023 .

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THE GODEL INCOMPLETENESS THEOREM FROM A LENGTH-OF-PROOF PERSPECTIVE

JOHN DAWSON

1. Introduction. The Godel Incompleteness Theorem is one of the most profound and sensational results of twentieth-century mathematics. Its appearance in 1931 ([9], translated in [5, pp. 5-381 and [11, pp. 596-6161), shattered the hopes of those committed to Hilbert's formalist program and added impetus to the development of mathematical logic. Yet it is doubtful even today that many non-logicians appreciate the significance of Godel's theorem or understand its basic content.

In part, this lack of understanding may reflect a lingering belief that foundational matters are not really relevant to the concerns of the "working" mathematician. Aside from such prejudice, however, most expositions of the Incompleteness Theorem have focused on the paradoxical nature of its proof, with the result that the significance of the theorem has tended to become lost amid technical intricacies. Some may even regard the theorem as an abstruse curiosity, like the Paradox of the Liar, to which it is closely related.

It is the aim of the present article to examine Godel's theorem from a wholly different point of view that illuminates its meaning while largely ignoring its proof. Along the way we shall survey a number of important results in logic and consider recent progress toward the solution of an outstanding open problem.

2. Logical background. Although the impact of Godel's theorem extends beyond the con- fines of number theory, we shall restrict attention to that original context. At the outset, we must distinguish between the informal Peano axioms familiar to non-logicians and their logical formalization in first-order number theory (to which Godel's theorem applies). Following Landau [141, we take the former in the form:

1. Zero is a natural number. 2. Associated to each natural number is a unique natural number called its successor. 3. Zero itself is not the successor of any natural number. 4. Natural numbers having equal successors are themselves equal. 5. (The induction axiom) Any set of natural numbers containing zero, and containing the

successor of every number it contains, contains all natural numbers. On the basis of these axioms, together with definitions of the arithmetical operations, we can proceed by rules of informal logic (as Landau does) to derive all the basic theorems of elementary arithmetic. Further, as is well known, we can prove that any two structures satisfying all the axioms must be isomorphic; to do so, however, we must make essential use of the induction axiom, which alone among the axioms refers to sets of numbers rather than to numbers themselves.

In formalizing the axioms it is desirable to avoid set-theoretic notions. This can be done by rephrasing the induction axiom in terms closer to Peano's original formulation: "Any property true of zero and true of the successor of a number whenever true of the number itself is true of all natural numbers." We interpret "property" to mean "property expressible by a formula" of a suitable formal language X, and correspondingly we replace Peano's single induction axiom by an infinite induction schema, introducing an induction axiom for each particular formula of our language.

John Dawson received his Ph.D. in 1972 under D. W. Kueker and A. R. Blass at the University of Michigan. Since then he has been associated with the Pennsylvania State University system, where he is now an assistant professor. His research interests are in foundations and axiomatic set theory.-Editors

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THE GODEL INCOMPLETENESS THEOREM 741

The "suitable formal language" can be constructed in a variety of ways, but in any case we understand that it shall be a first-order language. That is:

1. The symbols of e shall consist of a countably infinite number of distinct variable symbols, zero or more constant and function symbols, the equality symbol and zero or more other relation symbols, and finitely many logical connective and quantifier symbols (say -, A, V, --, *-, V, and 3). It is understood that each relation or function symbol applies to only finitely many arguments. Parentheses may also be included to enhance readability.

2. The syntax of e shall be as follows: Terms ("nouns") are obtained inductively by applying zero or more function symbols to variables, constants, or previously defined terms. Atomic formulas ("simple sentences") are obtained by taking terms as arguments of relation symbols. More complex formulas are built up inductively from atomic formulas by applying the connectives and quantifiers.

3. The semantics of e shall be as follows: A structure 6f for e shall consist of a non-empty set A (regarded as the domain over which the variable symbols are allowed to vary) together with a distinguished element of A corresponding to each constant symbol of X, an n-ary relation on A corresponding to each n-ary relation symbol of e (with the identity relation corresponding to the equality symbol), and an n-ary function from A to A corresponding to each n-ary function symbol of C.

Note that in first-order languages, variables are always interpreted as ranging over elements of structures, never over sets of such elements. Note also that although from a set-theoretic point of view functions can be treated as a special kind of relations, in formal logic function and relation symbols are distinguished because they play syntactically different roles. We shall see in ?6 that this distinction can be critical in some contexts.

Formal languages are vehicles for the precise expression of mathematical theories; their great virtue is that they permit a rigorous distinction to be made between the fundamental mathemati- cal notions of truth and provability. Thus the semantical notion of truth is defined relative to structures: given an assignment of specific elements of a structure to correspond to the variable symbols of the language, the relations, functions, and constants of the structure provide a natural framework for interpreting the meaning (and hence the truth or falsity) of the atomic formulas. The truth or falsity of more complex formulas can then be defined inductively (a crucial idea due to Tarski [24], translated in [28, pp. 152-278]) by interpreting the meaning of the connectives and quantifiers according to the rules of classical logic. A formula is true in the structure 6 if it is true there regardless of the particular assignment of elements to its variable symbols; it is logically valid if it is true in every structure for the language.

To define the concept of proof, we first select a subset (perhaps all) of the logically valid formulas as logical axioms. We then construct a particular first-order theory by selecting another set of formulas (perhaps empty) as proper (or non-logical) axioms. Finally we specify a set of rules of inference (finitary truth-preserving functions from tuples of formulas to formulas). A proof in the theory then consists of a finite sequence of formulas, each of which is either an axiom or the result of applying a rule of inference to previous formulas in the list. The length ("number of lines") of the proof is the number of formulas in the sequence.

The choice of a particular language and of the set of proper axioms is dictated to some extent by theyparticular mathematical theory to be formalized. For number theory we assume that the corresponding formal language C has the binary relation symbol <, the binary function symbols + and *, the unary function symbol S (for successor), and the constant symbol 0. Following Shoenfield [22, pp. 22 and 204], we take as proper axioms:

1. (Vx)-I(Sx=O) 2. VxVy(Sx = Sy--x =y) 3. Vx(x+O=x) 4. VxVy(x + Sy = S(x +y)) 5. Vx(x. O = O)

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742 JOHN DAWSON (November

6. VxVy(x.Sy=(x.y) + x) 7. (Vx) - (x < ) 8. VxVy(x<Sy*x<yVx=y) 9. (Induction schema) For every formula 4>(x) in which the variable symbol x is free

(unquantified), the induction axiom ((D(O)ANtx((D(x)->(Sx)))-tx(D(x)- Inessentially, we may for convenience expand e5 to include constant symbols n for each natural number n, adding as axioms the definitions

ni= s...SO. n times

We shall refer to the resulting formalization of Peano arithmetic as PA.

3. Limitations of formalization: Completeness and incompleteness. We posit that the aim of mathematical research is to ascertain what is true about certain mathematically interesting structures, while the role of proofs is secondary, a means to an end. In particular, the formal definition of truth in purely semantic terms suggests the possibility of dispensing with proofs altogether. This can indeed be done for the logic of connectives (propositional logic), where truth tables provide an effective means for deciding the validity of an arbitrary propositional statement. However, when quantifiers and relations are introduced, it is no longer clear how to determine whether a statement holds in all possible interpretations: a fundamental result in first-order logic (Church's Undecidability Theorem) states that there is no effective procedure (algorithm) for determining whether an arbitrary formula of PA is logically valid. Indeed, the same is true for any first-order language containing, in addition to the equality symbol, at least one n-ary relation symbol for some n > 2. (See [15, p. 293].)

Church's theorem has far-reaching implications. It justifies the use of proofs in establishing arithmetical truth, yet at the same time it poses a problem for the selection of an adequate set of logical axioms-for in any satisfactory proof system, it must be possible to recognize the axioms (a condition known as recursive axiomatizability), and this will not be possible if the logical axioms are taken to consist of all logically valid formulas. So the question arises: Is it possible to choose a recursive set of logical axioms and a finite set of rules of inference from which all logically valid formulas can be derived as theorems?

Tentative solutions to this problem were proposed by Hilbert and his followers during the early decades of this century. The capstone to their efforts was provided by Godel in his doctoral dissertation ([8], translated in [11, pp. 582-591]): he showed that a scheme can be so chosen that in any first-order theory, the theorems coincide exactly with those formulas true in all models of (that is, structures satisfying) the proper axioms. This result, the Gddel Complete- ness Theorem, is a corollary to Godel's result that every consistent first-order theory (one in which no contradiction is provable) has a model. For if a formula is not provable in a given theory, its negation can be consistently added as a new axiom; a model for the augmented theory is then a model of the original theory in which the formula in question is false. (Strictly speaking, Godel obtained his results only for languages with countably many symbols; exten- sions to uncountable languages were made later by Mal'cev and others.)

The Completeness Theorem seemingly vindicated formalist hopes. But, like the notion of logical validity, it contained a weakness: reference to all models of a set of axioms. Unfor- tunately, it is seldom apparent how all models of a theory behave. Indeed, the discovery of non-Euclidean geometries became a turning point in the history of mathematics because it awakened recognition of the existence of non-standard models of axiomatic systems. Modem consistency and independence proofs in set theory (work in which G6del's name again figures prominently) are based on the same idea.

Still, the problem of unintended interpretations may appear irrelevant to number theory in view of the informal proof that Peano's axioms characterize the natural numbers up to

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1979] THE GODEL INCOMPLETENESS THEOREM 743

isomorphism. But the Completeness Theorem itself implies the existence of non-isomorphic models of formal number theory. For, since proofs are finite in length, a theory is consistent if every finite subset of its axioms is consistent. Hence by completeness we obtain the Compactness Theorem for first-order logic: A first-order theory has a model if each of its finite subtheories does. In particular, adjoining to PA a new constant symbol c together with the axioms c #iI for each natural number n gives a consistent theory whose models satisfy PA but are not isomorphic to the natural numbers. (This argument is due to Henkin; the existence of non-standard models of PA was first established by Skolem some ten years prior to Godel's work.)

The mere existence of non-standard models did not of itself dash the hope of discovering exactly what is true of the natural numbers, since isomorphism is a sufficient but not a necessary condition that two structures satisfy the same sentences of PA. It remained conceivable that all models of PA possessed the same first-order properties. But in 1931, only a year after the Completeness Theorem, Godel astounded the mathematical world by giving an example of a sentence of PA true in the natural numbers but unprovable in PA. Similar examples can be constructed in all consistent, recursively axiomatizable extensions of PA. This (First) Incomplete- ness Theorem thus established that the formulas true in all models of formal number theory form a proper subset of those true of the natural numbers; the special properties that distinguish the natural numbers from non-standard "impostors" cannot be discovered by means of formal proofs in the given deductive system.

One may still ask: Is there an alternative to the use of formal proofs in establishing truth in number theory? Paradoxically, the answer is yes, since G6del's proof of the Incompleteness Theorem exemplifies the possibility of demonstrating the truth of a formula via an informal proof of its formal undecidability (see ?5 below). However, the potential for discovering number-theoretically significant truths by this method has only very recently begun to be realized (an exciting advance due to J. B. Paris [18]). More to the point, Godel's techniques also yield that there is no effective procedure for deciding the truth or falsity (in the natural numbers) of an arbitrary formula of PA. This analogue of Church's theorem stands in sharp contrast to a positive result of Presburger [19], who had shown only two years earlier that there is an effective decision procedure for arithmetic without multiplication. The interested reader may consult [2, pp. 220-228], [6, pp. 188-192], or [15, pp. 237-240] for further details.

4. Universal statements and the induction axiom. In attempting to understand how a statement can be true yet unprovable, it is natural to study the relative complexity of first-order formulas. The most obvious measure of complexity is in terms of quantifier structure. A standard theorem of first-order logic states that every formula is logically equivalent to one in prenex form, that is, to a formula of the form Qoxo... Qnxn(D, where the Q's denote a sequence (possibly empty) of quantifiers and D is a quantifier-free formula. The most significant classification of prenex forms is according to the number of alternations of quantifier. In particular, formulas with prenex equivalents Vxo... Vxn,D are called universal (or Hll) and those with prenex equivalents 3xo. * 3xn,D are called existential (or El). Formulas with no free variables are said to be closed, and we shall refer to closed formulas of the form 1(Dl) as numerical instances of the formulas ?(x), 3x?, and VxD. (Closed formulas are also called sentences.)

It is reasonable to hope that PA is at least strong enough to prove every closed, quantifier- free formula that is true in the natural numbers. In fact, every true closed existential formula is a theorem of PA. (For a proof of this assertion see [22, pp. 209-21 1]. The result is especially pertinent to G6del's Second Incompleteness Theorem, not considered here, concerning the impossibility of internally formalized consistency proofs.) Trouble first arises when we consider universal formulas. Indeed, if we extend PA by adjoining new function symbols for certain (primitive) recursive functions, adding as axioms the definitions of these functions in PA, then in the extended theory Godel's canonical example of a true but unprovable sentence is a I formula. In PA itself there are true unprovable sentences of the form Vxo... Vxn(P(xo,..., xJ) #

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744 JOHN DAWSON [November

Q(xo..., x)), where P and Q denote polynomials in several variables with positive integer coefficients. That is, there exist Diophantine equations having no positive integer solutions, whose unsolvability cannot be proved in PA. (The existence of such equations follows from Matiyasevic's celebrated negative solution (1970) of Hilbert's Tenth Problem-see [4] for a complete account.)

The main tool for proving universal statements in PA is the induction schema, which we may regard as the source both of the strength and the weaknesses of the theory. Without induction, PA would be emasculated. Moreover, no finite set of induction axioms can replace the full induction schema, as Ryll-Nardzewski showed in [21] using non-standard models. In a sense, even the full schema contains too few formulas, since there are only countably many formulas of PA, but uncountably many properties (subsets) of the natural numbers to which (informal) induction should apply. This fundamental conflict underlies the apparent contradiction between the existence of non-standard models and the informal proof of the categoricity of Peano's axioms.

We prefer to regard the induction axiom as a sort of compression principle, since it provides a means of proving in finitely many steps the truth of infinitely many numerical instances of a number-theoretic statement. As such, however, the induction axioms are of restricted scope, since to apply them one must somehow establish the premise Vx(D(x)-A(Sx)). Of course, if VxD is true, then so is Vx(D(x)-A(Sx)), but the problem confronting the theorem prover is to justify independently the general passage from n to n + 1, so as to conclude Vx(D by induction.

Now there is a rather obvious way around this difficulty: namely, extend the concept of proof to include infinite well-ordered sequences of formulas and introduce the infinitary rule "From 1(0),1(1),..., deduce VxD." This so-called w-rule is obviously sound, but just as obviously worthless from a practical standpoint, since it is not clear how we can ever know each of 1(0), (D),... apart from already knowing VxD. The w-rule has nonetheless been recommended as a useful principle for introducing induction to high school students [27], and it has a most important philosophical significance. For it turns out that the resulting w-logic is the proper vehicle for investigating the structure of the standard natural numbers, in the sense that the theorems provable in PA using w-logic coincide exactly with the formulas true in IN. This special case of the w-Completeness Theorem (see ?5 below) may be paraphrased as follows.

METATHEOREM: Any formula true in the natural numbers but unprovable in (ordinary) PA must be true by virtue of an infinite number of "special cases."

It is reasonable to suspect that there may be universal statements of PA that hold in N merely through the dovetailing of an infinite number of inherently different instances. What is remarkable is that the w-Completeness Theorem implies that this is the only reason for the proof-theoretic inadequacy of PA.

It is our contention that the w-Completeness Theorem provides the best approach to understanding Godel's Incompleteness Theorem. The next section is devoted to a more detailed consideration of this idea.

5. w-completeness and omission of types. The w-Completeness TheorenL grew out of con- cepts employed by Godel in his 1931 paper, yet it did not appear until twenty-five years later. The history of the evolution of the ideas involved is an interesting subject deserving critical study, for it is punctuated by repeated instances of discovery, dormancy, rediscovery, and application. We pause here to offer a capsule summary of the course of events.

In 1927, Tarski introduced the concepts of w-consistency and w-completeness (cf. [28, footnote 2, p. 279]): A theory in the language of number theory is W-consistent if whenever each of 1(O), D(1),... is provable, we cannot also prove 3x( '1); if the provability of 4(0), 4(i),... actually implies the provability of VxO (without the w-rule), then the theory is w-complete. Godel invoked w-consistency as a hypothesis in his original statement of the Incompleteness

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1979] THE G6DEL INCOMPLETENESS THEOREM 745

Theorem, and Tarski noted some further consequences of w-consistency and w-completeness in 1933 ([25], translated in [28, pp. 279-295]). But in 1936 Rosser ([20], reprinted in [5, pp. 231-235]) showed that by appeal to a comparison of proof lengths (of P and -9), the assumption of w-consistency in Gbdel's theorem could be replaced by ordinary consistency.

Whether Rosser's result was responsible for stifling interest in the w-concepts is unclear; in any event the first forms of the w-Completeness Theorem were enunciated by Orey [16] in 1956 and Henkin [12] in 1957. A period of "ripening" then ensued during which the theorem quietly reappeared in various guises. Eventually the several variants coalesced into the form:

Every consistent, c-complete theory in the language of number theory has a model whose domain consists precisely of the elements interpreting the constant symbols h (a so-called w-model). (*)

There followed another period of benign indifference. Then, with the sudden growth of model theory (currently the most active and exciting research area in logic), the theorem underwent a remarkable metamorphosis, emerging as the Omitting Types Theorem. In the latter form, the theorem has become an important tool for model construction (see [3, pp. 78-84]). We shall not need this more general version, but its curious name deserves some words of explanation. A (1 -) typer is a set of formulas in a single free variable x which is finitely satisfiable (that is, 3x(ai(x)A *. Aan(x)) is consistent for any finite subset (a1(x),...,Ua(x)} of r). A type is realized in a structure g if some assignment to x causes all its formulas to be satisfied in C; otherwise ? is said to omit the type. A type F is principal with respect to a theory T if some formula ?(x) consistent with T generates r, that is, if T U (3x1'(x)} is consistent and the formula Vx(QI(x)-*9(x)) is provable in T for every 9 in r. The Omitting Types Theorem asserts the existence of models omitting non-principal types. In particular, an c-model is a model omitting the type {x fh: n <co}. (At this point the reader should recall our construction of a non-standard model of PA.)

To conclude this section, we will use (*) to justify our earlier assertion that the theorems provable in PA using w-logic coincide with the formulas true in N. (For a proof of (*) itself, see [26, p. 27].) Additionally, we will show that any universal sentence neither provable nor disprovable in (ordinary) PA must be true in N. Toward those ends, we introduce the following notation. Given a theory T and a sentence 9, we write TF to mean "T proves 9 in ordinary logic" and TF,,9 to mean "T proves 9 in w-logic." We also write To for {9: TFh,,, }. Finally, we let Th N (the theory of N) denote the set of all closed formulas true in the standard natural numbers. Suppose then that a ETh N but a (PAO. (Note that we certainly have PAM CTh N.) Then the theory T= PA" U { -- a) is consistent. Moreover, T is W-complete. (For if THI(O), TF 0(1),... yet T fails to prove Vx(D(x), then by the Deduction Theorem of propositional logic, PA m a-A(O), PAOF m a-(1),..., but PA" does not prove -m a-'Vx0(x). However, PA"f- Vx(-i a-m I)(x)), and since m a is closed, Vx( m a-A(x)) is logically equivalent to -m a--x(x).) Hence by (*), T has an w-model. But since T extends PA, any w-model of T is isomorphic to N. This is a contradiction, since -m a e T but a eTh N.

Finally, suppose a is any universal statement undecidable in PA. Then a E Th N. For otherwise N would satisfy 3xo... 3xJ(m 1), where Vx0.. . Vx,* 4 is a prenex equivalent of a. But then 3xo0 * 3x,(n ? ) would be a true closed existential formula, so PA would prove 3xo. *3x,(, ) and thereby refute a.

6. Proof lengths-A conjecture of Kreisel and a theorem of Parikh. Throughout the model- theoretic "gestation period" described in the preceding section, a parallel development was taking place in proof theory. In one year (1936) three important papers appeared, all based on considerations of the length or complexity of proofs in formal systems. One, Rosser's improve- ment of the Incompleteness Theorem, has already been mentioned. The others were Gentzen's consistency proof for arithmetic ([7], translated in [23, pp. 132-213]), the first of its kind,

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7AA /46 _T1JOHN DAWSON [November

employing a transfinite induction on the complexity of derivations; and a brief and little-noticed article by Godel himself ([10], translated in [5, pp. 82-83D. The latter, titled simply "On the Length of Proofs," pointed out that, by passing to systems of "higher type" (allowing sets of integers, sets of sets of integers, etc.), not only can new theorems be proved, but "it becomes possible to shorten extraordinarily infinitely many of the proofs already available."

Translated only in 1965, Godel's length-of-proof paper was largely ignored until after the advent of the computer revolution, when concern for efficiency of computation led to Manuel Blum's creation of computational complexity theory, and Godel's proof-shortening result was resurrected to become the progenitor of a whole class of speed-up theorems (see [1, pp. 253 and 261-263D. Somewhat earlier (1955) Kreisel and Wang [13] also studied proof shortening. They considered various measures of "proof of length < n," including proof in at most n lines, and obtained results related to Godel's Second Incompleteness Theorem.

The possibility of some kind of synthesis of the model-theoretic and proof-theoretic perspec- tives must have suggested itself at about this time. To see how this might be possible, let us return to the ideas of ?4. There we saw how a universal statement could fail to be provable despite the provability of each of its numerical instances. In particular, it can happen that the lengths of proofs of the instances 4D(n) depend on n. Thus, given ?(x), let PL(Q) = {im: For some n, the shortest proof of 1(,i) requires >m lines). Using Godel's techniques, one can construct (in a recursive extension of PA) a true universal statement Vx(D for which PLQ() contains all natural numbers; it follows that VxD will not be provable in PA, since the schema Vx4D-$('r), for any term i, is a part of the underlying logic of any first-order theory. On the other hand, what of the converse problem? If there is a uniform bound m to the lengths of shortest proofs of the instances 4(n-) (a situation we abbreviate by (Vn)(PAFQ(n-))), is it necessary that VxI be provable in PA? A conjecture attributed to Kreisel asserts that this is indeed the case.

Kreisel's conjecture is certainly a bold one, since even if each 1(nf) is provable in at most m lines there is no reason to think that the individual proofs might not still incorporate an infinite number of different arguments. The conjecture has gained greater credence, however, from a remarkable recent result of Parikh [17]. Parikh's theorem applies not to PA, but to a seemingly inessential variant which, following Parikh, we denote by PA*. PA* differs from PA only in that the binary function symbols + and * are replaced by ternary relation symbols A and M, with corresponding axioms added stating that A and M represent functions. It follows that each formula 4 of PA has a translation 4V in PA* having the same meaning, and that PASO if and only if PA*F4*, so that PA and PA* prove the "same" theorems. Proofs in PA* tend to be more "cumbersome" (in particular, longer) than those in PA, but the terms of PA* are much simpler, consisting just of the successor function S applied some finite number of times to either 0 or a variable symbol. As a result, Parikh was able to establish that if (3m) (In) (PA*F,m4(ni)), then PA*FVx4, that is, the analogue of Kreisel's conjecture holds for PA*.

Parikh's result comes maddeningly close to settling Kreisel's conjecture, but the complexity of terms in PA blocks any obvious extension of Parikh's proof. It is certainly true that if the translates *(nQ) of each ?(ni) all have proofs in PA* of lengths <m, then VxZ* (which is just (Vx4)*) will be provable in PA*, and hence Vx4 will be provable in PA. But it is not clear that boundedness of proof lengths in PA implies any corresponding boundedness for proof lengths of translates in PA*. With Parikh's theorem we have in a sense come full circle, for Parikh's proof depends, of all things, on Presburger's early decidability result (cf. ?3 above). We have only to make a final speculation. The mathematical literature already contains examples of proofs obtained by computer verification of myriads of cases (most notably the recent solution of the Four-Color Problem), while the history of number theory is rife with similar instances of unsolved conjectures (Fermat's Last "Theorem," Goldbach's Conjecture, etc.) proved by a variety of methods to hold in a great many particular cases. Could some of these famous conjectures be natural examples of the undecidability phenomena we have been discussing?

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1979] THE TWENTIETH INTERNATIONAL MATHEMATICAL OLYMPIAD 747

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DEPARTMENT OF MATHEMATICS, THE PENNsYLvANA STATE UNIERSITY, YORK CAMPUS, YORK, PENNSYLVANIA 17403.

THE TWENTIETH INTERNATIONAL MATHEMATICAL OLYMPIAD

SAMUEL L. GREITZER

The Twentieth International Mathematical Olympiad took place in Bucharest, Romania, on July 6 and 7, 1978. Thanks to the Office of Naval Research, our team had its three-week training session at the U.S. Naval Academy, Annapolis. The Army Research Office provided funds for our travel, and on June 30 we started our journey. Unfortunately, a long delay at Dulles Airport resulted in our arriving at Frankfurt too late to make our connecting flight to Bucharest, where we arrived one day late.

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