Fabrice Pataut INCOMPLETENESS , CONSTRUCTIVISM AND TRUTH

Although Gödel proved the first incompleteness theorem by intuitionistically respectable means, Gödel’s formula, true although undecidable, seems to offer a counter-example to the general constructivist or anti-realist claim that truth may not transcend recognizability in principle. It is argued here that our understanding of the formula consists in a knowledge of its truth-conditions, that it is true in a minimal sense (in virtue of a reductio ad absurdum) and, finally, that it is recognized as such given the consistency and ω-consistency of P . The philosophical lesson to be drawn from Gödel’s proof is that our capacities for justification in favour of minimal truth exceed what is strictly speaking formally provable in P by means of an algorithm. Received June 20, 1998 © 1998 by Nicolaus Copernicus University Does Gödel’s first incompleteness theorem have consequences for the question whether intuitionistic semantics should be preferred to classical semantics and, given that the acceptance of semantic principles entails the acceptance of corresponding logical laws, for the question whether intuitionistic logic should be preferred to classical logic? Although some non-constructive proofs of the theorem have been proposed after the publication of Gödel’s 1931 result, e.g. by Boolos (1989), it may seem obvious that Gödel’s original proof cannot have any bearing on the issue of the choice of logic, for it remains conspicuously neutral between the classical and the intuitionistic standpoints. Gödel thought it worthwhile to remind his readers that his proof was indeed constructive. He pointed to the fact that the first incompleteness result had been obtained “in an intuitionistically unobjectionable manner” (Gödel [1931: 189], 1986a: 177) and offered as a warrant for his claim that “all existential statements [Existentialbehauptungen] occurring in the proof [were] based upon Theorem V [i.e. the theorem immediately preceding the first incompleteness theorem], which, as is easily seen, is unobjectionable from the intuitionistic point of view” (Gödel loc. cit.: note 45a). In Kleene’s terminology, Theorem V states that every primitive recursive relation is numeralwise expressible in P , where P is the system obtained from Whitehead and Russell’s Principia Mathematica, without the ramification of the types, taking the natural numbers as the lowest type and adding their usual Peano axioms (Kleene 1986: 132). When expressed formally, without reference to any particular interpretation of the formulas of P , and in Gödel’s own terminology, which favours the indirect talk of ‘Gödel’ numbers and concepts applying to those numbers rather than a direct talk of the formal objects, Theorem V claims that: Gödel’s ([1931], 1986a) Theorem V For every recursive relation R(x1, . . . , xn) there exists an n-place relation sign r (with the free variables u1, u2, . . . , un) such that for all n-tuples of numbers (x1, . . . , xn) we have R(x1, . . . , xn) → Bew[Sb(r u1...un Z(x1)...Z(xn) )], R(x1, . . . , xn) → Bew[Neg(Sb(r u1...un Z(x1)...Z(xn) ))]. © 1998 by Nicolaus Copernicus University Incompleteness, Constructivism and Truth 65 Gödel gives an outline of the proof and notes, on this occasion, that Theorem V is itself “of course, [. . . ] a consequence of the fact that in the case of a recursive relation R it can, for every n-tuple of numbers, be decided on the basis of the axioms of the system P whether the relation R obtains or not” (Gödel op. cit.: [186n39], 171n39). This, it must be noted, can also be decided by means of procedures which remain unobjectionable from the intuitionistic standpoint. One may object that, as far as a choice in favour of a given semantics is concerned, either classical or otherwise, it hardly matters whether or not Gödel’s proof is intuitionistically safe logically speaking, for, although the acceptance of semantic principles normally entails the acceptance of corresponding logical laws, the converse does not hold.1 If this conception of the relation between semantic principles and logical laws is correct, Gödel could very well have used, say, the law of excluded middle in carrying out his proof without thereby committing himself to the principle of bivalence (every statement is either true or false) ; or he could have used the law of double negation elimination without thereby committing himself to the principle of stability (every statement which is not false is true). If the remark applies to fundamental logical laws and fundamental semantic principles quite generally, and not only to excluded middle and double negation elimination, then the non-constructive proofs of the theorem should not imply any semantic claim which a constructivist or intuitionist would have to reject. The problem is that they do indeed imply such claims. Boolos’ proof, in particular, establishes the existence of an undecidable statement of arithmetic, just like Gödel’s ; but, unlike Gödel’s, it does not provide an effective procedure for producing it. Let a correct algorithm M be an algorithm which may not list a false statement of arithmetic. A truth omitted byM is just a true sentence of arithmetic not listed byM . Boolos’ proof establishes the existence of such a true statement, but the statement is recognized to be true only classically and not constructively. The problem, now, is whether the rejection of a given logical law entails the rejection of the corresponding semantic principle. Of course, if the acceptance of, say, bivalence across the board entails the endorsement of excluded middle, then, by contraposition, the rejection of excluded middle entails the rejection of bivalence. If acceptance goes one way, from the semantic to the logical, then rejection must go the other way, from the logical to the semantic. Maybe we would like to treat logical laws and semantic 1 The point has been made, e.g., by Michael Dummett in Dummett (1978: xix). © 1998 by Nicolaus Copernicus University