On the distribution of independent formulae of number theory

• Authors: David A. Plaisted
• Venue: STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
• Year: 1980

It follows by Gödel's incompleteness Theorem [6] that any effective sound system of logic for elementary arithmetic must be incomplete. We show that in any effective sound system of logic for elementary arithmetic, there exist valid unprovable formulae that are quite small relative to the complexity of the logical system. Also, such formulae are quite dense. In fact, the situation is about as bad as it could possibly be. That is , no infinite axiom system for elementary arithmetic can be much more compact than a listing of all the valid formulae. The unprovable formulae we construct express predicates in the classes &Sgr2 and &pgr; 2 of the Kleene arithmetic hierarchy [12]. The construction yields a set of short formulae, at least one of which must be valid and unprovable, but the construction does not tell us which one is valid and unprovable. We also construct small valid unprovable formulae expressing a relation in the class &pgr; 1 of the Kleene arithmetic hierarchy. These latter formulae are not as small. We do not know how small the independent formulae corresponding to the class &pgr; 1 are. The constructions are based on the concept of a restricted oracle, first introduced in [10] and further developed in [11]. The proofs make use of the recent result of Matijasevic [9] concerning the relationship between recursively enumerable sets and Diophantine equations.