Diophantine Sets and Algorithmic Undecidability

Abstract

In §4 of Chapter V we showed that enumerable sets are the same thing as projections of level sets of primitive recursive functions. The projections of the level sets of a special kind of primitive recursive function—polynomials with coefficients in Z⁺—are called Diophantine sets. We note that this class does not become any larger if we allow the coefficients in the polynomial to lie in Z. The basic purpose of this chapter is to prove the following deep result: 1.2. Theorem (M. Davis, H. Putnam, J. Robinson, Yu. Matiyasevič). All enumerable sets are Diophantine. The plan of proof is described in §2. §§3–7 contain the intricate yet completely elementary constructions that make up the proof itself; these sections are not essential for understanding the subsequent material, and may be omitted if the reader so desires.