Diophantine induction
Let \(I\exists^-_ 1\) be the subsystem of Peano arithmetic obtained by restricting the induction scheme to diophantine formulas without parameters. The main result of the paper says that \(I\exists^-_ 1\vdash IE^-_ 1+E\vdash\) Matijasevič's theorem, where \(IE^-_ 1\) is the scheme of parameter-free bounded existential induction and E is an \(\forall \exists\) axiom expressing the existence of a function of exponential growth. This answers the question of Z. Adamowicz if \(I\exists_ 1\) does prove Matijasevič's theorem. It is also connected with the problem: \(``I\Delta_ 0\vdash\) Matijasevič's theorem?'', which is still unsolved. The theorem is proved by means of a conservation result relating parameter and parameter-free induction schemes, which is of independent interest. The paper is concluded by some consequences of these results to the structure of countable nonstandard models of \(IE^-_ 1\).
scientific article; zbMATH DE number 4134037
scientific article