Variants of Robinson's essentially undecidable theoryR

Abstract

Cobham has observed that Raphael Robinson's well known essentially undecidable theoryR remains essentially undecidable if the fifth axiom scheme$\left( {x \leqq \bar n \vee \bar n \leqq x} \right)$ is omitted. We note that whether the resulting system is in a sense “minimal essentially undecidable” depends on what the basic constants are taken to be. We give an essentially undecidable theory based on three axiom schemes involving only multiplication and less than or equals.