There is no maximal decidable expansion of the $\langle \mathbb{N} ,\{ < \} \rangle$ structure

TL;DR

The paper proves that any decidable structure on natural numbers with the standard order definable in it can be expanded nontrivially while preserving decidability, showing no maximal decidable expansion exists.

Contribution

It establishes that structures with decidable theories and definable order have nontrivial decidable expansions, indicating no maximal such structure exists.

Findings

Decidable structures with definable order can be nontrivially expanded.

No maximal decidable expansion of the structure exists.

The result applies to structures on natural numbers with the standard order.

Abstract

We are going to prove that if the theory of a structure $\mathcal M=\langle \mathbb{N}, \Sigma \rangle$ is decidable and the standard order $<$ on natural numbers $\mathbb{N}$ is definable in $\mathcal M$, then there is a nontrivial decidable expansion of $\mathcal M$