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

Abstract

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

found an error in the proof of statement 2, page 3