We have a structure $R=(\mathbb{N},<)$ and I want to find another structure $M$ that is elementarily equivalent to $R$ s.t. $<^M$ has a desending chain.
I used compactness theorem. I extended the language by adding infinitly many new constant symbols $a_1,a_2,...$ and let $\Sigma=\{\alpha_{ij}:i,j\in\mathbb{N}\}$ where $\alpha_{ij}=a_i<a_j$. Now, let $A=\text{Th} R\cup\Sigma$. Every finite subset of $A$ is satisfiable in $(\mathbb{N},<)$ hence $A$ has a model say $M$. So, there is a desending chain in this new structure $M$.I restrict $M$ to the original langauge. I want to show that both $M,R$ are elementarily equivalent. clearily, If $\phi\in \text{Th} R$ then $\phi\in\text{Th}M$. But I can't show the converse.
So my question is, How to show that $\text{Th}M\subseteq\text{Th}R$.
If $R\not\vDash\varphi$, then $R\vDash\neg\varphi$, so $M\vDash\neg\varphi$ by what you’ve already shown, and therefore $M\not\vDash\varphi$. Thus, if $M\vDash\varphi$, then $R\vDash\varphi$.