LO is the theory of linear ordering. Suppose T a theory, which contains at least the symbol {<} in her language and T $\vDash$ LO. Suppose T has an infinite model. Prove that there's a model M for T for which we can find an order-preserving embedding for the rational numbers.
Conclude that the theory for ($\mathbb Z $,<) has a model for which we can find such an embedding with the rational numbers.
For the first part, I think that it's enough to use Upwards Löwenheim-Skolem Theorem with the set C = $\mathbb Q$. But I don't know how to come to the conclusion. Can somebody help me?
Thanks in advance Silke
Add a constant symbol $c_q$ for each $q\in\Bbb Q$. For each $p,q\in\Bbb Q$ with $p<q$ let $\varphi_{p,q}$ be $c_p<c_q$, and consider $T'=T\cup\{\varphi_{p,q}:p,q\in\Bbb Q\text{ and }p<q\}$. Show that $T'$ is finitely satisfiable and hence satisfiable.