Solution to exercise 2.5.9 of Marker's Model theory

Question

I'm trying to solve this exercise 2.5.9 of David Marker's model theory. The exercise is:

Take three L-structures: $\cal{M}_0 \subset\cal{M}_1 \subset\cal{M}_2$. If $\cal{M}_0 \prec \cal{M}_2$ and $\cal{M}_1 \prec \cal{M}_2$, then show that $\cal{M}_0 \prec \cal{M}_1$.

I know that this is the coherence property of abstract elementary classes. However, I have no idea how to prove the exercise. Does it use the compactness theorem?

Answer 1

Just use the definition: let $a$ be a tuple from $M_0$, and let $\varphi$ be a formula. We want to show $\mathcal{M_0}\models \varphi(a)$ iff $\mathcal{M_1}\models\varphi(a)$

Since $\mathcal{M_0}\prec \mathcal{M_2}$, and $a\in M_0$, $\mathcal{M_0}\models \varphi(a)$ iff $\mathcal{M_2}\models\varphi(a)$. Since $\mathcal{M_1}\prec \mathcal{M_2}$, and $a\in M_1$, $\mathcal{M_2}\models \varphi(a)$ iff $\mathcal{M_1}\models\varphi(a)$. And we're done!

Answer 2

Use the Tarski Vaught test. Suppose $\varphi(x,y)$ is a formula in the language, and $a \in M_0$ (I'm abusing notation wrt tuples and such). Then if there is $b \in M_1$ such that $M_1 \vDash \varphi(b, a) $, then since $M_1 \prec M_2$, $M_2 \vDash \varphi(b, a)$, and then by the Tarski vaught test between $M_0, M_2$, there is $b' \in M_0$ such that $M_2 \vDash \varphi(b', a)$, but then $b' \in M_1$ as well, so $M_1 \vDash \varphi(b', a)$. Thus the Tarski Vaught test is satisfied.