I came across a sentence in page 196 Chang & Keisler's model theory book that I don't understand. It says: Every complete theory has the joint embedding property.
Def. A theory $T$ has joint embedding property if for every $M,N\models T$ then there is a model $K\models T$ such that $M,N$ are isomorphically embeddable in $K$.
Why is it true? Can anybody guid me with this?
HINT: Let $L$ be the language of $T$. Consider the new language $L'=L\sqcup\{c_a: a\in M\}\sqcup\{d_b: b\in N\}$ gotten by adding new constants to $L$ corresponding to the elements of $M$ and $N$ (assuming WLOG that $M$ and $N$ have disjoint domains).
Now let $T'$ be the $L'$-theory consisting of $T$ together with the atomic diagram of $M$, for the $c_a$s, and the atomic diagram of $N$, for the $d_b$s. E.g. if $M\models a_1Ra_2$ then we put "$c_{a_1}Rc_{a_2}$" into $T'$.
Exercise: If $T'$ has a model $K$, then $M, N$ embed into (the reduct of) $K$ (to $L$).
This is immediate.
Exercise: $T'$ has a model.
This uses completeness of $T$, and compactness of first-order logic. By compactness, we just need to show that every finite subset of $T'$ has a model. But we can show that any finite configuration occuring in $M$ also occurs in $N$, since this is witnessed by an existential formula.