Robinson's Consistency Theorem for first order languages

Question

Is there a simple proof for the case of first order languages for this theorem?

Let $L_1$,$L_2$ be first order languages and $L$=$L_1$ $\cap$ $L_2$.
Let $T_1$, $T_2$ be consistent $L_1$,$L_2$-theories respectively.
Let $T$=$T_1\cap T_2$ be a complere $L$-theory.

Why $T_1\cup T_2$ is consistent?
And is there an example that $T_1\cup T_2$ is not consistent when T is not complete?

Answer 1

This is a short answer. (I am afraid it requires some background knowledge.)

Let $M_1\models T_1$ and $M_2\models T_2$ be saturated structures of the same cardinality. The reduced to $L$ of $M_1$ and $M_2$ are saturated models of the same complete theory $T$, hence there is an $L$-isomorphism $f:M_1\to M_2$. There is a (unique) expansion of $M_2$ that makes $f$ an $L_1$-isomorphism. This expansion is a model of $T_1\cup T_2$.

Answer 2

You can prove it by constructing two elementary chains of models $M_n\models T_1$, $N_n\models T_2$, so that $M_n\preceq M_{n+1}$ as $L_1$-structures and $N_n\preceq N_{n+1}$ as $L_2$-structures, while $M_n\preceq N_n\preceq M_{n+1}$ as $L$-structures. Then $\bigcup M_n=\bigcup N_n$ is a model of $T_1\cup T_2$ (as the union of an elementary chain).

For that, you need to prove the following lemma: if $T$ is a complete $L$-theory and $T'$ is a consistent extension of $T$ in $L'$, then for every $M\models T'$ which is an elementary $L$-structure of some $N$, we can find an $L'$-structure $M'$ which is an $L$-elementary extension of $N$ and an $L'$-elementary extension of $M$. For that, it is enough for the union of the $L$-elementary diagram of $N$ and the $L'$-elementary diagram of $M$ to be consistent. This is a standard application of compactness (using completeness of $T$). For simplicity, you might assume that $T'$ is complete.

Once you have the lemma, the recursive construction is straightforward: start with any model $M_0\models T_1$, extend $L$-elementarily to some $N_0\models T_2$ (more or less the same way as in the proof of the lemma), and then apply the lemma successively, alternating $T'$ between $T_1$ and $T_2$.