Explain what is meant by a structure for a first-order language and by a model for a first-order theory. If $T$ is a first-order theory whose axioms are all universal sentences (that is, sentences of the form $\left(\forall x_{1} \ldots x_{n}\right) p$ where $p$ is quantifier-free), show that every substructure of a $T$-model is a $T$-model.

Now let $T$ be an arbitrary first-order theory in a language $L$, and let $M$ be an $L$-structure satisfying all the universal sentences which are derivable from the axioms of $T$. If $p$ is a quantifier-free formula (with free variables $x_{1}, \ldots, x_{n}$ say) whose interpretation $[p]_{M}$ is a nonempty subset of $M^{n}$, show that $T \cup\left\{\left(\exists x_{1} \cdots x_{n}\right) p\right\}$ is consistent.

Let $L^{\prime}$ be the language obtained from $L$ by adjoining a new constant $\widehat{a}$for each element $a$ of $M$, and let

$$T^{\prime}=T \cup\left\{p\left[\widehat{a}_{1}, \ldots, \widehat{a}_{n} / x_{1}, \ldots, x_{n}\right] \mid p \text { is quantifier-free and }\left(a_{1}, \ldots, a_{n}\right) \in[p]_{M}\right\}$$

Show that $T^{\prime}$ has a model. [You may use the Completeness and Compactness Theorems.] Explain briefly why any such model contains a substructure isomorphic to $M$.