Proof Gödel's Completeness theorem in book of Prestel

Question

I am currently reading Mathematical Logic and Model Theory by Alexander Prestel and am stuck at the proof of the Completness Theorem (p. 41).

Via induction it is proven that $\cal{A}\models \varphi[h]$ iff $\varphi \in\Sigma^*$. However in the induction bases one only considers constant terms. Why is this sufficient and why aren't we considering terms with variables?

My first guess was, that this is enough since all the variables are replaced by elements of the universe in the given structure $\cal{A}$ according to the evaluation $h$ but I'm not sure if this is really the case.

Answer 1

See page 41 :

Let $Σ \subseteq \text {Sent}(L)$ and $\varphi \in \text {Sent}(L)$ ...

Thus, the theorem consider sentences only, i.e. formulas without free variables.

The proof of the theorem is based on the construction of a structure $\mathfrak A$ having as domain $A = \text{CT}/≈$, i.e.

the set of equivalence classes of constant terms of the language $L$.

This means that the terms used in the proof as elements of the domain of the interpretation will have no free variables, i.e. they must be either constants or built-up with function symbols from closed terms.