4.15 - Exercise - Mathematical Logic - Ebbinghaus

Question

First, I will present the notations of the exercise $4.15$, which can be founded at the bottom of the page $37$ of Ebbinghaus's Mathematical Logic:

Now, the question:

I know that it's necessary to prove doing induction by S-terms and I can see clearly how to prove if the S-term $t$ is a constant or is a $n$-ary function $f$ applied on $t_1, \cdots, t_n$ S-terms (it's basically by definition of direct product of the S-structures). My doubts are how to prove the base step if $t = x$ is a variable?

Thanks in advance!

Answer 1

Long comment

We have that : $\text{var}(t) \subset \{ v_0,\ldots, v_{n-1} \}$.

Thus, for the base case : $t := v_0$ we must have :

$t^{\mathfrak A}[g_0]=\langle \ t^{\mathfrak A_i}[g_0(i)] \ \mid \ i \in I \ \rangle$.

And this is trivial because the "object" $g_0$ of the direct product $\mathfrak A$ is an element of the domain $\Pi_{i \in I} A_i$, i.e. a "sequence" $\langle \ g_0(i) \in A_i \ \mid \ i \in I \ \rangle$.