Confusion about Satisfiability by a denumerable sequence. In "Introduction to Mathematical logic" by Elliot Mendelson , page $59$ . I couldn't understand the basic preliminary notations for the rigorous definition of Satisfiability.
For example , at a line they said,
"Moreover, instead of talking about the
$n$-tuples of objects that satisfy a wf that has $n$ free variables, it is much more
convenient from a technical standpoint to deal uniformly with denumerable
sequences. What we have in mind is that a denumerable sequence
$s = (s_1, s_2 , s_3 , ... )$ is to be thought of as satisfying a wf $\beta$ that has
${x_j}_1 , {x_j}_2 ,  ... , {x_j}_n$ as free variables (where $j_1 < j_2 < ... < j_n$ ) if the $n$-tuple
$\langle {s_j}_1 , {s_j}_2 ,  ... , {s_j}_n \rangle$ satisfies $\mathscr B$ in the usual sense."
I didn't quite understand why they used ${s_j}_1$ and ${x_j}_1$ instead of $s_1$ and $x_1$ . Maybe there is a critical detail I am missing out?
Also , Another passage for the rigorous definition:
"Let $M$ be an interpretation of a language $\mathscr L$ and let $D$ be the domain of
$M$. Let $\sum$ be the set of all denumerable sequences of elements of $D$. For a wf
$\mathscr B$ of $\mathscr L$ , we shall define what it means for a sequences $s= (s_1, s_2, ... )$ in $\mathscr L$ to
satisfy $\mathscr B$ in $M$. As a preliminary step, for a given s in $\sum$ we shall define a
functions $s^*$ that assigns to each term $t$ of $\mathscr L$ an element $s^{*}(t)$ in $D$."
They did say that for each $s = (s_1, s_2, ... )$ , there is a defined function $s^*$ , but even after further reading the text afterwards , I still did not understand how exactly $s^*$ is defined from $s$ . Can someone tell me where the gap in my understanding here is?
 A: A denumerable sequence is a function $s: \text {Var} \to D$, where $\text {Var}$ is the ordered list of variables of the language: $x_1,x_2,\ldots$ and $D$ is the domain of the interpretation.
What is e.g. $s(x_1)$? It is an element $d \in D$ such that: $s(x_1)=d$, and so on.
This function "induces" a function $s^* : \text {Term} \to D$ that assigns a reference to each term $t$ of the language, according to the following rule:
(i) if $t$ is a variable $x_i$, then $s^*(t)=s(x_i)$; as we know already, it is an element of $D$.
(ii) if $t$ is a constant symbol $c$, then its reference is fixed by the interpretation, i.e. an element $c^D \in D$. Thus: $s^*(c)=c^D$.
What is left?
The case (iii) with a compound term $f_n(x_1,\ldots,x_n)$.
In this case, we use the $n$-ary function $f_n^D$ on the domain $D$ assigned to the symbol $f_n$ by the interpretation, and we use $s$ to assign reference to the variable. Thus: $s^*[f_n(x_1,\ldots,x_n)]=f_n^D(s(x_1),\ldots,s(x_n))$.
Consider a simple example with the language of arithmetic, where the symbol $+$ is a binary function symbol, interpreted in the domain $D$ of natural numbers ($\mathbb N$) with the usual sum.
Let $s(x_1)=1$ and $s(x_2)=2$, and let $t$ the term: $+(x_1,x_2)$.
We have:

$s^*[+(x_1,x_2)]=+^D(1,2)=1+2$.


Regarding the first question, there is no "deep meaning".
The author is suggesting that the definition of satisfaction in terms of denumerable sequences (due originally to Alfred Tarski (1936)) is equivalent to the definition in terms of $n$-tuples currently used in model theory.
Consider a simple example with as $\mathscr B$ the formula: $\forall x_1 (x_1 < x_2)$.
In this case, the free variable in the formula is $x_2$ and thus what is relevant for the satisfaction of the formula by a sequence $s$ is the object assigned by $s$ to $x_2$.
Thus, the first quote above means: we have a denumerable sequence $s= (s_1,s_2,s_3,\ldots)$, that means a function: $s(x_1)=s_1, s(x_2)=s_2 \in D$, and so on.
We have that, regarding satisfiability, it is the same to consider the "full" sequence or the $n$-uple extracted from the sequence considering only the free variables occurring into $\mathscr B$: in this case: $⟨s_2⟩$.
