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I have expanded theory adding more basics in classical syntactic approach. I added a new subsection
Set-theoretic models for a first-order theory in syntactic approach
The basic concept is of a structure for a first-order language $L$: a set $M$ together with an interpretation of $L$ in $M$. A theory is specified by a language and a set of sentences in $L$. An $L$-structure $M$ is a model of $T$ if for every sentence $\phi$ in $T$, its interpretation in $M$, $\phi^M$ is true (“$\phi$ holds in $M$”). We say that $T$ is consistent or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for $T$ (in our universe). Two theories, $T_1$, $T_2$ are said to be equivalent if they have the same models.
Given a class $K$ of structures for $L$, there is a theory $Th(K)$ consisting of all sentences in $L$ which hold in every structure from $K$. Two structures $M$ and $N$ are elementary equivalent (sometimes written by equality $M=N$, sometimes said “elementarily equivalent”) if $Th(M)=Th(N)$, i.e. if they satisfy the same sentences in $L$. Any set of sentences which is equivalent to $Th(K)$ is called a set of axioms of $K$. A theory is said to be finitely axiomatizable if there exist a finite set of axioms for $K$.
A theory is said to be complete if it is equivalent to $Th(M)$ for some structure $M$.
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