Let $\mathcal{L}$ be a language. In David Marker's Model Theory: An Introduction, the set of $\mathcal{L}$-terms is defined to be the smallest set $\mathcal{T}$ containing all of the constant symbols of $\mathcal{L}$, the 'variable symbols' $v_1$, $v_2$,... and such that for each function symbol $f$ (in $\mathcal{L}$) and $t_1,...t_{n_f}\in\mathcal{T}$ (where $n_f$ is the positive integer associated to $f$), we must have that $f(t_1,...,t_{n_f})\in\mathcal{T}$.
I'm a little puzzled about where these 'variable symbols' have come from. A language is defined on the previous page to consist of function symbols, relation symbols and constant symbols, yet there is no mention of 'variable symbols' as being in any way intrinsic to the language $\mathcal{L}$.
Am I right in thinking that for each language $\mathcal{L}$, there is an associated sequence of variable symbols $(v_i)_{i\in\mathbb{N}}$, and that this is what is being referred to in the definition of the $\mathcal{L}$-terms? If so, then I don't understand why it would not be part of the definition of the language, like the other symbols.
Many thanks