I encountered the following passage when reading Yu. I. Manin’s A Course in Mathematical Logic for Mathematicians:
2.12. Proposition. The cardinality of the class of $\phi$-definable sets does not exceed
card(alphabet of $L$) + $\aleph_0$
Here and below, by “card(alphabet of $L$)” we mean the cardinality of the alphabet of $L$ without the set of variables.
Proof. If the language has $\le \aleph_0$ variables, then there are at most
card(alphabet of $L$)+ $\aleph_0$ formulas.
If, on the other hand, it has an uncountable set of variables, then we note that every definable set can be defined by a formula whose variables belong to a fixed countable subset of the variables that is chosen once and for all. [Last emphasis mine]
To clarify the context, let me add that there follows an immediate corollary:
If $M$ is infinite and card(alphabet of $L$) < $2^{\text{card} M}$, then “almost all” sets are undefinable.
Manin's definition of a $\phi$-definable set is as follows:
A set $S \subset M_r, r \ge 1$, is called $\phi$-definable (by the formula $P$ in $L$ with the interpretation $\phi$) if there exist variables $x_1 , . . . , x_r$ such that
$$|P|_\phi (\xi) = 1 \iff \langle x^{\xi}_1,...,x^{\xi}_r\rangle \in S$$
 for all $\xi$ in $M$. > [Viz. $\xi$ denotes a point in the interpretation class $\overline M$ for variables in $L$ .]
Back to the proof, I don’t get what the author means by "chosen once and for all", in my naive view to find or "produce" a $\phi$-definable set for a certain formula $P$ of degree $r$ all we need to do is simply gather or "prescribe" all ordered r-tuples (thus determining a subset of $M^r$) that satisfy $P$, i.e. that are in $\phi(P)$, which is again in $M$ (the set or class which the interpretation $\phi$ of $L$ is said to be in). So when there are uncountably many variables, $\overline M$ will be uncountable, meaning that there are uncountably many ways to assign values to certain finite r-tuple of variables, it seems when determining the cardinality of the class of $\phi$-definable sets for such an $L$ we will be dealing with at least the collection $\mathcal{C}$ of (all?) finite subsets of an uncountable set, with card($\mathcal{C}) \le $ the said number of formulas.
I admit that the whole line is totally confused, obviously I'm missing something, for example I'm not quite sure how the sum's first term, viz. card(alphabet of $L$), affects the whole (if it's countable then it makes no difference, but I find it abstract that it qualifies as a term at all).
Back to the main question, any enlightenment as to what rationale underlies the last sentence of the first citation will be appreciated.