For a given language L, with its set of variables $V$, function symbols $F$, relation symbols $R$. Once we are given the cardinality of $V$, $F$, $R$. Is there any way to define the cardinality of the set of expressions over this language?
I think maybe there can be a systematic way to find the cardinality (which may be very messy, but if it is such a way to deal with that, I really wish to learn about it also). And also I expect to learn (if there are) some useful tricks to determine it simply use the relation of the size of the set of expressions and the size of finite strings (which can be find much easily in this related question below).
Related question:
Logic: cardinality of the set of formulas
The cardinality of a language over a set of variables $V$ with $\#V= κ$
Both of above seems talked about the set of finite strings
So could someone give some way to determine the size of the set of expression? Any help will be appreciate.