First-order language.

Question

In Elliot Mendelson's "Introduction to Mathematical Logic" there is a definition of a first-order language $L$ which says, that it contains denumerably many individual variables. A first-order theory in the language $L$, by its definition, contain the same symbols as $L$.

As we know, the $\text{ZFC}$ set theory (as an example) is a first-order theory where we call those individual variables "sets". But then we have only countably many sets, since there are only countably many variables. This is definitely wrong. What do I miss?

Answer 1

Variables are variables and sets are sets, they belong to different worlds, the first ones are syntactic objects while the second ones are semantic. Even if you're interpreting your variables as denoting sets you don't need one variable for each set, so from the fact you have countably many variables in your language doesn't follow there are countably many sets

Answer 2

There is a difference between the symbols we use in a logic language and the kinds of objects we try to symbolize with them, and we usually don't have a symbol for every object in our domain. For example, Peano arithmetic, which is our language for the natural numbers, has a symbol only for the number 0.