What does "variable-free L-term" mean?

Question

Our lesson source is Lou van den Dries's Mathematical Logic Lecture notes. As a beginner, I do not understand the definition of $variable-free$ L-term. On pg.29, it says that "A term is said to be variable-free if no variables occur in it". As far as I understand, this means, variable-free term is a function symbol. But there are $variables$ in function interpretation unless the function is of arity 0. What am I missing?

Thanks in advance.

Answer 1

A term is said to be variable-free if no variables occur in it.

In the language of arithmetic, these are terms :

$x, y, x+y, 0, 0+x, 0+0, \ldots$.

Only : $0$ and $0+0$ are variable-free (i.e. "closed" terms) : they acts as "names" for objects in the domain.

A terms is variable-free if it is a constant (like : $0$) or built-up with an $n$-ary function symbol $f^n$ (like the binary : $+$) and $n$ variable-free terms $t_i$.