What exactly is $L$-terms in model theory?

Question

I got confused after seeing the inductive definition of $L$-terms in model theory. So I do get that there are variables and constants, and when function $f$ is applied to the term, the resulting thing is also a $L$-term.

But what does this really mean? Let's say we have constant 1,2,3 and variable $x,y,z$. Let $f$ be $f = x*3*y+2*z$ where $*$ and $+$ is in $L$. Then does this mean that the whole $x*3*y+2*z$ is a $L$-term? Or am I being confused about what function is? (so I should only treat function as sets or something like that?)

Answer 1

You are a bit confused about what functions and function symbols are. The thing to keep in mind is that when you are defining terms, you have not yet assigned interpretations (i.e. meanings) to any of your symbols yet. Therefore, function symbols are not the same as functions. They are just symbols with an associated arity. Functions appear in model theory when you assign interpretations to the symbols in your language, but you are not there yet because you haven't even gotten to the point where formulas have been defined. Say our language has the variables $x$, $y$, and $z$, function symbols $*$ (binary), $+$ (binary), and $f$ (ternary), constant symbols $1$, $2$, and $3$, and relation symbol $=$.

Then constants and variables are terms: $1$, $2$, $3$, $x$, $y$, and $z$. To get more terms, for each function symbol we define a map from the set of terms to itself: $F_*(t_1,t_2)=*(t_1,t_2)$ (which we write as $(t_1*t_2)$ as a convention), $F_+(t_1,t_2)=+(t_1,t_2)$ (which we write as $(t_1+t_2)$ as a convention), and $F_f(t_1,t_2,t_3)=f(t_1,t_2,t_3)$. The set of terms is generated by applying the term building functions $F_*$, $F_+$, and $F_f$ repeatedly to elements in $\{$1$, $2$, $3$, $x$, $y$, $z$\}$. So, for example, $(x+z)$ is a term, $((x+z)*2)$ is a term, and $f(1,(x+z),((x+z)*2))$ is a term.

Answer 2

Terms are linguistic objects. So one should not say constant, one should say constant symbol. Similarly, it is variable symbols, and function symbols.

In your informal examples, there are many missing parentheses.

Let us suppose that our language has, among possibly others, constant symbols $a$ and $b$, and variable symbols $x$ and $y$. Let us also suppose that the language has a unary function symbol $S$, and a binary function symbol $F$. The language also has the comma symbol "," and parentheses ( and ).

Then $a$, $b$, $x$, $y$ are terms. So are $S(a)$, $F(b,S(x))$, and $S(F(b,S(x)))$.

A term can be part of another term. For example, $S(x)$ is part of the term $F(b,S(x))$.

But some parts (subwords) of a term are not terms. For example, although $F(b,S(x))$ is a term, $F(b,$ is not.

Note that terms are to be viewed purely as strings of symbols. We are doing syntax. Semantics (interpretations, assigning meaning to expressions) comes later.

In the semantics, terms work much like nouns and pronouns in ordinary English. They denote objects, possibly incompletely specified objects, such as "her" in "her uncle."