Herbrand universe

Example

Let $L$ be a first-order language with at least one constant symbol; if it has none, one constant is added. The Herbrand universe $H(L)$ is the set of ground terms of $L$, the terms that contain no variables. It is the smallest set that contains every constant symbol of $L$ and contains $f(t_1, \ldots, t_n)$ whenever $f$ is an $n$-ary function symbol and $t_1, \ldots, t_n \in H(L)$.

With the function symbols acting by $f(t_1, \ldots, t_n) \mapsto$ the term $f(t_1, \ldots, t_n)$, the set $H(L)$ is the term algebra on the empty set of variables. So it is the absolutely free algebra on no generators. Check: every map from the empty set extends uniquely. Indeed, a homomorphism to any algebra $A$ must send each ground term to its value in $A$.

For $L$ with one constant $0$ and one unary symbol $s$, the universe is $\set{0, s(0), s(s(0)), \ldots}$, a copy of $\N$. For $L$ with one constant $a$ and one binary symbol $g$, it contains $a$, $g(a, a)$, $g(a, g(a, a))$, $g(g(a, a), a)$, and so on. There is one element for each binary tree.