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  "body": "\n## Example\n\nLet $L$ be a first-order language with at least one constant symbol; if it has none, one constant is added.\nThe *Herbrand universe* $H(L)$ is the set of ground terms of $L$, the terms that contain no variables.\nIt 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 $t_1, \\ldots, t_n \\in H(L)$.\n\nWith 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|3YYKGWJJND]] on the empty set of variables.\nSo it is the [[absolutely free algebra|TTTEV3NWWM]] on no generators.\nCheck: every map from the empty set extends uniquely.\nIndeed, a homomorphism to any algebra $A$ must send each ground term to its value in $A$.\n\nFor $L$ with one constant $0$ and one unary symbol $s$, the universe is $\\set{0, s(0), s(s(0)), \\ldots}$, a copy of $\\N$.\nFor $L$ 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.\nThere is one element for each binary tree.\n\n## Explanations\n\n### L1 human\n\nThe Herbrand universe is the list of all objects that a language can name without variables.\nStart from the constants and apply the function symbols again and again.\n[[Herbrand's theorem|VZ21F10WRS]] reduces the unsatisfiability of a universal sentence to that of finitely many instances built from ground terms in this universe.\nThis is why it is central in automated theorem proving.\n",
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