Let $X$ be a set. For example, $X$ may be the set of strings consisting only of specific characters. What we explain here is the phrasing peculiar to mathematics used when recursively defining subsets of $X$ and the like.

For example, using an element $c$ of $X$, non-negative integers $n_0,m_0,n_1,m_1$, and maps $F_0 \colon X^{n_0} \times X^{m_0} \to X$ and $F_1 \colon X^{n_1} \times X^{m_1} \to X$, one may write as follows:

We recursively define a subset $A$ of $X$ as follows:

• $c \in A$.
• For any $(a,t) \in A^{n_0} \times X^{m_0}$, we have $F_0(a,t) \in A$.
• For any $(a,t) \in A^{n_1} \times X^{m_1}$, we have $F_1(a,t) \in A$.

This is merely an example. We used $3$ conditions here, but in general there need not be exactly $3$, and relations may be used instead of maps. The general formulation of recursive definitions^{[1] 6.1.1 Definition} is quite abstract, so here we consider only the examples needed to solve this problem.

The important point is that this definition does not merely mean that $A$ is an arbitrary subset of $X$ satisfying all $3$ conditions; rather, it stands for the following definition:^{[1] See 6.1.2 Definition and the paragraph immediately after it}

We define the subset $A$ of $X$ as the intersection of all subsets $B$ of $X$ satisfying all of the following $3$ conditions:

• $c \in B$.
• For any $(a,t) \in B^{n_0} \times X^{m_0}$, we have $F_0(a,t) \in B$.
• For any $(a,t) \in B^{n_1} \times X^{m_1}$, we have $F_1(a,t) \in B$.

Taking $B = X$ does satisfy the $3$ conditions above, so the set of all $B$ satisfying the conditions is non-empty and the intersection is well-defined. It then follows that $A$ is the smallest among the $B$ satisfying the $3$ conditions.

Note in this way that, by convention, a recursive definition of a subset is not a mere list of conditions but also implies minimality. However, this convention applies only to recursive definitions. An ordinary definition of course does not imply minimality, and in the first place, self-reference like that of a recursive definition, when used in an ordinary definition, usually results in circular reasoning and makes the definition break down.

Now, since a $1$-ary relation on $X$ is equivalent to a subset of $X$, the same rule applies to recursive definitions of $1$-ary relations on $X$. Specifically, suppose we write a recursive definition of the $1$-ary relation "is a good term" as follows:

We recursively define when an element of $X$ is a good term as follows:

• $c$ is a good term.
• For any $(a,t) \in X^{n_0} \times X^{m_0}$, if every component of $a$ is a good term, then $F_0(a,t)$ is also a good term.
• For any $(a,t) \in X^{n_1} \times X^{m_1}$, if every component of $a$ is a good term, then $F_1(a,t)$ is also a good term.

This does not mean that the predicate "is a good term" denotes an arbitrary $1$-ary relation satisfying the $3$ conditions. The above definition is equivalent to the following definition:

We define an element $x$ of $X$ to be a good term if $x \in B$ holds for every subset $B$ of $X$ satisfying all of the following $3$ conditions:

• $c \in B$.
• For any $(a,t) \in B^{n_0} \times X^{m_0}$, we have $F_0(a,t) \in B$.
• For any $(a,t) \in B^{n_1} \times X^{m_1}$, we have $F_1(a,t) \in B$.

Based on the above, the recursive definition of a hydra in this problem is equivalent to the following definition:

We define a string $h$ consisting only of the $2$ kinds of characters ( and ) to be a hydra if, for any set $B$ consisting only of strings consisting only of ( and ), $h \in B$ holds whenever $B$ satisfies all of the following $3$ conditions:

• The empty string is an element of $B$.
• For any elements $h_0$ and $h_1$ of $B$, the string obtained by concatenating $h_0$ and $h_1$ in this order is an element of $B$.
• For any element $h$ of $B$, the string obtained by concatenating (, $h$, and ) in this order is an element of $B$.

This phrasing is considerably more complicated than the original one. To avoid such complicated phrasing, mathematics defines and adopts the concise phrasing introduced here.

Such phrasing is actually adopted, for example, in the references listed below:

• the recursive definitions of the $1$-ary relations "be an $\mathcal{L}$-term" and "be an $\mathcal{L}$-formula" in [1], etc.
• the recursive definitions of the set $T$ and the $1$-ary relation "be a principal term" in [2], etc.
• the recursive definitions of the sets $B(\alpha,\beta)$ and $\textrm{K}_{\kappa}(\gamma)$ in [3], etc.

and so on.

For example, when $h$ is (((())()())), $h[n]$ for $n = 1,2,3,4,5,6,7,8,9,10$ is as follows:

$h[1] =$ (((())()))
$h[2] =$ (((())())((())()))
$h[3] =$ (((())())((())())((())()))
$h[4] =$ (((())())((())())((())())((())()))
$h[5] =$ (((())())((())())((())())((())())((())()))
$h[6] =$ (((())())((())())((())())((())())((())())((())()))
$h[7] =$ (((())())((())())((())())((())())((())())((())())((())()))
$h[8] =$ (((())())((())())((())())((())())((())())((())())((())())((())()))
$h[9] =$ (((())())((())())((())())((())())((())())((())())((())())((())())((())()))
$h[10] =$ (((())())((())())((())())((())())((())())((())())((())())((())())((())())((())()))