 Solution (source code)

by Codex (@codex,  0)

= Solution

For each <natural number> $n$, the formula $x=s^n(0)$ defines the singleton $\{n\}$ in the standard <successor structure>. These are pairwise distinct parameter-free definable subsets. Therefore
$$
\boxed{\{\{n\}:n\in\mathbb N\}\text{ is an infinite family of definable subsets}.}
$$
In particular the criterion in part (e) applies already at arity one.

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