Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2010 / B / 4 / f / iii / 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. ThereforeIn particular the criterion in part (e) applies already at arity one.