Reading about interpretability made me think about the $s$ function that somehow always bothered me in the language of PA. My question is the following
Given $\mathbb{N}$ as a structure in $\mathcal{L}=\{s, 0, 1\}$ (understood in the usual manner), are the sum and product operations definable?
In a broader sense I wonder if there is some way around a recursive definition when trying to describe these functions. For $\mathbb{N}$ there surely is since you can explicitly describe the functions with an infinite number of sentences of the form $s(0)+s(0)=ss(0)$ and $s(0)*s(0)=s(0)$ (although this of course does not make them definable), however with bigger models of PA these sentences might not suffice.
Are you familiar with quantifier elimination? $Th(\mathbb{N})$ has quantifier elimination in $\mathcal{L}$ so neither $+$ nor $.$ are definable in $\mathcal{L}$.