 Definition 3.19: A set \(A \subseteq \mathbb{N}\) is implicitly definable in \(\mathcal{L}^2_a\) just in case there is an arithmetical formula \(\varphi(X)\) with \(X\) as its sole free set variable and no free number variables such that \(A\) is the unique set satisfying \(\varphi(X)\) in the standard model of \(\mathcal{L}^2_a\).