 Definition 3.17: The language \(\mathcal{L}^2_a\) of second-order arithmetic extends the language \(\mathcal{L}_a\) of first-order arithmetic with the binary relation symbol \(\in\), together with set variables \(X,Y,Z, \ldots\) and set quantifiers \(\exists X\) and \(\forall Y\). The standard model of \(\mathcal{L}^2_a\) is the structure \(\langle \mathbb{N},\mathcal{P}(\mathbb{N}),0,&lt;,s,+,\times,\in \rangle\). The intended range of the set quantifiers is thus \(\mathcal{P}(\mathbb{N})\) (i.e. the power set of \(\mathbb{N}\)) while the intended interpretation of \(x \in X\) is that the number \(x \in \mathbb{N}\) is a member of the set \(X\) where \(X \in \mathcal{P}(\mathbb{N})\).