 The foregoing results all pertain to the use of \(\mathcal{L}^2_a\)-formulas to describe sets of natural numbers. The initial steps connecting the analytical hierarchy to classical descriptive set theory are mediated by considering formulas \(\varphi(X)\) which define subclasses \(\mathcal{X} \subseteq \mathcal{P}(\mathbb{N})\). In this case, \(A \in \mathcal{X}\) may be identified with the graph of its characteristic function \(\chi_A(x)\)—i.e., as an infinite sequence whose \(n\)th element is 1 if \(n \in A\) and 0 if \(n \not\in A\). In this way a formula \(\psi(X)\) with a single free set variable may be understood as defining a subset of the Cantor space \(\mathcal{C} = 2^{\mathbb{N}}\) consisting of all infinite 0-1 sequences and a formula \(\psi(\vec{X})\) with \(X_1,\ldots,X_k\) free as defining a subclass of \(2^{\mathbb{N}} \times \ldots \times 2^{\mathbb{N}}\).