Starting from the analytic (or the co-analytic) sets and applying the operations of projection (from the product space \(\mathbb{R}\times \mathcal{N}\) to \(\mathbb{R}\)) and complementation, one obtains the projective sets. The projective sets form a hierarchy of increasing complexity. For example, if \(A\subseteq \mathbb{R}\times \mathcal{N}\) is co-analytic, then the projection \(\{ x\in \mathbb{R}: \exists y\in \mathcal{N}((x,y)\in A)\}\) is a projective set in the next level of complexity above the co-analytic sets. Those sets are called \(\mathbf{\Sigma}^1_2\), and their complements are called \(\mathbf{\Pi}^1_2\).