 For each axiom scheme \(\mathbf{Ax}\) we denote by \((\mathbf{Ax})_0\) the theory consisting of the basic arithmetical axioms plus the scheme \(\mathbf{Ax}\). By contrast, \((\mathbf{Ax})\) stands for the theory \((\mathbf{Ax})_0\) augmented by the scheme of induction for all \(\cL_2\)-formulae. An example for these notations is the theory \((\bPi^1_1-\bCA)_0\) which has the comprehension schema for \(\bPi^1_1\)-formulae.