 We denote by both \(\Sigma^1_0\) and \(\Pi^1_0\) the class of \(\mathcal{L}^2_a\)-formulas containing no set quantifiers (i.e., a so-called arithmetical formulas). An \(\mathcal{L}^2_a\) formula is in the class \(\Sigma^1_{n+1}\) if it is of the form \(\exists X \psi(X)\) where \(\psi \in \Pi^1_n\) and a relation is \(\Sigma^1_{n+1}\)-definable if it is defined by a \(\Sigma^1_{n+1}\)-formula. Similarly an \(\mathcal{L}^2_a\)-formula is in the class \(\Pi^1_{n+1}\) if it is of the form \(\forall X \psi(X)\) where \(\psi \in \Sigma^1_n\) and a relation is \(\Pi^1_{n+1}\)-definable if is defined by a \(\Pi^1_{n+1}\)-formula. A relation is \(\Delta^1_n\)-definable just in case it is definable by both a \(\Sigma^1_n\)- and a \(\Pi^1_n\)-formula.