 The notations \(\Sigma^0_n\), \(\Pi^0_n\), and \(\Delta^0_n\) are also standardly used to denote the classes of sets and relations which are definable by a formula in the corresponding syntactic class. For instance it follows from the second part of  Proposition 3.8  that \(\textit{PRIMES}\) is \(\Delta^0_1\) (since it is computable) and from the first part that \(\HP\) and \(K\) are \(\Sigma^0_1\) (since they are c.e.). It thus follows that their complements \(\overline{HP}\) and \(\overline{K}\) are both \(\Pi^0_1\). It is also not hard to see that \(\TOT\) is \(\Pi^0_2\) as the fact that \(\phi_x(y)\) is total may be expressed as \(\forall y \exists z \tau_1(x,y,z)\) by using the arithmetized formulation of the \(T\)-predicate introduced above. Similarly, \(\textit{FIN}\) is \(\Sigma^0_2\)-definable since the fact that \(\phi_x(y)\) is defined for only finitely many arguments is expressible as  \(\exists u \forall y\forall z(u &lt; y \rightarrow \neg \tau_1(x,y,z))\).