 When it comes to theories T, quite unlike to other areas of logic (e.g., model theory), results as those presented in this section depend not only on the set of axioms of T, but also on the way they are presented. When talking about a theory T we assume that T is given by a \(\Sigma^0_1\)-formula \(\psi(v_0)\) such that F is an axiom of T just in case \(\psi({\Corner{F}})\) holds; a \(\Sigma^0_1\)-formula is of the form \(\exists y_1\ldots\exists y_n\,R(y_1,\ldots y_n)\) with R primitive recursive. This consideration together with Kleene’s \({\cO}\) allows us to build a transfinite hierarchy of theories based on any suitable theory T. A consistency progression based on T is a primitive recursive function \(n\mapsto \psi_n\) that associates with every natural number n a \(\Sigma^0_1\)-formula \(\psi_n(v_0)\) that defines \(\bT_n\) such that PA proves: (i) \(\bT_0=\bT\); (ii) \(\bT_{{\rsuc}(n)}=\bT_n'\), and (iii) \(\bT_{{\rlim}(n)}=\bigcup_x\bT_{\{n\}(x)}\). So, finally we can formulate Turing’s completeness result.