Prenex normalization and the hierarchical classification of formulas

Abstract

Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes \(\textrm{E}_k\) and \(\textrm{U}_k\) introduced in [1] are exactly the classes induced by \(\Sigma _k\) and \(\Pi _k\) respectively via the transformation procedure in any first-order theory.