PRENEX NORMAL FORM THEOREMS IN SEMI-CLASSICAL ARITHMETIC

Abstract Akama et al. [1] systematically studied an arithmetical hierarchy of the law excluded middle and related principles in context first-order arithmetic. In that paper, they first provide a prenex normal form theorem as justification their semi-classical restricted to formulas. However, there are some errors proof. this we simple counterexample [1, Theorem 2.7], then modify it appropriate way which still serves largely justify hierarchy. addition, characterize variety theorems by logical The characterization results reveal our optimal. For results, establish new conservation on generalizes well-known fact classical arithmetic is $\Pi _2$ -conservative over intuitionistic