A generalization of a conservativity theorem for classical versus intuitionistic arithmetic

A basic result in Intuitionism is Π2-Conservativity. Take any proof p in Classical Arithmetic of some Π2-statement (some arithmetical statement ∀x.∃y.P (x, y), with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ∀x.∃y.P (x, y), with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = 0, we get the original conservativity result as particular case. This result was a byproduct of a semantical construction. J. Avigad, of Carnegie Mellon University, found a short, direct syntactical derivation of the same result, using H. Friedman’s A-translation. His proof is included here with his permission. Iniziato a Torino, il 2 Aprile 2003. Ultimo salvataggio: September 2, 2003. 1 Plan of the paper. In section 2 we quickly introduce Classical and Intuitionistic Arithmetic, and we state Generalized conservativity. In section 3 we introduce Friedman’s A-translation, and we explain how it was used to prove Π2-conservativity. In section 4 we use Friedman’s A-translation to prove Generalized Conservativity. Generalized Conservativity could look uninteresting, from an intuitionistic viewpoint. Indeed, it produces proofs using Excluded Middle, even if only over degree k formulas. We think of it instead as a