Prenex Normal Form Theorems in Semi-Classical Arithmetic

Abstract

Akama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is Π 2 -conservative over intuitionistic arithmetic.

Conservation theorems on semi-classical arithmetic

“…The proof of [6,Theorem 6.14] in that paper is similar to the former approach in the sense of using the Friedman A-translation. However, the proof has somewhat intricate structure in dealing with the Friedman A-translation of the inner part of Kuroda's negative translation.…”

“…In this section, we simulate Ishihara's proof of [8,Theorem 10] in the specific context of semi-classical arithmetic studied in [1,6] with some additional arguments. We first recall the translation studied in [8].…”

“…[9,Section 4]). In [6,Theorem 6.14], the authors showed a conservation result which generalizes the aforementioned conservation result on PA and HA in the context of semi-classical arithmetic (which lies between classical and intuitionistic arithmetic). In fact, the following is an immediate corollary of [6, Theorem 6.14]: Proposition 1.1.…”

“…That is, for classes Γ and Γ of HA-formulas, Γ ⊆ Γ denotes that for all ϕ ∈ Γ, there exists ϕ ∈ Γ such that FV (ϕ) = FV (ϕ ) and HA ϕ ↔ ϕ, and Γ = Γ denotes Γ ⊆ Γ and Γ ⊆ Γ. In this sense, one may think of Σ k and Π k as sub-classes of Σ k and Π k for all k > k (see [6,Remark 2.5]).…”

“…However, the proof has somewhat intricate structure in dealing with the Friedman A-translation of the inner part of Kuroda's negative translation. In Section 3, by extending the latter approach from [8,9] in the context of semi-classical arithmetic, we provide a much more structured proof of [6,Theorem 6.14]. As an advantage of the structured proof, we obtain an extended conservation result for much larger classes of formulas (see Theorem 3.17 and Remark 3.18).…”