Arithmetical independence results using higher recursion theory

Abstract

We extend an independence result proved in [arana2]. We show that for all n, there is a special set of Π_n sentences {φ_a}_{a∈ H} corresponding to elements of a linear ordering (H,<_H) of order type ω₁^CK(1+η). These sentences allow us to build completions {T_a}_{a∈ H} of PA such that for a<_H b, T_a ∩ Σ_n ⊂ T_b ∩ Σ_n, with φ_a∈ T_a, ¬φ_a ∈ T_b. Our method uses the Barwise—Kreisel Compactness Theorem.