Gödel’s second incompleteness theorem for Σn-definable theories

Abstract

Gödel's second incompleteness theorem is generalized by showing that if the set of axioms of a theory T ⊇ PA is Σ n+1 -definable and T is Σ n -sound, then T dose not prove the sentence Σ n -Sound(T ) that expresses the Σ n -soundness of T . The optimality of the generalization is shown by presenting a Σ n+1 -definable (indeed a complete ∆ n+1 -definable) and Σ n−1 -sound theory T such that PA ⊆ T and Σ n−1 -Sound(T ) is provable in T . It is also proved that no recursively enumerable and Σ 1 -sound theory of arithmetic, even very weak theories which do not contain Robinson's Arithmetic, can prove its own Σ 1 -soundness.

Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic

“…The goal is not to prove the sharpest or most general results (in fact some of the results follow from each other), but rather to exemplify how the hierarchical principles enter into more or less well-known proof methods. Even so, the results presented here improve on some results of Chao and Seraji [5], Kikuchi and Kurahashi [23], and Salehi and Seraji [40], and sharpen some of Blanck [3], Hamkins [18], Lindstr öm [28], and Woodin [48]. These sharpenings are in terms of gauging the amount of induction needed for the proofs, bringing the (in this particular sub-field largely ignored) fragments-ofarithmetic perspective to attention.…”

“…Introduction. There has been a recent interest in hierarchical generalizations of classic incompleteness results [5,23,27,40]. A sample result, generalizing the G ödel-Rosser incompleteness theorem, and independently proved by both Kikuchi and Kurahashi [23] and Salehi and Seraji [40], is: Theorem 1.1.…”

Current Research on Gödel’s Incompleteness Theorems

“…• T is Σ 0 n -definable if there is a Σ 0 n formula φ(x) such that n is the Gödel number of some sentence of T if and only if N |= φ(n). 21 • T is Σ 0 n -sound if for all Σ 0 n sentences φ, T φ implies N |= φ; T is sound if T is Σ 0 n -sound for any n ∈ ω. • T is Σ 0 n -consistent if for all Σ 0 n formulas φ with φ = ∃xθ(x) and θ ∈ Π 0 n−1 , if T ¬θ(n) for all n ∈ ω, then T φ.…”

“…Chao-Seraji [21] and Kikuchi-Kurahashi [74] give another generalization of G2 to arithmetically definable non-r.e. extensions of PA: for each n ∈ ω, any Σ 0 n+1 -definable and Σ 0 n -sound extension of PA cannot prove its own Σ 0 n -soundness (see [21,Theorem 2] and [74,Theorem 5.6]).…”

“…Chao-Seraji [21] and Kikuchi-Kurahashi [74] give another generalization of G2 to arithmetically definable non-r.e. extensions of PA: for each n ∈ ω, any Σ 0 n+1 -definable and Σ 0 n -sound extension of PA cannot prove its own Σ 0 n -soundness (see [21,Theorem 2] and [74,Theorem 5.6]). The optimality of this generalization is shown in [21]: there is a Σ 0 n+1 -definable and Σ 0 n−1sound extension of PA that proves its own Σ 0 n−1 -soundness for n > 0 (see [21,Theorem 3]).…”