GENERALIZATIONS OF GÖDEL’S INCOMPLETENESS THEOREMS FOR ∑_n-DEFINABLE THEORIES OF ARITHMETIC

Abstract

It is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic. We generalize Gödel’s incompleteness theorems for arithmetically definable theories. First, we prove that every ∑n+1-definable ∑n-sound theory is incomplete. Secondly, we generalize and improve Jeroslow and Hájek’s results. That is, we prove that every consistent theory having ∏n+1 set of theorems has a true but unprovable ∏n sentence. Lastly, we prove that no ∑n+1-definable ∑n -sound theory can prove its own ∑n-soundness. These three results are generalizations of Rosser’s improvement of the first incompleteness theorem, Gödel’s first incompleteness theorem, and the second incompleteness theorem, respectively.

Current Research on Gödel’s Incompleteness Theorems

“…Kikuchi-Kurahashi [74] and Salehi-Seraji [118] make contributions to generalize Gödel-Rosser's first incompleteness theorem to non-r.e. arithmetically definable extensions of PA.…”

“…From G1, we have: if T is a Σ 0 1 -definable and Σ 0 1 -sound extension of Q, then T is not Π 0 1 -decisive. Kikuchi and Kurahashi [74] generalize G1 to arithmetically definable theories via the notion of "Σ 0 n -sound". [118]).…”

“…Salehi and Seraji [118] generalize G1 to arithmetically definable theories via the notion of "Σ 0 n -consistent". Theorem 4.6 (Theorem 4.9 [74], Theorem 4.3 [118]). If T is a Σ 0 n+1definable and Σ 0 n -consistent extension of Q, then T is not Π 0 n+1 -decisive.…”

“…27 Similarly to G1, one can generalise G2 to arithmetically definable nonr.e. extensions of PA. Kikuchi and Kurahashi [74] reformulate G2 as: if S is a Σ 0…”

A Formally Verified Abstract Account of Gödel’s Incompleteness Theorems

“…Our abstract approach, based on generic syntax and provability and truth predicates, resembles the style of institution-independent model theory [9,14] and our previous work on abstract completeness [3] and completeness of ordered resolution [34]. Dimensions of generality that our formalized work does not (yet) explore include quantifier-free logics [17] and arithmetical hierarchy refinements [19]. Our syntax axiomatization is inspired by algebraic theories of the λ-calculi syntax [11,12,29].…”