Gödel–Rosser's Incompleteness Theorem, generalized and optimized for definable theories

Salehi, Saeed; Seraji, Payam

doi:10.1093/logcom/exw025

Cited by 9 publications

(28 citation statements)

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“…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. Theorem 4.6 is also optimal: the complete Σ 0 n−1 -sound and Σ 0 n+1 -definable theory constructed in the proof of Theorem 2.6 in [118] is also Σ 0 n−1 -consistent since if a theory is Σ 0 n -sound, then it is Σ 0 n -consistent. The proof of Theorem 4.6 cannot be constructive as the following theorem shows.…”

Section: Definition 44 ([74]mentioning

confidence: 99%

“…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.…”

Section: 2mentioning

confidence: 99%

“…If T is a Σ 0 n+1definable and Σ 0 n -sound extension of Q, then T is not Π 0 n+1 -decisive. Salehi and Seraji [118] point out that Theorem 4.5 has a constructive proof: given a Σ 0 n+1 -definable and Σ 0 n -sound extension T of Q, one can effectively construct a Π 0 n+1 sentence which is independent of T . The optimality of Theorem 4.5 is shown by Salehi and Seraji in [118]: there exists a Σ 0 n−1 -sound and Σ 0 n+1 -definable complete extension of Q for any n ≥ 1 (Theorem 2.6, [118]).…”

Section: Definition 44 ([74]mentioning

confidence: 99%

“…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.…”

Section: Definition 44 ([74]mentioning

confidence: 99%

“…The proof of Theorem 4.6 cannot be constructive as the following theorem shows. Theorem 4.7 (Non-constructivity of Σ 0 n -consistency incompleteness, Theorem 4.4, [118]). For n ≥ 3, there is no (partial) recursive function f (even with the oracle 0 n ) such that if m codes (the Gödel code) a Σ 0 n+1 -formula which defines an Σ 0 n -consistent extension T of Q, then f (m) halts and codes a Π 0 n+1 sentence which is independent of T .…”

Section: Definition 44 ([74]mentioning

confidence: 99%

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Current Research on Gödel’s Incompleteness Theorems

Cheng¹

2021

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We give a survey of current research on Gödel's incompleteness theorems from the following three aspects: classifications of different proofs of Gödel's incompleteness theorems, the limit of the applicability of Gödel's first incompleteness theorem, and the limit of the applicability of Gödel's second incompleteness theorem. §1. Introduction. Gödel's first and second incompleteness theorems are some of the most important and profound results in the foundations of mathematics and have had wide influence on the development of logic, philosophy, mathematics, computer science, as well as other fields. Intuitively speaking, Gödel's incompleteness theorems express that any rich enough logical system cannot prove its own consistency, i.e., that no contradiction like 0 = 1 can be derived within this system.Gödel [43] proves his first incompleteness theorem (G1) for a certain formal system P related to Russell-Whitehead's Principia Mathematica based on the simple theory of types over the natural number series and the Dedekind-Peano axioms (see [8, p. 3]). Gödel announces the second incompleteness theorem (G2) in an abstract published in October 1930: no consistency proof of systems such as Principia, Zermelo-Fraenkel set theory, or the systems investigated by Ackermann and von Neumann is possible by methods which can be formulated in these systems (see [147, p. 431]).Gödel comments in a footnote of [43] that G2 is corollary of G1 (and in fact a formalized version of G1): if T is consistent, then the consistency of T is not provable in T where the consistency of T is formulated as the arithmetic formula which says that there exists an unprovable sentence in T. Gödel [43] sketches a proof of G2 and promises to provide full details in a subsequent publication. This promise is not fulfilled, and a detailed proof of G2 for first-order arithmetic only appears in a monograph by Hilbert and Bernays [59]. Abstract logic-free formulations of Gödel's incompleteness theorems have been given by Kleene [76] ("symmetric form"), Smullyan [121] ("representation systems"), and others. The following is a modern reformulation of Gödel's incompleteness theorems.

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Section: Definition 44 ([74]mentioning

confidence: 99%

“…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.…”

Section: 2mentioning

confidence: 99%

Section: Definition 44 ([74]mentioning

confidence: 99%

Section: Definition 44 ([74]mentioning

confidence: 99%

Section: Definition 44 ([74]mentioning

confidence: 99%

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Current Research on Gödel’s Incompleteness Theorems

Cheng¹

2021

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Gödel’s Second Incompleteness Theorem: How It Is Derived and What It Delivers

Salehi

2020

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The proofs of Gödel (1931), Rosser (1936), Kleene (first 1936 and second 1950), Chaitin (1970), and Boolos (1989) for the first incompleteness theorem are compared with each other, especially from the viewpoint of the second incompleteness theorem. It is shown that Gödel’s (first incompleteness theorem) and Kleene’s first theorems are equivalent with the second incompleteness theorem, Rosser’s and Kleene’s second theorems do deliver the second incompleteness theorem, and Boolos’ theorem is derived from the second incompleteness theorem in the standard way. It is also shown that none of Rosser’s, Kleene’s second, or Boolos’ theorems is equivalent with the second incompleteness theorem, and Chaitin’s incompleteness theorem neither delivers nor is derived from the second incompleteness theorem. We compare (the strength of) these six proofs with one another.

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Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic

Blanck¹

2021

The Review of Symbolic Logic

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There has been a recent interest in hierarchical generalizations of classic incompleteness results. This paper provides evidence that such generalizations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “ $\mathrm {T}$ is $\Sigma _n$ -ill” is a canonical example of a $\Sigma _{n+1}$ formula that is $\Pi _{n+1}$ -conservative over $\mathrm {T}$ .

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Gödel–Rosser's Incompleteness Theorem, generalized and optimized for definable theories

Cited by 9 publications

References 9 publications

Current Research on Gödel’s Incompleteness Theorems

Current Research on Gödel’s Incompleteness Theorems

Gödel’s Second Incompleteness Theorem: How It Is Derived and What It Delivers

Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic

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