A Formally Verified Abstract Account of Gödel’s Incompleteness Theorems

Popescu, Andrei; Traytel, Dmitriy

doi:10.1007/978-3-030-29436-6_26

Cited by 5 publications

(3 citation statements)

References 33 publications

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“…9). This article extends our CADE 2019 conference paper [43] with a significantly more finegrained and self-contained presentation of the results, which includes lemmas and detailed proof sketches. Given the existence of formal proofs in Isabelle, one may question the usefulness of paper proof sketches; however, we believe these are important for reaching out to a wider audience-perhaps interested in following the reasoning behind our fine-grained results discovered with the help of Isabelle, but not willing to read and understand Isabelle scripts.…”

Section: Introductionmentioning

confidence: 91%

Distilling the Requirements of Gödel’s Incompleteness Theorems with a Proof Assistant

Popescu

Traytel

2021

J Autom Reasoning

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We present an abstract development of Gödel’s incompleteness theorems, performed with the help of the Isabelle/HOL proof assistant. We analyze sufficient conditions for the applicability of our theorems to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosser’s variation of the first theorem, Jeroslow’s variation of the second theorem, and the Świerczkowski–Paulson semantics-based approach. As part of the validation of our framework, we upgrade Paulson’s Isabelle proof to produce a mechanization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation.

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Section: Introductionmentioning

confidence: 91%

Distilling the Requirements of Gödel’s Incompleteness Theorems with a Proof Assistant

Popescu

Traytel

2021

J Autom Reasoning

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“…If we move to Gödel's incompleteness theorems, the first one has been formalized in the Boyer-Moore theorem prover by Shankar in 1986 [29] and in Coq by O'Connor in 2003 [22]. Both incompleteness theorems have been formalized in Isabelle/HOL by Paulson in 2013 [23] and by Popescu and Traytel in 2019 [25].…”

Section: A History Of Formalized Completeness Proofsmentioning

confidence: 99%

Formalizing Henkin-Style Completeness of an Axiomatic System for Propositional Logic

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2023

Lecture Notes in Computer Science

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“…• Paulson [45,46,47] formalized in Isabelle/HOL Gödel's Incompleteness Theorems, but this does not include a completeness proof. • Popescu and Traytel present a formalization of Gödel's Incompleteness Theorems [51]. • Jensen, Hindriks and Villadsen [29,30] also present an approach to formalize in Isabelle/HOL a verification framework for agent programs.…”

Section: Related Workmentioning

confidence: 99%

Interactive Theorem Proving for Logic and Information

Villadsen¹,

From²,

Jensen³

et al. 2021

Studies in Computational Intelligence

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Automated reasoning is the study of computer programs that can build proofs of theorems in a logic. Such programs can be either automatic theorem provers or interactive theorem provers. The latter are also called proof assistants because the user constructs the proofs with the help of the system. We focus on the Isabelle proof assistant. The system ensures that the proofs are correct, in contrast to pen-and-paper proofs which must be checked manually. We present applications to logical systems and models of information, in particular selected modal logics extending classical propositional logic. Epistemic logic allows intelligent systems to reason about the knowledge of agents. Public announcements can change the knowledge of the system and its agents. In order to account for this, epistemic logic can be extended to public announcement logic. An axiomatic system consists of axioms and rules of inference for deriving statements in a logic. Sound systems can only derive valid statements and complete systems can derive all valid statements. We describe formalizations of sound and complete axiomatic systems for epistemic logic and public announcement logic, thereby strengthening the foundations of automated reasoning for logic and information.

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A Formally Verified Abstract Account of Gödel’s Incompleteness Theorems

Cited by 5 publications

References 33 publications

Distilling the Requirements of Gödel’s Incompleteness Theorems with a Proof Assistant

Distilling the Requirements of Gödel’s Incompleteness Theorems with a Proof Assistant

Formalizing Henkin-Style Completeness of an Axiomatic System for Propositional Logic

Interactive Theorem Proving for Logic and Information

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