A Mechanised Proof of Gödel’s Incompleteness Theorems Using Nominal Isabelle

An Isabelle/HOL formalisation of Gödel’s two incompleteness theorems is presented. The work follows Świerczkowski’s detailed proof of the theorems using hereditarily finite (HF) set theory (Dissertationes Mathematicae 422 , 1–58, 2003 ). Avoiding the usual arithmetical encodings of syntax eliminates...

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Bibliographic Details
Published in:Journal of automated reasoning Vol. 55; no. 1; pp. 1 - 37
Main Author: Paulson, Lawrence C.
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.06.2015
Springer Nature B.V
Subjects:
Artificial Intelligence
Computer Science
Mathematical Logic and Formal Languages
Mathematical Logic and Foundations
Symbolic and Algebraic Manipulation
Isabelle/HOL
Nominal syntax
Gödel’s incompleteness theorems
Formalisation of mathematics
ISSN:0168-7433, 1573-0670
Online Access:Get full text
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Summary:An Isabelle/HOL formalisation of Gödel’s two incompleteness theorems is presented. The work follows Świerczkowski’s detailed proof of the theorems using hereditarily finite (HF) set theory (Dissertationes Mathematicae 422 , 1–58, 2003 ). Avoiding the usual arithmetical encodings of syntax eliminates the necessity to formalise elementary number theory within an embedded logical calculus. The Isabelle formalisation uses two separate treatments of variable binding: the nominal package (Logical Methods in Computer Science 8 (2:14), 1–35, 2012 ) is shown to scale to a development of this complexity, while de Bruijn indices (Indagationes Mathematicae 34 , 381–392, 1972 ) turn out to be ideal for coding syntax. Critical details of the Isabelle proof are described, in particular gaps and errors found in the literature.
Bibliography:SourceType-Scholarly Journals-1
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ISSN:0168-7433
1573-0670
DOI:10.1007/s10817-015-9322-8