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

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Paulson, LC 


An Isabelle/HOL formalisation of G"odel's two incompleteness theorems is presented. The work follows 'Swierczkowski's detailed proof of the theorems using hereditarily finite (HF) set theory. 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 is shown to scale to a development of this complexity, while de Bruijn indices 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.



Godel's incompleteness theorems, Isabelle/HOL, Nominal syntax, Formalisation of mathematics

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Journal of Automated Reasoning

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Springer Science and Business Media LLC
Jesse Alama drew my attention to Swierczkowski, the source material for this ´ project. Christian Urban assisted with nominal aspects of some of the proofs, even writing code. Brian Huffman provided the core formalisation of type hf. Dana Scott offered advice and drew my attention to Kirby. Matt Kaufmann and the referees made many insightful comments.