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 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.

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