On the Choice of Gödel Numberings

The choice of Gödel numberings is a source of indeterminacy in metamathematics. For instance, Gödel’s second incompleteness theorem fails under certain Gödel numberings. Nevertheless, it is widely believed that our standard metamathematical theorems hold for all reasonable Gödel numberings. This article investigates what makes a Gödel numbering reasonable and develops a general framework for addressing philosophical questions about Gödel numberings and, more generally, the arithmetisation of syntax.