# Self-referential Gödel numberings and uniformity

## Léon Probst

In arithmetic, self-reference relies, among other things, on a correct choice of Gödel numbering. Interestingly, one can build self-reference directly into a numbering, which allows us to get the diagonal lemma for free, i.e. bypassing the usual arithmetic of the syntax. For this reason, self-referential numberings are often considered inadequate in a philosophical interpretation of certain metamathematical theorem. However, it is difficult to impose a precise criterion on the numberings (such as monotonicity) to exclude these numberings. Most common criteria fail in this respect or seem unmotivated (cf. [1]).

This talk introduces *uniformity* as a new criterion for numbering. First, we show that standard numberings (on the common conceptions of syntax), as well as other standard codings (e.g. of ordered pairs), are indeed uniform. Second, we show how our criterion excludes the non-standard numberings, including the deviant and self-referential ones. Finally, we discuss its philosophical motivations and some of its implications.

Joint work with Balthasar Grabmayr.

## Bibliography

- Grabmayr, Balthasar & Visser, Albert,
*{Self-Reference Upfront: A Study of Self-Referential Gödel Numberings}*,,vol. 16 (2023), no. 2, pp. 385–424.*Review of Symbolic Logic*