Self-reference Upfront: A Study of Self-referential G\"odel Numberings

TL;DR

This paper investigates how different formalisation choices, especially self-referential numberings, influence the ability to formalise self-reference in arithmetic, revealing the importance of coding methods in self-referential reasoning.

Contribution

It provides a detailed analysis of self-referential numberings and demonstrates their impact on formalising self-reference in arithmetic, highlighting the sensitivity to coding choices.

Findings

Self-referential numberings enable strong self-reference in weak languages.

Formalising self-reference depends heavily on the coding apparatus used.

Sensitivity to coding choices affects the formal study of self-referential truth principles.

Abstract

In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study of certain principles of self-referential truth.