Concerning the Representability of Self-Reference in Arithmetic

TL;DR

This paper investigates the limitations of representing self-reference in arithmetic, revealing that certain self-referential terms lead to non-terminating expressions under standard encoding methods.

Contribution

It provides a detailed analysis showing that self-referential terms in arithmetic cannot be finitely represented using conventional encoding functions, highlighting fundamental limitations.

Findings

Self-referential terms produce non-terminating expressions.

Standard encoding methods cannot finitely represent certain self-reference.

The results challenge assumptions about representability in arithmetic.

Abstract

Terms in arithmetic of the form s in the formula s=t(< s >), with t a term with one free variable and < s > a numeral denoting the G\"odel number of s, are examined by writing the explicit definition of the encoding functions whose representation they include. This is first done with a specific encoding function and system of encoding and then examined more generally. The surprising result of each such construction, involving conventionally defined substitution or diagonalization functions and using conventional systems of encoding, is shown to be a non-terminating symbolic expression.