Concerning the Representability of Self-Reference 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.