  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.
