### Accidental and other fixpoints

That A is a fixpoint for B(x) does not necessarily mean that A "says
of itself that it (i.e. its Gödel number) has the property B(x)" in
any reasonable sense. For example, if the sentence "0=0" has the Gödel
number 889283477878, "0=0" is by definition a fixpoint for
"x=889283477878", but "0=0" is not self-referential.
The fixpoint A for B(x) shown to exist in the proof of the diagonal
lemma is self-referential in the following sense: A has the form
B(t), where t=#A is provable in T. It is only through the use of such
self-referential formulas that we know that every B(x) in fact has a
fixpoint.

Nevertheless, in the proof of Gödel's theorem, the fact that the
Gödel sentence is self-referential is not used, but only the fact that
it is a fixpoint for the predicate "x is not a theorem of T".