  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.
