Self‐reference and the Languages of Arithmetic

I here investigate the sense in which diagonalization allows one to construct sentences that are self‐referential. Truly self‐referential sentences cannot be constructed in the standard language of arithmetic: There is a simple theory of truth that is intuitively inconsistent but is consistent with Peano arithmetic, as standardly formulated. True self‐reference is possible only if we expand the language to include function‐symbols for all primitive recursive functions. This language is therefore the natural setting for investigations of self‐reference.