Fifty years of self-reference in arithmetic

It is now fifty years since Hans Hahn first presented an abstract of the then unknown Kurt Godel to the Vienna Academy of Sciences. The rest, as it is said, is history. Much of this history is well-known and I do not propose to repeat the usual platitudes. On any golden anniversary, however, it is natural to look back and I am not one to rebel against nature. On this occasion I will sing the hitherto unsung song of diagonalisation. While self-reference is one of the more outstanding features of GodePs work and self-reference in arithmetic has had some no-table success, this success is neither so widely known nor so great that my message should bore the average reader. One of the great curiosities of my topic is how long it took (perhaps better: is taking) for the subject to develop. Even the most obvious and central fact—the Diagonalisation Theorem—seems to have had difficulty in surfacing. It is not to be found in many of the basic textbooks (e.g., Kleene [20], Mendelson [25], Shoenfield [38], Bell and Machover [1], and Manin [24]) and it is only stated in its most rudimentary form in most others (e.g., Boolos and Jeffrey [31, Enderton [4], and Monk [26]). The two most substantial expositions of Incompleteness Theory (Mostowski [28] and Stegmuller [47]) offer no explicit statement of the Diagonalisation Theorem in any form. Indeed, it is only in a recent more advanced exposition (Boolos [2]) that the full Diagonalisation Theorem has finally graced the pages of a book. Yet, diagonalisation in arithmetic is fifty years old and was stated in proper generality in print in 1962 [27]—long before most of the available textbooks were written. Perhaps, before writing another word, I should outline the history of the Diagonalisation Theorem. While I have not made an exhaustive search of the literature, I can report that a cursory examination of the more important papers yields the following development: