This is really a question for Jeff, but I hope others will be interested as well. Here is the thing: next week NewAPPS will be hosting a symposium on a text by Paul Livingstone which presents a comparative analysis of Gödel’s incompleteness theorems, and Graham Priest and Derrida (yes indeed!) on diagonalization. It is an interesting text, even though some of the more metaphysical claims seem a bit over-the-top to me (as I will argue in my contribution to the symposium). But anyway, so I’ve been thinking about the whole Gödel thing again and how wide-ranging the conclusions drawn by Livingstone really are, and this got me thinking about the conditions that a system must satisfy for the Gödel argument to go through. It is clear that containing arithmetic is a sufficient condition for the argument to go through, as arithmetic allows for the encoding which then gives rise to the Gödel sentence.
But my question now is: is containing arithmetic a necessary condition in a system for the Gödel argument to go through? It seems to me that the answer should be negative. In fact, I recall that many years ago I heard Haim Gaifman saying that any other suitable encoding technique would be enough for the argument to run; arithmetization is just a particularly convenient encoding method. So my first question is whether this is indeed correct, i.e. that containing arithmetic is not a necessary condition for a Gödel incompleteness argument to take off. The second question is whether there are interesting examples of systems that can be proved to be incomplete by a Gödel argument even though they do not ‘contain arithmetic’ (I have the feeling that even the concept of ‘containing arithmetic’ might need to be clarified).