Simplicity and Incompleteness

Gregory Chaitin’s “information-theoretic” incompleteness theorem (Chaitin, 1974a, 1974b) has been praised as “a dramatic extension of Gödel’s incompleteness theorem” with “devasting import” (Davis, 1978); accordingly, various ambitious philosophical conclusions from this result are common in the literature. In another paper (Raatikainen, 1998), I have shown, conclusively, that the received interpretation of Chaitin’s theorem is in all respects false, thus strengthening the earlier criticism by Michiel van Lambalgen (1989). In addition, I observed that one can, in fact, obtain Chaitin’s theorem as an easy consequence of the undecidability of the Halting problem. In this paper, I shall use a similar simple argument with respect to the undecidability of consistency to derive an analogous result. I do not intend to draw any strong philosophical conclusion from my own little result, either. Still, whatever real logical interest there is in Chaitin’s result, my result below is at any rate just as strong as it, and has moreover various evident advantages when compared to it; viz. my result is unquestionably simpler and less relative (cf. below Section 5); moreover, the undecidable problem in question has a more immediately logical character and a clear intuitive content. There is a certain parallel between the setting of the present work, and that of Gödel (1972) and Feferman and Solovay (1990).