Measuring Non-computability: Can we relate this to the 3 dimensions of space? John Small (Mindoro Marine Ltd, Faversham, Kent United Kingdom) P1
Certain propositions which are undecidable from a set of axioms can become decidable when the axioms are extended. The amount of information that has to be added to make an undecidable proposition become decidable is a measure of the degree of non-computability. In some sense this is an analog of tipping the light cone, which makes events outside the light cone come within the light cone. The self-referential logic used in proofs of the non-existence of a general algorithm that can solve the halting problem depend on parallel transport on the sphere S1, but other higher dimensional spheres exist which also allow parallelisation, S3 and S7. This suggests that the measure of non-computability might have more than one dimension. It may be possible to show that space must have 3 dimensions. Pursuing this idea does not yet yield answers, but it does yield better questions which is almost as good.