On 05 May 2009, at 22:31, Jason Resch wrote:
> > On Sun, May 3, 2009 at 10:56 AM, Bruno Marchal <marc...@ulb.ac.be> > wrote: >> >> >> With just arithmetic, when we stop to postulate a primitive or >> ontological material world, all primitive ad-hocness is removed, >> given >> that the existing internal interpretations are all determined, with >> their relative frequency, by addition and multiplication rules, and >> physics will be defined by the (absolute) probability of relative >> computations (here = probability of relative number theoretical >> relations. > > > Bruno, > > In other posts I have seen you mention that the rule of succession is > not enough, that addition and multiplication are needed. Why is it > that it stops at multiplication, and not exponentiation or tetration? > Is it enough to say some form of iteration + succession are required? > (e.g. a for loop with succession gives addition, a for loop with > addition yields multiplication, etc.) > > Jason > It is due to the fact that, when formalized (in first order logic, say) Turing Universality begins with addition and multiplication (you don't even need succession). Then you can define exponentiation, tetration, etc. All partial recursive function can then be defined. Succession + addition, or succession + multiplication, are not Turing Universal, and leads indeed to decidable theories. For the "ontology" we need no more than a universal system. it determines the universal dovetailing. For the "epistemology" we need succession, addition, multiplication and the axioms of induction. This gives a notion of universal system together with its internal "self-aware" substructures played by the Lobian machine and their consistent extensions (the believer in induction), simulated by the universal systems. Those internal machines will develop far beyond "simple induction" though. The general internal view (the first person plenitude) is not axiomatisable. Iteration and succession? I don't think so. You need induction. With induction it is Turing Universal, but not without, I think. It could depend how you formalized the iteration rule, but without induction and staying in first order logic, that would astonish me. The crazy thing, not so simple to prove, is that even without induction, addition + multiplication is Turing universal. You bypass the role of induction by defining finite sequence through Gödel bata function and an ingenuous use of the Chinese Lemma. Far easier to prove, without induction, is that addition+multiplication +exponentiation is Turing universal, but thanks to Godel' beta function you can eliminate exponentiation. If you know "Godel's original "Godel's numbering" you can guess why. Bruno > > http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
