On 27 Aug 2009, at 08:19, marc.geddes wrote: > But is there a form of math more powerful than algebra? Yes, > Category/ > Set Theory! Unlike algebra, Category/Set theory really *can* fully > reason about itself, since Sets/categories can contain other Sets/ > Categories. Greg Cantor first explored these ideas in depth with his > transfinite arithmetic, and in fact it was later shown that the use of > transfinite induction can in theory bypass the Godel limitations. (See > Gerhard Gentzen)
Zermelo Fraenkel theory has full transfinite induction power, but is still limited by Gödel's incompleteness. What Gentzen showed is that you can prove the consistency of ARITHMETIC by a transfinite induction up to epsilon_0. This shows only that transfinite induction up to epsilon_0 cannot be done in arithmetic. Algebra escapes Gödel's limitation by being to weak. Gödel's limitation applies to *any*effective and rich theory, like category theory or set theory. I agree with your critics on Bayesianism, because it is a good tool but not a panacea, and it does not work for the sort of credibility measure we need in artificial intelligence. Not sure about what you say about Bohm's formulation of QM. In my opinion he uses the many worlds, and selects one world by reintroducing particles or singularities in the field. This introduces zombie with no body, yet they talk and act like us. (and it is Georg Cantor, not Greg). 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 -~----------~----~----~----~------~----~------~--~---
