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/
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Received on Thu Aug 27 2009 - 09:35:17 PDT