Dear Juergen: At 4/11/01, you wrote: >Hal, Chaitin just says "you cannot prove 20 pound theorems with 10 pound >axioms".

Please refer to Chaitin's "The Unknowable" generally and page 25, Chapter V, and note 10 at the bottom of page 97 in particular. >But the infinite cascade of all provable theorems of number >theory collectively does not convey more information than the axioms. Do you consider the following as being a reasonable AIT expression of a theorem cascade "j"? 1) Pj(i) = {Rules(Pj(i -1)) + Self-delimiter(i)} is the shortest AIT program that computes Tj(i) where: Tj(i) = the "i"th theorem in the cascade Self-delimiter(i) = the required AIT self contained length of Pj(i) Pj(i -1) = the compressed form of Tj(i - 1) Rules(Pj(i -1)) = the rules of the FAS acting on the previous theorem as data If so and the cascade is the only way to arrive at any Tj(i) [which I think is inherent in the idea of theorem cascade] then each theorem Tj(i) in the cascade is more complex than its preimage Tj(i - 1) since Pj(i) contains Pj(i - 1) and thus Tj(i - 1) as data plus its own Self-delimiter's bit string making it longer than Pj(i -1). Actually the increase in complexity with each step just accelerates because the Self-delimiter has to grow in length. With the right class of isomorphism this can be interpreted as a universe with an accelerating expansion. > > But what if you declare theorems which say the same thing to be the > > _same theorem_, rewritten? > > Duraid > >Since a theorem is a symbol string, different strings cannot be the same >theorem. Yes and to quote Chaitin's note on page 97 referenced above "More precisely, the number of N-bit strings X such that H(X) < [N + H(N) - K] is less than 2^(N - K + c)". >Perhaps you mean you can easily derive certain theorems from >others stating "the same thing, rewritten". But this is vague - you can >derive all theorems from the axioms stating "the same thing, rewritten". Then perhaps you will agree that we need not add to the above those strings shorter than N since they are just "the same thing, rewritten" without a long boring prefix. Hal