Dear Juergen: You demonstrate my point.

At 4/12/01, you wrote: >Hal, here is an infinite chain of provable unique theorems: >1+1=2, 2+1=3, 3+1=4, 4+1=5, ... First these are not theorems they are proof chains ending in theorems. For example: "4 + 1 =" is a proof chain and the theorem proved is: "5" is a number. The base or data for the proof chain is "4 is a number" which has just been proven by the previous step in the cascade. Now notice that the data or base theorem [a well formed base 10 string] for each proof chain in the cascade counts up in value. Therefore it must inexorably increase in length. Since most strings of length L are not compressible and have a complexity on the order of L + H(L) eventually the cascade will encounter a base theorem string that makes the proof chain itself too complex to fit into a number theory of a given finite complexity. The cascade must then stop or number theory must become more complex. Cascades do not have an internal stop rule. Here we have a contradiction. To cure it the applicable FAS must become more complex. Below I say the same thing in a more formal way. xxxxxxxxxxxxx Let us first arbitrarily call this theorem cascade #27 in number theory. So an individual theorem in cascade #27 is identified as T27(i). Let us look at say T27(4) in this cascade. Its decompressed proof program looks like: (1) P27(4) = {RULES27(T27(3)) + Self-delimiter27(4)} computes T27(4) [which is the base ten string "5"] Where: RULES27 = Add 1 to the base ten string that is the current data T27(3) = is the data for the proof program of T27(4) [which is just the base ten string "4"] The most compressed version of its proof program would be: (2) P27'(4) = {RULES27(P27'(3)) + Self-delimiter27'(4)} computes T27(4) [which is the base ten string "5"] Where: P27'(3) = the max compression of T27(3) Now let cascade #27 continue. Eventually the complexity of the data: P27'(i - 1) plus the complexity of RULES27 plus the complexity of Self-delimiter27'(i) when combined into P27'(i) makes P27'(i) exceed the complexity of number theory if number theory has a finite complexity. Then what happens? This cascade has no internal stop but it must stop [there is no rule of the cascade saying it can hop over unacceptably complex n(i) and even so the hop computation would become more and more complex as the base string gets more complex] or the FAS must get more complex. If the FAS gets more complex where does the added information come from? Does that not look a lot like: "If a FAS is consistent and finite and doing arithmetic it is incomplete"? So Godel is already a corollary of Turing and perhaps of Chaitin as well. Going a step further when looking at universes in general I currently see no need for the new information to preserve a FAS. It just gets a running cascade to the next iteration at which point more new information may be needed. So perhaps an essential dash of random oracle finishes the entree we call our universe. Hal