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



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