Here is a revised version of my comments on this subject.  I think it fixes 
several aspects of what I have had to say earlier.

Standalone deterministic evolving universes:

Such a universe is describable as a concatenation of single output programs 
of the form:

          Rj(Aj) -> B; Rj(B) -> C; ... Rj(F) -> G; Rj(G) -> H; Rj(H) -> I; 

"Rj" is a subset of the rules of an N-bit FAS.  "Aj" is an axiom of that 
FAS.  There are no potential branches in or out.  It is an elegant proof 
and defines the complexity of each of its objects B,C,...F,G,H,I ...

Because the proof is everywhere elegant each successive object is more 
complex than its preimage.  This is the type of theorem cascade I was 
trying to identify.

It has a problem: It will - if it does not stop - eventually outrun the 
complexity of its N-bit FAS.  Since the proof is known to be elegant this 
is [as I understand it] a contradiction of Chaitin's result.

However, if it stopped at complexity N + c as required by Chaitin [again I 
think] the end state would be highly but finitely complex.  Highly but 
finitely complex states are going to have some consequent under the rules 
of the cascade.  It can not be both "provably highly but finitely complex" 
and "absent operative correlations" under the same set of finite rules.  It 
might reach an absence of operative correlations when it becomes infinitely 
complex and the domains of any correlations fall below the resolution of 
the still finite Rj of the cascade.  So the cascade can not stop prior to 
an infinity complex end state.

This seems to be a contradiction.  It can be cured if the FAS is able to 
spontaneously take in additional complexity from an outside source.  I call 
such a source a Superverse.  The new information would act to replace the 
latest object of the cascade with a new axiom acceptable to the Rj and the 
cascade proceeds.

I use this to argue in favor of non deterministic standalone universes.

Now this universe is actually an isomorphism linked to the successive 
strings [objects] B, C etc.  The isomorphism has the same Rj as its laws of 

This should continue to hold even if one shifts to a universal dovetailer 
running all universes simultaneously.  Individual isomorphisms would still 
jump to the string currently containing as the latest sub string [string 
structure:  ....;sub string;sub string;latest substring] that is the 
successor state of that isomorphism under its individual Rj.  Thus each 
individual isomorphism remains a single output program or elegant proof 
under its Rj.  A non deterministic result is sustained.

Random histories are just isomorphisms with Rj = "do not care" and they 
jump as easily as any isomorphism.

Well that is the current state of the argument.


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