Dear Juergen:

I am not so much interested in provability as I am in whether or not the 
"noise" in a universe's evolution is pseudorandom or random and forging an 
Everything that was as free of information [selection] as possible.  I try 
to use incompleteness in various forms to show that as far as an individual 
universe is concerned the noise comes from outside that universe.

I believe that many proposals on this list are more in agreement than we 
might at first glance think.  Below I use the approach of producing numbers 
using the null set to try to demonstrate this.

Using the symbol * to represent the null set and {} to represent sets with 
elements:

                                                     horizontal 
the
                                                    on this 
page      isomorphically
                                                    isomorphic 
linked
anisomorphic                                     links                  string

         *            < 
::: >           *                  =                    0
                                         {*}                 = 
       1
                                       {*,{*}}               = 
       10
                                    {*,{*},{*,{*}}}          = 
      11
                                           :                   : 
           :
                                           :                   : 
           :
                                                              = 
11001110111...0110
                                                                                  ||
                                                                      active 
and inactive
                                                                    page 
"vertical" isomorphic
                                                                      links 
to this string
                                                               :
                                                               :
                                                               = 
11100110111...0011
                                                                                  ||
                                                                       active 
and inactive
                                                                    page 
"vertical" isomorphic
                                                                       links 
to this string
                                                                  : 

                                                                  :
                                                                etc.

The page horizontal isomorphic links on the right hand side are the usual 
ones.

The page vertical isomorphic links are the ones I have used in my 
model.  Each of these vertical isomorphic links [there can be more than one 
per string] uses a portion of the string to which it links to define its 
self contained FAS.  The rest of the string determines the associated state 
of the vertical isomorphism.  The current state of a vertical isomorphism 
is the active link for that isomorphism.   All inactive links are either 
past or future states of isomorphisms.  The self contained FAS of a 
particular vertical isomorphism determines which links are acceptable 
immediate successor states of that isomorphism.  Depending on the nature of 
the FAS there could be more than one acceptable immediate successor state 
for that isomorphism.

The symbol "< ::: >" indicates that the isomorphism tree structure - "The 
Everything" -  vanishes upon occasion and the anisomorphic null set - "The 
Nothing" - resumes.  This is the E/N alternation.  Neither the anisomorphic 
null set nor the isomorphism tree since they both contain no information 
can internally address the unavoidable question of their own 
durability.   This bilateral incompleteness drives the E/N 
alternation.  The alternation since it destroys any record of the previous 
mix of active/inactive vertical isomorphic links causes a new random 
active/inactive mix each time the isomorphism tree resumes.  This avoids a 
"selected" structure to The Everything.

In order for a vertical isomorphic link to transfer to another string both 
the current link and an acceptable successor link must be simultaneously 
active.  The transfer inactivates the prior link.  Vertical isomorphic 
links driven active or inactive by the E/N alternation absent an active 
acceptable successor is simply in stasis.  For a transfer to take place 
both the current and acceptable successor link must be simultaneously 
active.  It is the transfer that is an "event" to an isomorphism.

Are UD's or other such string generating machines vertical isomorphic 
links?  I think so if I understand them correctly.  Simply concatenate all 
the output strings of a UD so that successor vertical link shifts are just 
based on very localized regions of the overall string.

I also think this process is similar to "observer moments" if "observer 
moment" means active links.

Since the null set just "is" then the process may satisfy the idea that the 
Everything just "is".
It is the isomorphic tree that undergoes change.

Now I see nothing wrong with FAS that have a "do not care" content to their 
rules determining acceptable successor links.

My goal is to try to show that universes [vertical isomorphic links] that 
can support SAS have at least some "do not care" content in the rules.  To 
do this I turn to Chaitin and explore the viability of deterministic 
cascades.  By deterministic I mean each state has only one possible prior 
and one possible successor - the computational exercise is by definition 
elegant.  However, the complexity of the strings to which such sequences 
form isomorphic links must relentlessly increase .  This produces a 
conflict between the idea of cascade and the idea of a limit on the 
complexity of the string that the finite self contained FAS can determine 
is acceptable using only elegant programs.

This conflict is resolved by an injection of complexity into the FAS from 
"outside".

It is also reasonable that some of these FAS are subject to the 
incompleteness of Godel.

So it is just Chaitin and Godel that interest me as far as "proof" is 
concerned.

Other than the above I really can not find a simple thing in your post 
below with which I disagree.

Yours

Hal

At 5/4/01, you wrote:

>Which are the logically possible universes?  Max Tegmark mentioned
>a somewhat vaguely defined set of ``self-consistent mathematical
>structures,'' implying provability of some sort. The postings of Bruno
>Marchal and George Levy and Hal Ruhl also focus on what's provable and
>what's not.
>
>Is provability really relevant?  Philosophers and physicists find
>it sexy for its Goedelian limits. But what does this have to do with
>the set of possible universes?
>
>I believe the provability discussion distracts a bit from the
>real issue. If we limit ourselves to universes corresponding to
>traditionally provable theorems then we will miss out on many formally
>and constructively describable universes that are computable in the
>limit yet in a certain sense soaked with unprovable aspects.
>
>Example: a never ending universe history h is computed by a finite
>nonhalting program p. To simulate randomness and noise etc, p invokes a
>short pseudorandom generator subroutine q which also never halts. The
>n-th pseudorandom event of history h is based on q's  n-th output bit
>q(n) which is initialized by 0 and set to 1 as soon as the n-th element
>of an ordered list of all possible program prefixes halts.  Whenever q
>modifies some q(n) that was already used in the previous computation of
>h, p appropriately recomputes h since the n-th pseudorandom event.
>
>Such a virtual reality or universe is perfectly well-defined.  At some
>point each history prefix will remain stable forever.  Even if we know p
>and q, however, in general we will never know for sure whether some q(n)
>that is still zero won't flip to 1 at some point, because of Goedel etc.
>So this universe features lots of unprovable aspects.
>
>But why should this lack of provability matter? It does not do any harm.
>
>Note also that observers evolving within the universe may write
>books about all kinds of unprovable things; they may also write down
>inconsistent axioms; etc. All of this is computable though, since the
>entire universe history is.  So again, why should provability matter?
>
>Juergen Schmidhuber           http://www.idsia.ch/~juergen/toesv2/

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