Re: Provable vs Computable

2001-06-06 Thread juergen

from George Levy, 1 Jun 2001:
 A purely mechanical model no matter how complicated, including random
 variables, cannot replicate the results generated by Quantum mechanics +
 probability theory. This is exactly what Bell's inequality implies. In
 fact Bell proved his inequality using Quantum theory and probability.
 Therefore Juergens' erector (fr: meccano) set approach using pseudo-random
 generators, would definitely violate Bell's inequality theorem, and would
 not be phenomenally or experimentally equivalent to quantum mechanics.

I do not agree, of course.  Since you keep insisting on this, I suggest
you clearly write down all implicit assumptions and why exactly you
believe Bell's inequality is not compatible with pseudorandomness and
algorithmic TOEs.

http://www.idsia.ch/~juergen/everything/html.html
http://www.idsia.ch/~juergen/toesv2/




Re: Provable vs Computable

2001-06-01 Thread George Levy


John and Hal, Bruno and all everythingers, sorry for the delay guys, I
was travelling and had lots of work. Bruno, I just scanned your post quickly.
It seems to me we are going in the right direction but I shall need time
to digest what you wrote. I shall reply to you later
Let me first reply to John and Hal because it is the shortest reply.
Let's go back to the original Juergens' post
[EMAIL PROTECTED] wrote:
> 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.
I replied:
>Such a universe would violate Bell' inequality theorem. Quantum randomness
>cannot be simulated by hidden variables. We have to move beyond
>realism..to get a model of objective reality we must first develop
a
>model of consciousness.
A purely mechanical model no matter how complicated, including random
variables, cannot replicate the results generated by Quantum mechanics
+ probability theory. This is exactly what Bell's inequality implies. In
fact Bell proved his inequality using Quantum theory and probability.
Therefore, Juergens' erector (fr: meccano) set approach using
pseudo-random generators, would definitely violate Bell's inequality theorem,
and would not be phenomenally or experimentally equivalent to quantum mechanics.
Some of his (our) choices are:
1) Quantum mechanics + probability -> Bell's inequality and give up
on a mechanical hidden variable, on pseudo random generators, and more
generally, on realism.
2) Something else of power equivalent to Quantum mechanics in describing
natureGood Luck!!! I do not believe the route to this solution
is the erector set technique. Many a 19th and early 20th century physicist
has broken a tooth on that bone!
George

jamikes wrote:
George, thanks for your reply, which is almost as
convoluted and
hard-to-follow as was my question. You wrote:
> I am not restricting anything. I am only saying that Juergens has
to
choose
> between violating Bell's inequality theorem and all that this
implies, or
not
> and all that this implies. My stand is that we shouldn't.
> George
>
So ;let me rephrase the question:
is your stand that if an imaginary universe would violate eg.
Bell's
theorem, it should be excluded from consideration as a possibility,
- or -
we should rather conclude that Bell's theorem (or any other fundemntal
"human" rule) has a limited validity and does not cover every possible
universe?
John



Re: Provable vs Computable

2001-05-22 Thread Hal Ruhl

Dear George:

I do not see how the aspect of Juergen's approach he cited at the 
initiation of this part of the thread causes a dilemma re Bell's 
inequality.  As I understand it the history h is not THE history until the 
applicable portion of h stops changing.  But p and q are both non 
halting.  No part of h may settle in a finite time.  So what? The generator 
q is not in h.  The fact that p may recompute virtually any amount of h 
depending on what happens to q(n) is not a non locality in h but rather a 
substitution of a new h for the old h.  The old h is no longer relevant.

However, p itself being global seems to me to allow the violation of Bell's 
inequality in its universe.  My formal system has the same feature as 
does a UD IMO.  As far as I know the violation of Bell's inequality was 
already well established in our universe.

Hal

At 5/22/01, you wrote:


jamikes wrote:

  George Levy [EMAIL PROTECTED] wrote
   Saturday, May 05, 2001 :
 
  (SNIP Jurgen's remark about such a universe whatever, my remark is not
  topical, rather principle:)
 
   Such a universe would violate Bell' inequality theorem. Quantum 
 randomness
   cannot be simulated by hidden variables. We have to move beyond
   realism..to get a model of objective reality we must first develop a
   model of consciousness.
  
   George
 
  Can you restrict a universe according to its compliance with or 
 violation of
  a theory, no matter how ingenious, or vice versa? Are WE the creators who
  has to perform according to some rules/circumstances of human logic or
  computability?
  John Mikes
  

I am not restricting anything. I am only saying that Juergens has to choose
between violating Bell's  inequality theorem and all that this implies, or not
and all that this implies. My stand is that we shouldn't.

George




Re: Provable vs Computable

2001-05-22 Thread jamikes

George, thanks for your reply, which is almost as convoluted and
hard-to-follow as was my question. You wrote:

 I am not restricting anything. I am only saying that Juergens has to
choose
 between violating Bell's  inequality theorem and all that this implies, or
not
 and all that this implies. My stand is that we shouldn't.
  George

So ;let me rephrase the question:
 is your stand that if an imaginary universe would violate eg. Bell's
theorem, it should be excluded from consideration as a possibility,
- or -
we should rather conclude that Bell's theorem (or any other fundemntal
human rule) has a limited validity and does not cover every possible
universe?
John




Re: Provable vs Computable

2001-05-22 Thread George Levy



jamikes wrote:

 George Levy [EMAIL PROTECTED] wrote
  Saturday, May 05, 2001 :

 (SNIP Jurgen's remark about such a universe whatever, my remark is not
 topical, rather principle:)

  Such a universe would violate Bell' inequality theorem. Quantum randomness
  cannot be simulated by hidden variables. We have to move beyond
  realism..to get a model of objective reality we must first develop a
  model of consciousness.
 
  George

 Can you restrict a universe according to its compliance with or violation of
 a theory, no matter how ingenious, or vice versa? Are WE the creators who
 has to perform according to some rules/circumstances of human logic or
 computability?
 John Mikes
 

I am not restricting anything. I am only saying that Juergens has to choose
between violating Bell's  inequality theorem and all that this implies, or not
and all that this implies. My stand is that we shouldn't.

George




Re: Provable vs Computable

2001-05-22 Thread Brent Meeker

Well I thought the whole point was to restrict the universe (that we're
in) by the anthropic principle.  But if the anthropic principle is to
meant to include all intelligent beings, then some theory will be
necessary to say in what respects the universe could differ and still
produce intelligent beings.

Have you read Tegmark's paper, quant-ph/9907009v2 10 Nov 1999, which
shows that entangled quantum probabilities are not necessary for
consciousness - only ordinary randomness.

Brent Meeker 
As far as the laws of mathematics refer to reality, they are not   
certain, and as far as they are certain, they do not refer to reality. 

-- Albert Einstein

On 22-May-01, George Levy wrote:
 
 
 jamikes wrote:
 
 George Levy [EMAIL PROTECTED] wrote
  Saturday, May 05, 2001 :
 
 (SNIP Jurgen's remark about such a universe whatever, my remark is
 not
 topical, rather principle:)
 
 Such a universe would violate Bell' inequality theorem. Quantum
 randomness
 cannot be simulated by hidden variables. We have to move beyond
 realism..to get a model of objective reality we must first
 develop a
 model of consciousness.
 
 George
 
 Can you restrict a universe according to its compliance with or
 violation of
 a theory, no matter how ingenious, or vice versa? Are WE the
 creators who
 has to perform according to some rules/circumstances of human logic
 or
 computability?
 John Mikes
 
 
 I am not restricting anything. I am only saying that Juergens has to
 choose between violating Bell's inequality theorem and all that this
 implies, or not and all that this implies. My stand is that we
 shouldn't.
 
 George
 
Regards




Re: Provable vs Computable

2001-05-21 Thread jamikes


George Levy [EMAIL PROTECTED] wrote
 Saturday, May 05, 2001 :

(SNIP Jurgen's remark about such a universe whatever, my remark is not
topical, rather principle:)

 Such a universe would violate Bell' inequality theorem. Quantum randomness
 cannot be simulated by hidden variables. We have to move beyond
 realism..to get a model of objective reality we must first develop a
 model of consciousness.

 George

Can you restrict a universe according to its compliance with or violation of
a theory, no matter how ingenious, or vice versa? Are WE the creators who
has to perform according to some rules/circumstances of human logic or
computability?
John Mikes





Re: Provable vs Computable

2001-05-06 Thread Hal Ruhl

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 

Re: Provable vs Computable

2001-05-06 Thread Hal Ruhl

Dear Juergen:

My little isomorphism tree did not transmit very well.

The second column heading should read:  horizontal on this page isomorphic 
links
The third should read: the isomorphically linked string

All the binary strings should be under the third column.

Yours

Hal



 * 
:::*  =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 

Re: Provable vs Computable

2001-05-05 Thread Saibal Mitra

George Levy wrote:


 [EMAIL PROTECTED] wrote:

  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.

 Such a universe would violate Bell' inequality theorem. Quantum randomness
 cannot be simulated by hidden variables. We have to move beyond
 realism..to get a model of objective reality we must first develop a
 model of consciousness.

 George

I disagree. Hidden variables are indeed excluded, but that doesn't mean that
deterministic models proposed by J├╝rgen or 't Hooft are in conflict with
Bell's theorem. In the case of the model proposed by 't Hooft, you have a
universe that is very chaotic. Quantum mechanics arises in a statistical
description of the theory. Particles such as electrons, photons etc. don't
describe the degrees of freedom of the original deterministic theory, but
rather they arise only in the statistical description of this theory. In
other words: Mach was right in not believing that atoms exist.

In the case of the two slits experiment,
a hidden variable theory would tell you through what particular slit an
elecron travelled, and this is not possible. Okay, but does the electron
exist in the first place? I think not. The electron is just a mathematical
tool that allows you to calculate probabilities and is unphysical, just like
virtual particles and ghosts in Feynman diagrams. Why believe in electrons,
but not in the Fadeev-Popov ghost?

Saibal




Re: Provable vs Computable

2001-05-05 Thread Marchal

scerir wrote:

Juergen Schmidhuber 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?

Many people think that if a formal statement is neither provable nor 
refutable, then it should be considered neither true, nor false. 
But it is not that way that we - normally - use the term true. 
Somebody wrote: Suppose that I have a steel safe that nobody
knows the combination to. If I tell you that the safe contains 100 
dollars - and it really does contain 100 dollars - then I'm telling the 
truth, whether or not anyone can prove it. And if it doesn't contain 100 
dollars, then I'm telling a falsehood, whether or not anyone can prove it.
(A multi-valued logics can deal with statements that are either definitely 
true or definitely false, but whose actual truth value may, or may not,
be known, or even be knowable.).


That is basicaly the difference between classical logic 
with gap between proof and truth, and intuitionistic
logic, or constructive logic, which equivote truth and provability.
And that is something which will be translated in the language of
the machine ... It is part of the proof I explain currently.


I have begin the explanations of logic with classical logics.
But other logics are fundamental in the derivation (mainly
intuitionist and quantum logics).


Bruno




Re: Provable vs Computable

2001-05-05 Thread Marchal

Juergen Schmidhuber 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.

You know that Hal Ruhl doesn't distinguish computability and provability, 
so
it is open for me if his approach is nearer your's or mine.

The difference relies more between averaging on the (local) set
of consistent extensions defined in the whole UD*, or  finding a priori
defining universes probabilities, or Universal prior.

I communicate with the sound lobian machine because I agree (in 
arithmetic)
with the laws of exclude middle, and all classical logic.
Provable plays the role of thoroughly verifiable scientific
communication. (BTW *you* were the guy asking for formalisation!). Now
we are working at the metalevel (as George aptly remarks) and we will 
interview the machine on its self-reference abilities. Recall the goal 
consists in translating  UDA in a language interpretable by a sound UTM.
And UDA is a self-referential thought experiment.

It will happen that the incompleteness phenomenon will force us to
take into account the nuance between []p and ([]p  p) and 
([]p  p) in the discourse of the machine. They will correspond to 
provable p, 
knowable p, probability(p)=1. 
Knowable will give rise to intuitionist logic and probability 1 will give
quantum logic.

Probability(p) 1 will really be no more that 1) there is consistent 
extension,
2) p is true in all those consistent extension. (Only in an ideal frame
we have []p - p, remember that in the cul-de-sac world []p is always 
true).


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?


Wait and see. Remember I told George we have not yet really 
beging the proof. The hard and tiedous thing is to arithmetise
the provability predicate.
I will define knowledge and observable (in the UDA sense)
 *in* the language of the machine and I will show that the observable
propositions obeys some quantum logic. I will only consider the case
observable with a probability one. This will give a concrete 
purely arithmetical interpretation of a quantum logic.
The probabilities are taken on the set of relative consistent 
(UD accessible) extensions, and by consistent I just mean -[]-
with [] Goedel's provability predicate. (so you can guess the
role of provability).
The UD will be translated in the form of the set of 
all (true) \Sigma_1 sentences.


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?


It has to do with the origin of the belief in universe(s) once
we bet we do survive digital substitution. 


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.


Actually, provability is just a step in my derivation (and we have still
not begin to discusse it! nor to define it). We have just seen some 
modal logic which have a priori nothing to do with provability.
You are still anticipating.

It is a good thing you are open to unprovable aspects, and it makes 
weirder you are not open to uncomputable aspects. (Although I know
provability is relative and computability is absolute (Church's Thesis)
Do you really believe than one of us limit universes to sets of provable
theorems. I am myself just defining the local *discourse* of a 
machine-scientist.



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.


I have no problem with that. As you should know from our earlier
discussion. Remember that the big role in my work comes from 
(G* minus G), which
is a logic of the *unprovable* statements. G ang G* will be defined
formally soon, but you can also consult Solovay 1976 or Boolos ...
By logic here I mean a well defined set (of formulas) logically
closed for modus ponens.


Note also that observers evolving within the universe ...


The UDA shows that such an expression has no meaning.
The movie graph (or Maudlin's Olympia) illustrates how non trivial
the mind-body problem is with comp.
I am aware there is something very hard to swallow here. 
But it is a consequence of comp. 


 ...may write
books about all kinds of unprovable things; they may also write down
inconsistent 

Re: Provable vs Computable

2001-05-04 Thread George Levy



[EMAIL PROTECTED] wrote:

 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.

Such a universe would violate Bell' inequality theorem. Quantum randomness
cannot be simulated by hidden variables. We have to move beyond
realism..to get a model of objective reality we must first develop a
model of consciousness.

George




Re: Provable vs Computable

2001-05-04 Thread scerir

Juergen Schmidhuber 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?

Many people think that if a formal statement is neither provable nor 
refutable, then it should be considered neither true, nor false. 
But it is not that way that we - normally - use the term true. 
Somebody wrote: Suppose that I have a steel safe that nobody
knows the combination to. If I tell you that the safe contains 100 
dollars - and it really does contain 100 dollars - then I'm telling the 
truth, whether or not anyone can prove it. And if it doesn't contain 100 
dollars, then I'm telling a falsehood, whether or not anyone can prove it.
(A multi-valued logics can deal with statements that are either definitely 
true or definitely false, but whose actual truth value may, or may not,
be known, or even be knowable.).
- scerir





Re: Provable vs Computable

2001-05-04 Thread jamikes



 Juergen Schmidhuber 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?

scerir writes an enjoyable version on the last part of the quote.

Let me address the first part about possible universes. Of course Juergen
was
cautious and included logically in his phrase.
 Logically most likely refers to human (on this list: even mathematical)
logic.
Do we really think that human (math) logic is the restrictive principle for
nature?
What we see (what we want to see?) seems to point to that, but do we see'em
all?
Isn't possible what we don't see or understgand or realize?
Didn't our horizon (logic, math) increase over some time? Are we at the end?
In considering plenitude/multiverse, does it make sense to select part of it
(maybe a small, unimportant segment only)?
Even if we cannot develop knowledge about the rest, we should not deny its
possibility of existence. The farthest from this list would be a closed
mind!
John Mikes




Re: Provable vs Computable (post not done)

2001-05-04 Thread Hal Ruhl

Sorry, some how my mailer decided I wanted to send this.
Clearly it is not done.


Dear Juergen:

I am not so much interested in provability as I am in whether or not the 
noise in a universes history is pseudorandom or random and forging an .




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/