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From: Hal Ruhl <hjr.domain.name.hidden>

Date: Tue, 31 Oct 2000 22:21:38 -0800

I have been asked to compress my model so as to maximize its

accessibility. The following is an attempt to do so. The result is rather

cryptic and has little effort to "prove" anything. It is frequently just

an application of Occam's Razor.

BEGINNINGS

Start with:

1) Nothing - an empty finite consistent Formal Axiomatic System - an fc-FAS.

2) Nothing is incomplete because it can not address the meaningful question

of its own stability - it is an ifc-FAS.

3) To address the incompleteness Nothing becomes Something.

4) To address the question of the stability of Nothing, Something must have

a bifurcation to implement the concept of "Yes" vs "No", or "True" vs

"False", or "Is" vs "Is Not", or affirmation vs negation. [As in "Nothing

is unstable." is a well formed statement Yes or No.]

5) Something [in my view] takes the form of a growing, seething, foamy

fractal of such bifurcations that are mathematically isomorphic to 1's and

0's I call a superverse.

6) Some [perhaps all] alignments by happenstance of these bifurcations

within the Superverse are isomorphic to finite strings of N bits.

7) Some such N bit strings are isomorphic to fc-FAS's.

8) Some such fc-FAS are incomplete or ifc-FAS. [Open ended alignments?]

9) Such ifc-FAS are composed of :

a) A finite theorem set [no more than 2^[N + c] as per AIT.]

b) Each theorem has a finite AIT complexity less than or equal to N +

c bits

and thus

b) A finite symbol set or alphabet.

c) A finite set of axioms

d) A finite set of rules

10) Some such ifc-FAS are isomorphic to mathematical objects U that are

themselves isomorphic to universes.

11) All such universes have physical realization.

12) A low complexity class of such ifc-FAS have just one axiom.

13) The proposal is that our physical universe is isomorphic to a

mathematical object U that is described by an ifc-FAS in this class. More

precisely a currently realized physical configuration of the universe - a

CRPC(i) is isomorphic to a current string configuration - a U(i) that is

described by a currently expressed theorem of the ifc-FAS - a p(i).

MID STAGE DYNAMICS

14) The only dynamic available in such a ifc-FAS is a recursive enumeration

as follows:[viewing p(i) as an AIT shortest possible program]

p(i) = {R acting on p(i - 1), PL(i)} will compute U(i) isomorphic to

CRPC(i).

where R is the rule set of the ifc-FAS, PL(i) is the self delimiter of AIT and

p(1) = {R acting on [single axiom], PL(1)} will compute U(1) isomorphic to

CRPC(1) and

[single axiom] describes U(0) isomorphic to CRPC(0).

15) Dynamic results so far:

a) p(i) gets longer with each step of the recursion.

b) Thus the information content of U(i) is higher than U(i - 1) isomorphic

to the Second Law.

16) The ifc-FAS as stated has a finite theorem set each with a finite

amount of information. I think the simplest [perhaps only] isomorphic

physical realization consistent with the above is a structure composed of

discrete points. The points are isomorphic through the mathematical object

U to the alphabet of the ifc-FAS.

17) At the moment we have a potential contradiction. Successive U(i) have

been shown to contain ever more information. However, if the recursion is

to stop with a theorem in the current ifc-FAS then this theorem must have

no consequent other than itself when acted on by the rules. We just

described that theorem. It is inherent in the rules and thus contains less

information than any theorem in the recursion. Wherein lies I believe the

origin of much confusion over the origin and nature of the Second Law. How

to resolve this? Well the ifc-FAS is incomplete and all that is necessary

is to add ever more complex theorems to the ifc-FAS. The easiest way to do

this while preserving "meaningfulness" [history] across the event seems to

be to increase the size of the alphabet - add more points. This I call a

perturbation of the ifc-FAS.

18) The points would tend to form a structure because:

a) They attract each other since the ifc-FAS would like to merge back into

the Superverse as a destruction of the bifurcation alignment.

b) They repel each other because if such fractures of alignments could take

place it would destroy the assumed consistency of the ifc-FAS. From the

above the merger of two points would sent the recursion back to some

theorem that did not contain them from which step it would then proceed

down some other recursion. Thus the ifc-FAS would have a mechanism that

would allow the proof of both B and Not B from A - it would be inconsistent.

19) The simplest dynamic structure that could contain SAS seems to me to be

a 3 space in which the points are restricted to a fixed region of the

structure. The dynamic would entail some points following some pattern of

relocations within their region - I call it a dance. Some dances would

have configurations that allowed them to move through the grid of

points. The simplest dances look like "particles". Thus the addition of

points to incrementally resolve the incompleteness as a way of adding ever

more complex theorems to the ifc-FAS would be [at least a small part of the

perturbation] isomorphic to adding "particles" to the physical realization

- that is produce quantum fluctuations. SAS would be huge associations of

dances - a super dance.

20) Since the recursion steps need not take place in any fixed pattern

"time" would be an illusion of SAS.

21) "Motion" does not exist. "Forces" would be an illusion. Only

quantified relocations of points upon a recursion step have physical

realization.

22) The system is naturally quantified and consistent with the postulates

of relativity. For developments of things like "time" dilation see my full

draft at:

http://www.connix.com/~hjr/model01.html

23) Due to the perturbations it is discontinuously computational i.e. non

deterministic.

24) Right now the most interesting candidate I have found for a point

structure possibly yielding our universe is face centered cubic - again see

my full draft.

25) Some super dances may have to seek out and absorb other dances in order

to maintain their structure. Due to the inherent quantification of the

system dances influence the gird over a limited range. This process

therefore requires levels of blind seeking - choices of alternates based on

incomplete information. Such choices can be based on a coupling to the

perturbations as a source of randomness. Thus my definition of life as

being such super dances.

26) This is applied to institutions by looking at how such super dances

create rule sets. Again see my full draft.

END POINT DYNAMICS

27) Looking at Turing's use of Cantor's diagonal argument I believe that

when the FAS finally gets to be countably infinite it is as big as it can

get - it is complete. However, there is in general still no decision

procedure so it must now be inconsistent. This allows the previously

forbidden merger of points and the alignment melts back into the superverse.

Hal

Received on Tue Oct 31 2000 - 19:33:08 PST

Date: Tue, 31 Oct 2000 22:21:38 -0800

I have been asked to compress my model so as to maximize its

accessibility. The following is an attempt to do so. The result is rather

cryptic and has little effort to "prove" anything. It is frequently just

an application of Occam's Razor.

BEGINNINGS

Start with:

1) Nothing - an empty finite consistent Formal Axiomatic System - an fc-FAS.

2) Nothing is incomplete because it can not address the meaningful question

of its own stability - it is an ifc-FAS.

3) To address the incompleteness Nothing becomes Something.

4) To address the question of the stability of Nothing, Something must have

a bifurcation to implement the concept of "Yes" vs "No", or "True" vs

"False", or "Is" vs "Is Not", or affirmation vs negation. [As in "Nothing

is unstable." is a well formed statement Yes or No.]

5) Something [in my view] takes the form of a growing, seething, foamy

fractal of such bifurcations that are mathematically isomorphic to 1's and

0's I call a superverse.

6) Some [perhaps all] alignments by happenstance of these bifurcations

within the Superverse are isomorphic to finite strings of N bits.

7) Some such N bit strings are isomorphic to fc-FAS's.

8) Some such fc-FAS are incomplete or ifc-FAS. [Open ended alignments?]

9) Such ifc-FAS are composed of :

a) A finite theorem set [no more than 2^[N + c] as per AIT.]

b) Each theorem has a finite AIT complexity less than or equal to N +

c bits

and thus

b) A finite symbol set or alphabet.

c) A finite set of axioms

d) A finite set of rules

10) Some such ifc-FAS are isomorphic to mathematical objects U that are

themselves isomorphic to universes.

11) All such universes have physical realization.

12) A low complexity class of such ifc-FAS have just one axiom.

13) The proposal is that our physical universe is isomorphic to a

mathematical object U that is described by an ifc-FAS in this class. More

precisely a currently realized physical configuration of the universe - a

CRPC(i) is isomorphic to a current string configuration - a U(i) that is

described by a currently expressed theorem of the ifc-FAS - a p(i).

MID STAGE DYNAMICS

14) The only dynamic available in such a ifc-FAS is a recursive enumeration

as follows:[viewing p(i) as an AIT shortest possible program]

p(i) = {R acting on p(i - 1), PL(i)} will compute U(i) isomorphic to

CRPC(i).

where R is the rule set of the ifc-FAS, PL(i) is the self delimiter of AIT and

p(1) = {R acting on [single axiom], PL(1)} will compute U(1) isomorphic to

CRPC(1) and

[single axiom] describes U(0) isomorphic to CRPC(0).

15) Dynamic results so far:

a) p(i) gets longer with each step of the recursion.

b) Thus the information content of U(i) is higher than U(i - 1) isomorphic

to the Second Law.

16) The ifc-FAS as stated has a finite theorem set each with a finite

amount of information. I think the simplest [perhaps only] isomorphic

physical realization consistent with the above is a structure composed of

discrete points. The points are isomorphic through the mathematical object

U to the alphabet of the ifc-FAS.

17) At the moment we have a potential contradiction. Successive U(i) have

been shown to contain ever more information. However, if the recursion is

to stop with a theorem in the current ifc-FAS then this theorem must have

no consequent other than itself when acted on by the rules. We just

described that theorem. It is inherent in the rules and thus contains less

information than any theorem in the recursion. Wherein lies I believe the

origin of much confusion over the origin and nature of the Second Law. How

to resolve this? Well the ifc-FAS is incomplete and all that is necessary

is to add ever more complex theorems to the ifc-FAS. The easiest way to do

this while preserving "meaningfulness" [history] across the event seems to

be to increase the size of the alphabet - add more points. This I call a

perturbation of the ifc-FAS.

18) The points would tend to form a structure because:

a) They attract each other since the ifc-FAS would like to merge back into

the Superverse as a destruction of the bifurcation alignment.

b) They repel each other because if such fractures of alignments could take

place it would destroy the assumed consistency of the ifc-FAS. From the

above the merger of two points would sent the recursion back to some

theorem that did not contain them from which step it would then proceed

down some other recursion. Thus the ifc-FAS would have a mechanism that

would allow the proof of both B and Not B from A - it would be inconsistent.

19) The simplest dynamic structure that could contain SAS seems to me to be

a 3 space in which the points are restricted to a fixed region of the

structure. The dynamic would entail some points following some pattern of

relocations within their region - I call it a dance. Some dances would

have configurations that allowed them to move through the grid of

points. The simplest dances look like "particles". Thus the addition of

points to incrementally resolve the incompleteness as a way of adding ever

more complex theorems to the ifc-FAS would be [at least a small part of the

perturbation] isomorphic to adding "particles" to the physical realization

- that is produce quantum fluctuations. SAS would be huge associations of

dances - a super dance.

20) Since the recursion steps need not take place in any fixed pattern

"time" would be an illusion of SAS.

21) "Motion" does not exist. "Forces" would be an illusion. Only

quantified relocations of points upon a recursion step have physical

realization.

22) The system is naturally quantified and consistent with the postulates

of relativity. For developments of things like "time" dilation see my full

draft at:

http://www.connix.com/~hjr/model01.html

23) Due to the perturbations it is discontinuously computational i.e. non

deterministic.

24) Right now the most interesting candidate I have found for a point

structure possibly yielding our universe is face centered cubic - again see

my full draft.

25) Some super dances may have to seek out and absorb other dances in order

to maintain their structure. Due to the inherent quantification of the

system dances influence the gird over a limited range. This process

therefore requires levels of blind seeking - choices of alternates based on

incomplete information. Such choices can be based on a coupling to the

perturbations as a source of randomness. Thus my definition of life as

being such super dances.

26) This is applied to institutions by looking at how such super dances

create rule sets. Again see my full draft.

END POINT DYNAMICS

27) Looking at Turing's use of Cantor's diagonal argument I believe that

when the FAS finally gets to be countably infinite it is as big as it can

get - it is complete. However, there is in general still no decision

procedure so it must now be inconsistent. This allows the previously

forbidden merger of points and the alignment melts back into the superverse.

Hal

Received on Tue Oct 31 2000 - 19:33:08 PST

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