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