The following is an effort to remove some errors from my description of my
idea.
1) POSTULATE: All counting numbers exist with countably infinite copies in
a seething foamy fractal.
Call this fractal Everything or as I prefer to call it the Superverse S.
In this fractal every universe Uj is isomorphic to a self sorting sequence
Uj(i) of these numbers that, upon chance encounters between numbers, they
self sort according to:
(1) Pj(i) = {Rj(Uj(i - 1)) + PLj(i)} has the same length in bits as
the AIT complexity of Uj(i).
Where:
Rj is the fixed rule set of Uj and Rj(Uj(i - 1)) is Rj acting as a
comparison filter based on Uj(i - 1).
Uj(i - 1) is the previous configuration.
PLj(i) is the program length self delimiter.
This is similar to the following expression from AIT:
(2) P'j(i) = {"Expression" + "Data" + "Self Delimiter"} is the
shortest program that can compute Uj(i).
Here P'j(i) is the actual shortest program that can compute Uj(i), its
length in bits is the complexity of Uj(i) so it has the same length as
Pj(i). "Expression" is identical to Rj, but the "Data" may not be Uj(i -
1) and the "Self Delimiter" may not be PLj(i).
The first number encountered that passes the Rj mediated compare becomes
the successor to Uj(i - 1) i.e. Uj(i). the model is that some Rj's can
perform the compare in a partially non deterministic "do not care" fashion
and thus Rj can act as a source of "true noise" since the "do not care"
sections of the filter will admit any sub string. The
consequences of this will be discussed below.
The relation (1) is like a compressed recursion notation and contains a
finite Formal Axiomatic System [f-FAS] that defines Uj. The structure of
the f-FAS is:
a) It initially has a single axiom Aj unique to a particular family of
universes which is in Pj(1) as Rj(Aj). It serves as Pj(0). It is thus the
compressed form of Uj(0) and initiates the recursion.
b) It has a fixed set of rules = Rj which form the comparison filter.
c) It has an alphabet = differently sequenced strings of bits in Uj that I
call types of "regions". The axiom Aj contains the entire initial alphabet
of regions and simultaneously the initial configuration for a particular
universe.
Thus as stated the equation (1) contains the entire N-bit, f-FAS.
Each "region" type is modeled as isomorphic to a physical structure that
has existence such that the location of a discrete isolated space point u
inside a small fixed portion of a space grid is coded by each region
type. [In our case I believe the grid to be 3D and most likely Face
Centered Cubic.] the points can not leave its region but it can relocate
within it.
Uj(i) contains a theorem of this f-FAS giving a specific arrangement of
"regions" - a configuration of a particular universe. The structure of
Uj(i) using English letters to represent different "region" types would
look like the following.
(3) Uj(i) =
aaxcvsyplmnjhkkyufpoiiimjkhlyhhhhnkmlpmneidhsu...qaeeerwetgbfvvvdcsdxazjfirjnfjgkkeirejqzq
To compress this idea I will call each of these a type of sub string u. In
this limited example there are 26 types of sub string u. For example a =
the u 00000100000000, and b = the u 00100000000000. The location within
the string Uj(i) of a particular u identifies the particular grid region it
codes and the location of the 1 in each u codes the location of the
discrete space point u within that region.
2) If there were no "true noise" in the system due to the nature of Rj [a
fully deterministic Rj] then (1) would be a recursion as follows
(4) Pj(i) = {Rj(Uj(i - 1)) + PL(i)} computes [proves] Uj(i).
Since Pj(i) must be longer than Pj(i - 1) [It contains Rj and PLj(i) in
addition to the data Uj(i - 1)] unless Uj(i -1) is empty the sequence can
not stop with a finite length Pj(i). Consequently Uj(i) must also grow in
length as i counts up. All "regions" as described above have roughly equal
complexity so increased length of Uj(i) means increased
complexity. Since Rj is potentially a source of "true noise" if not fully
deterministic then this accumulation of information just accelerates.
To sustain this growth in complexity of Uj(i) the fc-FAS - as per Chaitin -
must also increase in complexity. It does this by randomly [nothing within
the Pj(i) can control the process] adding additional alphabet elements -
new types of "regions". This is also equivalent to "true noise" as well as
to starting the sequence over with a new axiom which is now just Uj(i).
3) Is this f-FAS also consistent? Even though information can come from
"outside" through the nature of the Everything, the possible dynamics of Rj
and the necessary growth in the complexity of the FAS and ultimately reside
in Uj(i) the FAS is none the less as structured incapable of proving both A
and not A so it is consistent - an fc-FAS.
Thus there are two potential sources of "true noise" in the model.
4) How does the potential "true noise" nature of Rj impact the logic system
during the sorting process? Relation (1) is really saying that there is
not necessarily enough data in Uj(i - 1) but the required additional bits
to make Pj(i) the same length as P'j(i) will reside in PLj(i). The
consequent is that as a comparator based process Rj would have to be
applied to a string longer than Uj(i - 1) if we are to get to P'j(i). So
Uj(i) is longer - it has more "regions" - than Uj(i - 1). Thus in the
following:
(5) Pj(i + 1) = {Rj(Uj(i)) + PLj(i + 1)} has the same length in bits
as the AIT complexity of Uj(i + 1),
Uj(i) acts like a new initiating axiom.
5) When Uj(i) gets to have a countably infinite number of bits the sequence
ends since there are no larger numbers in the Superverse.
The fractal contains all universes potentially an infinite number of times.
6) Alternate beginning: No numbers exist. This is the "Nothing". This is
unstable logically. It contains no answer to its own stability which it
must address. It decays into the above fractal which is sufficient to
contain the answer.
Once Nothing becomes Everything in an attempt to answer its own stability,
the question of the stability of Nothing is no longer meaningful to
Everything neither is "Nothing?" itself, so Everything is now stuck as
Everything.
The relation (1) is just spontaneous self replication with mutation, in
another word Life - as a minimum it is present in every universe as itself.
Received on Sat Dec 02 2000 - 15:46:49 PST
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