Sorry I think an unedited version of this may have got out by mistake. I have not seen it get posted so perhaps it did not. Again if it did I am sorry.
Dear Juergen: At 4/10/01, you wrote: >Hal, you wrote: > > > I believe that attempting an extensive detailed formal description of > > the Everything is the wrong approach. IMO - at least in this case - the > > more information used to describe, the smaller the thing described. > >I was not able to follow this, but informal and vague descriptions of >everything have been around for millennia. Now it is time to go beyond >this. One cannot scientifically discuss nonformal statements or theories. I did not say I had no formal description. I have a small one which I have attached. >You did refer to formal describability when you repeatedly wrote that >if a universe is a theorem cascade in an N-bit formal axiomatic system >"then Chaitin's incompleteness dictates that the cascade must stop," >and there are no more than 2^[N + c] theorems. For some reason this >statement even made it into your "FAQ" list. > >Counter example: number theory is describable by finitely many bits of >axioms yielding an infinite cascade of provable theorems. "FAQ" is a poor name for what we can do on this list at this time. The "FAQ" I am attempting is - as stated several times - a comparison of various approaches. The above is one of mine. That being said, how are you treating numbers? The finiteness of a FAS is not just a description of its axioms but of all its components that contribute to the complexity of the program describing its theorems which - as I understand it - is the complexity of which Chaitin speaks. [I will look for the reference in his books and post it.] If numbers are just various strings of two binary bits giving a small alphabet then I suspect Chaitin's work applies. Hal ------------------------------------------------------------ The rest of this is at: http://www.connix.com/~hjr/style2.html Parts of: PATTERN BASED MODEL OF THE EVERYTHING VERSION J DRAFT OF 4/08/01 BY HAL RUHL FILE FAQstyle2j.html SECTION I DEFINITIONS #1 1) Pattern: A pattern is a geometric structure that could vary in at least one characteristic from location to location within the pattern. Patterns must be at least a set of more than two zero dimensional isolated discrete points. 2) Information: There are two types Relative and Absolute: Relative information is the structure of a given pattern, bit string or other such structure both within the structure itself which would be just the relative position and style of its elements, and between structures in a larger whole which would be the relative styles, quantities, and positions of elements/patterns. Absolute information is based on mathematical structure and selection. An example of mathematical structure absolute information would be the resolution of : The Everything is stable; true or false. An example of absolute information based on selection would be the selection of a single pattern, bit string or other such structure out of a collection of such structures. As an example of total information the selection of a particular bit string consisting of some count of all zero's would be a maximum of selection based absolute information while the string selected contains no internal relative information. SECTION II THE LOWEST LEVEL In this model the lower level of existence contains an expression of no information. Rather like 345/345 is an expression of an identity element. There are two polar extremes at this lower level: the Everything and the Nothing. The Everything expresses pan-isomorphism, the Nothing expresses anisomorphism, hence neither is fully describable. However, selected pieces can be described. The Nothing, since it contains no information, can not resolve the question of its own stability which it must address. The only option is to test the issue with the only perturbation available - it becomes the Everything. The interesting thing is that the Everything being a perturbation from the Nothing does not contain the Nothing. Subtracting the Nothing from "all information" to produce the Everything is the smallest selection [perturbation] I can think of. The idea being that no information and all information are identical but the Everything is not quite all information. For this reason I prefer the name Superverse. The Everything also having almost no information similarly can not resolve the question of its own stability and must also test the issue by becoming the Nothing. Both poles - being mutually exclusive - destroy history whenever manifest, thus a random Everything/Nothing [E/N] alternation is established. Existence at this level is sustained but has bipolar expression. SECTION III THE UPPER LEVEL Pan-isomorphism is the upper level of existence. It is manifest only when the Everything is manifest so it is not continuous. A portion of the pan-isomorphism expressing Everything is describable as a seething foamy fractal pattern of infinite repeats of all patterns. Part of the seething is due to the differences between successive Everything manifestations. Since the E/N alternation destroys history, each Everything manifest has a randomly selected pattern of patterns. Any further seething that may be present during an Everything manifestation may be thought of as generating historic information, but it is relative information and not absolute information. It is not germane to the question of the Everything's stability. Thus it does not preclude the E/N alternation and is destroyed by the alternation. Individual patterns in this portion of the everything have multiple interpretations. Each such interpretation produces a pair: [pattern;interpretation]. These pairs are linked to isomorphisms which can be states of universes. These isomorphic links can be self sequenced by rules internal to the individual isomorphism. The links of a sequence are either past, active, or possible. Looking just at the present and successor links we shorten this to active and possible. Speculation as to past links may or may not be incorporated into the rules of a given isomorphism. Thus we write these isomorphic links as {[pattern;interpretation];[possible/actual isomorphism]}. The sequencing is a migration of active isomorphic link from one [pattern;interpretation] pair to another. The rules can have a non deterministic content. Such rules define the family of viable link successors with a procedure that can be partially "do not care". Since the Everything is in effect a seething pattern of patterns the first acceptable "possible" link encountered by any search scheme is random for rules with a non deterministic content. This link becomes the next active one. SECTION IV MATHEMATICAL DESCRIPTION PART 1 In order to focus on a particular class of these link shift rules which may contain our universe the model makes the following associations: Some of these isomorphic links can be represented by strings Uj() consisting of at most a countable infinity of binary bits. Every evolving universe Uj is isomorphic to a self sorting sequence Uj(i) of these links that, upon an encounter between an active link and an acceptable "possible" link in the fractal, they switch states thus they self sequence according to [written post shift]: (1) Pj(i) = {Rj(U'j(i - 1)) + PLj(i)} is the shortest program that computes Uj(i). Where: Rj is the current rule set of Uj and Rj(U'j(i - 1)) is Rj acting as a comparison filter based on U'j(i - 1). U'j(i - 1) is the previous link's representing string if Rj was fully deterministic. [See below] PLj(i) is the program length self delimiter. Uj(i) is the current link's representing string. The structure of U'j(i - 1) depends on the nature of Rj. If Rj is fully deterministic then U'j(i - 1) is the same as Uj(i - 1). This causes the Algorithmic Information Theory [AIT] complexity [Note #1] of Uj(i) to increase as i counts up. This can be more clearly seen by rewriting (1) as: (2) Pj(i) = {Rj(Pj(i - 1)) + PLj(i)} is the shortest program that computes Uj(i). since Pj(i - 1) is the most compressed form of Uj(i - 1). Pj(i) has more bits than Pj(i - 1) since it contains Pj(i - 1) as data. If Rj is partly non deterministic but still needs all of Uj(i - 1) as data then U'j(i - 1) is more complex than Uj(i -1) and the AIT complexity of Uj(i) increases even faster. If Rj becomes even more random then not much can be said of the relation between U'j(i - 1) and Uj(i - 1) The first link encountered while the fractal is seething that passes the Rj mediated compare becomes the successor to Uj(i - 1) i.e. Uj(i). Each Uj(i) is a stand alone isomorphism that contains a finite semi-consistent Formal Axiomatic System [fsc-FAS] that defines Uj. The structure of the fsc-FAS is: a) It initially has a single axiom Aj unique to a particular family of universes which is in Pj(1) as the initial data provided to Rj i.e. as Rj(Aj). It is Uj(0) and initiates the recursion. b) It has a set of rules = Rj which form the comparison filter. Rj is a particular evolving universe's laws of physics. The rules as stated need not be fully deterministic. This is the source of the "semi" modifer to "consistent". While Uj(i) is a theorem of the FAS in the standard sense, when the rules Rj act on Uj(i) to compute Uj(i + 1), Uj(i) may activate a "do not care" operator in Rj. 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. SECTION V DEFINITIONS #2 3) Granular-discrete space-time: A discrete space in this thread a 3-space in which isolated points are confined to fixed regions [granules] that have a fixed relative location with respect to other regions in a space grid structure. The grid structure currently preferred in this thread is Face Centered Cubic [with spherical surfaced regions]. Within each region the associated point can assume a finite number of discrete locations. Each region with a particular point location can be thought of as isomorphic to a particular alphabet element of a FAS such as an "a" region or a "b" region etc. A finite string of these alphabet elements [such as abcryopaabctgjk...abcfg] isomorphic to state of a finite physical universe. Different such strings are then isomorphic to different states of that physical universe. The dimensions within a region may be continuous but a given FAS can only contain a finite number of alphabet elements i.e. regions. A binary bit string can be constructed which designates a region by a string of 0's with a single 1 buried in it some where. The "a" region might be "00000000000000001000" and the "b" region "00000100000000000000". This would allow a finite string to locate each of the finite number of points in such a finite granular-discrete space-time with infinite precision without the need to actually display the associated infinite strings. Again time is quantified and subjective to a given universe. SECTION VI MATHEMATICAL DESCRIPTION PART 2 Each "region" type is modeled as isomorphic to an interpretative structure that has physical meaning 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 their own region but can relocate within it. The isomorphism need not give rise to an actual physical existence of a universe within the Everything as this would be redundant to the existence of the pattern itself. Just the applicable interpretive isomorphism need exist as an overlay on the pattern. A given pattern can have many such overlays depending on which section of the pattern is considered to be the rules of succession by the particular isomorphism. 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 express this idea as an expanded binary string 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: 00000010000000000000000000, and b = the u: 00000000000000100000000000. The location within the string Uj(i) of a particular u identifies the particular grid region it codes and the location of the 1 within each u codes the location of the discrete space point u within that region. The result is a granular-discrete space-time. The granules are the regions and for starters have a spherical surface. The point within a region is discrete but may be located by continuous dimensions. However, the finite complexity FAS can only contain so many alphabet elements and thus the space-time is discrete. Nevertheless the bit string scheme above allows the FAS to locate points with infinite precision absent a need to display the actual dimensional coordinates of any of the discrete points. Time in any given universe is also discrete and runs at some ratio of the Nothing-Everything oscillation rate. This ratio need not be constant, the Nothing-Everything oscillation rate itself can not be constant or it represents absolute information, and time is subjective to a particular universe. SECTION VII TRUE NOISE If there were no "true noise" in the system due to the nature of Rj [a fully deterministic Rj] then (1) would be a recursive enumeration or theorem cascade in an N-bit, fc-FAS. Chaitin's incompleteness [The maximum AIT complexity of a theorem of an N-bit, fc-FAS is (N + 1).] dictates that the cascade must stop since there are no more than 2^[N + c] theorems in the FAS. But as was shown above such cascades produce monotonically more AIT complex theorems. For the cascade to stop it must end on a theorem that has no consequent under the rules of the cascade. If region n is the neutral region then all regions of the granular-discrete space-time would be the neutral one. If this is region "n" in (3) then Uj(i) becomes: (4) Uj(i) = nnnnnnnnnnnnnnnnnnnnnnnnnn...nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn = m repeats of n This is a very low AIT complexity theorem. Thus we have a contradiction. Call this the "Halting Contradiction". A random oracle can provide new information to the FAS [say new alphabet elements - new types of regions] as necessary and thus repeatedly increase N eliminating the contradiction. 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). The sequence may stop when N becomes countably infinite if this destroys the consistency of the FAS. Consequently Uj(i) must also grow in length as i counts up since there are only 2^N - N-bit strings. Since Rj is a source of "true noise" if not fully deterministic then this accumulation of information just accelerates as the sequence progresses since Rj is applied to ever longer Uj(i). This constitutes further use of the random oracle at theorem complexities below that which triggers the "Halting Contradiction". Further one must consider that those Rj [FAS] that form Godelian incomplete systems will be prone to the need to resolve that type of undecidable which are seen as additional inputs from a random oracle. Thus there are so far three potential sources of "true noise" in the model. I happen to believe that all SAS require some true noise influx into their universe. I have so far argued the possible presence of true noise from the perspective of Chaitin's and Godel's incompleteness and an added just plain "do not care" component in the rules of link succession of a particular universe. But is such noise built into any SAS? If we consider ourselves to be SAS do we allow that any SAS - to be a SAS - must be able to run any possible experiment including thought experiments we can? If we do then the idea that we perhaps can construct an isomorphism between the S/N alternation and the scanner/transporter/duplicator [STD] thought experiment [see the everything list FAQ and archive] may be another source of true noise. Considering that the U'j(i - 1) to Uj(i) operation of the Rj is a deterministic one [given the data on what just happened our current state is predictable but the data may not have been deterministicly arrived at] the S/N alternation looks like the STD operating at the level of the Superverse itself. However, any SAS in any particular universe can not identify its actual operation. Thus the SAS is faced with a fundamental undecidable. How does the potential "true noise" impact the logic system during the sequencing process? Relation (1) is really saying that there is not necessarily enough data in Uj(i - 1). The consequent then is that each new U'j(i - 1) would be a new initiating axiom for the cascade whenever a true noise event takes place. ----------------------------------
