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Solutions to the "General Grand Unification Problem,"
and the Questions "How Did Our Universe Come Into Being?"
and "Of What is Empty Space Composed?"

Robert A. Herrmann
Mathematics Department
U. S. Naval Academy
572 Holloway Rd.
Annapolis, MD 21402-5002

Presented before the MAA, Western Maryland College, 12 NOV 1994.


1. History.

[Note: References for this section appear on pages 4 - 5.] In August 1979, I was to give a paper before a gathering of mathematicians at the Summer Meeting of The American Mathematical Society. [1] This summer meeting is a joint meeting with the Mathematical Association of America and was being held at the University of Minnesota at Duluth. I boarded an airplane at Chicago for the last leg of my trip, and who should be setting next to me at the window seat but John Wheeler, the Joseph Henry Professor of Physics emeritus at Princeton. Professor Wheeler was to deliver an invited talk before the Mathematical Association of America on what he believed was a fundamental difficulty with the philosophy of science associated with physical theories.

We discussed various things as we passed over some magnificent Minnesota thunderheads. Looking out of the window Professor Wheeler made the curious but simple statement, "We can't do that." What does this statement mean and who are the "we"?

After more discussion, the meaning of his comment was clear. First, the "we" are physicists and what they can't do was illustrated by the thunderhead. The thunderhead is assumed to be a collection of Natural systems. You have the water droplets, or crystals. Then their paths of motion caused by internal forces. The various electric potential differences produced by such motion and hundreds of other factors that scientists claim contribute to the overall behavior we observed from our window.

A Natural system is a set or arrangement of physical entities that are so related or connected as to form an identifiable whole. Many Natural systems have an associated physical theory. These theories are used to predict behavior for each Natural system. BUT many of these theories use different methods to predict behavior. Indeed, one often has to restrict a particular theory to specific areas of application. When this is done you often have a boundary between two theories - an overlapping region -- where one or the other but not both apply for the methods used in one of the theories are inconsistent with the methods used in the other theory. The hundreds of Natural systems (now called subnatural systems) that comprise the Natural system called a thunderhead are then put together in some manner to obtain the entire thunderhead one observes. With respect to the universe as a whole, the differences between various predicting theories are profound. What Professor Wheeler meant by the ". . . can't do that" with respect to our universe is the acknowledgment that physicists had not found a general unifying theory that predicts all of the behavior of all subnatural systems that comprise the entire Natural system called our universe. Science needed a theory that would be totally consistent, and a theory that would replace the piecemeal approach. Of course, such a unifying theory did not even exist for the behavior of the thunderhead we observed.

I restated this problem in the following manner.

Does Nature really combine subnatural systems together, in a way that we can understand, to produce an entire Natural system or does it use an entirely different method so that inconsistencies are somehow avoided? Is there something else required, something more basic than science has yet described and that's needed to combine all the Natural systems together that comprise our universe? Indeed, how can a universe that's perceived by humans to have order and harmony really be a product of chaos?
By the way, I added that last question to this problem since it is certainly relevant.

After our discussion, it occurred to me that something discovered in October 1978, and now called ultralogics [2], might possibly lead to an answer to these questions. But I also knew that the answer might be rather startling.

The standard approach is a piecemeal "bottom-up" approach beginning with the "bottom," so to speak, of the scientific hierarchy (or "domain of explanation"). Supposedly, if you can find a unification of the fundamental forces (or interactions), then this would lead to an "upward" process that would eventually unify all physical theories. As discussed above, this is a doubtful assumption. Moreover, many biologists and philosophers of science believe that there are emergent properties of organisms that cannot be fully understood as products of DNA or chemistry and, thus, do not follow directly from the four fundamental forces. Furthermore, the concept of "time" as described within some quantum physical models specifically contradicts the concept as it is described within certain cosmological theories.

On the other hand, they may be a top-down approach that answers the Wheeler questions. This means to find some scientific theory that gives processes which yield cosmologies, cosmologies that contain all of the processes that control the behavior of all Natural systems.

The "deductive-world model," (i.e. the D-world model) was the first constructed (1978) and is interpreted in a linguistic sense. D-world model properties will not be discussed directly within the following pages. Many of these properties can be found in sections 1 - 5 of the book Ultralogics and More [2]. However, some of what appears in the next sections does have the requisite linguistic interpretation associated with the D-world model. The actual model used to solve these problems is generally termed the metamorphic-anamorphosis model (i.e. the MA-model). Further, the intuitive concepts and mathematical methods that determine the MA-model properties are very similar to those used for the additional D-world model conclusions. Thus becoming familiar with the MA-model conclusions will aid in ones comprehension of the D-world model.

The basic construction of the MA-model uses the following philosophical stance as described by deBroglie.

. . . the structure of the material universe has something in common with the laws that govern the workings of the human mind. [3, p. 143]
It was determined that an appropriate cosmogony would depend upon the properties of a logic-like operator and subsidiary concepts associated with the nonstandard physical world (i.e. NSP-world) [4, part III]. The construction of the MA-model began in August 1979 and a series of announcements [5 - 10] relative to its mathematical construction appeared in the Abstracts of papers presented to the American Mathematical Society. In particular, [5] mentions the logic-like operators.

In 1982, papers by Bastin [11], and Wheeler and Patton [12] were discovered in The Encyclopaedia of Ignorance. I mention that [12] does not contain an important appendix that appears in the original 1975 paper. The paper by Bastin describes the discreteness paradox and the paper [13], using a few MA-model procedures, presents a solution to this paradox. Next the requirements for an acceptable cosmogony stated in [12] were compared with MA-model properties. A major difficulty in showing that the MA-model meets all of these requirements is in the language used to give physical-like meaning to the abstract entities. After considerable reflection, the prefix "ultra" was decided upon. The use of this prefix is consistent with its use in the construction of the nonstandard structure. A relatively explicit construction of such a structure uses the concepts of the ultrafilter, the ultraproduct and the ultralimit. In 1987, a paper justifying the fundamental mathematical theory of nonstandard consequence operators was published [14].

Although a few technical aspects associated with the MA-model were yet to be fully justified, two papers were written in 1986 [15], [16] announcing the solution to the general grand unification problem and the pregeometry problem of Wheeler and Patton. To properly prepare paper [16], the original Wheeler and Patton paper [17] was utilized. An appendix in [17] describes how Wheeler and his colleagues at Princeton tried to construct a pregeometry from the statistics of very long propositions and very many propositions, where the term "proposition" refers to the propositional (i.e. the sentence) logical calculus. They failed to achieve a solution, but Wheeler left open the possibility that a solution could be obtained using concepts from the area of Mathematical Logic. This is exactly what has occurred.

Beginning in about 1989, a project was instituted to justify fully all of the MA-model concepts, among other aspects of the NSP-world, and to present them in monograph form. The basic monograph [2] was completed in 1991 and the final result that completes the justification process was published in 1993 [18].

In the next section is given a very brief, general and mostly nonmathematical description of how the MA-model and the language of the NSP-world solve the problems mentioned in the title. Presented in the third and technical section titled "The Mathematics," with additional comments, is most of the actual mathematics that, when interpreted, describes the properties of the MA-model [2] that correspond to the nonmathematical descriptions.

The following quotations are taken from [17] and are all relative to the Patton and Wheeler cosmogony requirements.

(1) Five bits of evidence argue that geometry is as far from giving an understanding of space as elasticity is from giving a understanding of a solid. [17, p. 539]

(2) They also suggest that the basic structure is something deeper than geometry, that underlies both geometry and particles ("pregeometry"). [17, p. 539]

(3) For someday revealing this structure no perspective seems more promising than the view that it must provide the universe with a way to come into being. [17, p. 539]

(4) It brings us into closer confrontation than ever with the greatest questions on the book of physics: How did our universe come into being? And of what is it made? [17, p. 540]

(5) Tied to the paradox of the big bang and collapse is the question, what is the substance out of which the universe is made? [17, p. 543]

(6) But is it really imaginable that this deeper structure of physics should govern how the universe came into being? Is it not more reasonable to believe the converse, that the requirement that the universe should come into being governs the structure of physics? [17, p. 558]

(7) It is difficult to avoid the impression that every law of physics is "mutable" under conditions sufficiently extreme, . . . . [17, p. 568]

(8) It is difficult to believe that we can uncover this pregeometry except as we come to understand at the same time the necessity of the quantum principle, with its "observer-participator," in the construction of the world. [17, p. 575]

(9) . . . . a guiding principle, is what we seek. [17, p. 575]

As the MA-model pregeometry is discussed, I will refer to these quotations where applicable. Relative to (9), the first half of the guiding principle is the deBroglie statement viewed from the NSP-world. The second half of the guiding principle is relative to the philosophy of realism and an observation made by the originator of nonstandard analysis Abraham Robinson. First recall that Newton believed that infinitesimal measures were real measures associated with objectively real entities. Berkeley and Leibniz did not accept Newton's belief. Robinson at the end of his very first published paper on nonstandard analysis made the following statement relative to the modern concepts of how mathematical models are used to predict indirectly Natural system behavior.

For phenomena on a different scale, such as are considered in Modern Physics, the dimensions of a particular body or process may not be observable directly. Accordingly, the question whether or not the scale of non-standard analysis is appropriate to the physical world really amounts to asking whether or not such a system provides a better explanation of certain observable phenomena than the standard system. . . . The possibility that this is the case should be borne in mind.
Fine Hall,
Princeton University [19, p. 440]
Robinson is referring to infinitesimal measures in this quotation. Since the publication of [19], nonstandard analysis has been applied to entities that are not infinitesimal in character. The second half of the guiding principle is an extension of this Robinson statement to the NSP-world. Thus the acceptance of the NSP-world as a viable realism depends upon whether or not it provides a "better" rational explanation of certain observable phenomena than the standard world model. The philosophy of what constitutes "a better explanation" is left to individual choice.

References

[1] Herrmann, Robert. A. (1979). Perfect maps on convergence spaces I, Notices of the AMS, (26)(5): A-475.

[2] Herrmann, Robert. A. (1991). Ultralogics and More, Institute for Mathematics and Philosophy, P. O. Box 3268, Annapolis, MD 21403-0268.

[3] deBroglie, L. (1963). In March, Arthur and Ira M. Freeman, The New World of Physics, Vintage Books, New York.

[4] Herrmann, Robert A. (1989). Fractals and ultrasmooth microeffects, J. Math. Physics, 30(4) (April):805-808.

[5] Herrmann, Robert A. (1981). Mathematical philosophy, Abstracts of papers presented to the AMS, Vol. 2(6). #81T-03-529:527.

[6] Herrmann, Robert A. (1983). A useful *-real valued function, Abstracts of papers presented to the AMS, Vol. 4 (4), #83T-26-280:318.

[7] Herrmann, Robert A. (1984). Nonstandard consequence operators I, Abstracts of papers presented to the AMS, Vol. 5 (1), #84T-03-61:129.

[8] Herrmann, Robert A. (1984). Nonstandard consequence operators II, Abstracts of papers presented to the AMS, Vol. 5 (2), #84T-03-93:195.

[9] Herrmann, Robert A. (1984). D-world alphabets I, Abstracts of papers presented to the AMS, Vol. 5 (4), #84T-03-320:269.

[10] Herrmann, Robert A. (1984). D-world alphabets II, Abstracts of papers presented to the AMS, Vol. 5 (5), #84T-03-374:328.

[11] Bastin, T., (1977). A clash of paradigms, in The Encyclopaedia of Ignorance, Duncan and Westen-Smith, eds. Pergamon, New York.

[12] Wheeler, J. A. and C. M. Patton. (1977). Is physics legislated by cosmogony? in The Encyclopaedia of Ignorance, Duncan and Westen-Smith, eds. Pergamon, New York.

[13] Herrmann, Robert A. (1983). Mathematical philosophical and developmental processes, Nature and System, 5(1/2):17-36.

[14] Herrmann, Robert A. (1987). Nonstandard consequence operators, Kobe J. Math. 4(1):1-14.

[15] Herrmann, Robert A. (1986). A solution to the grand unification problem, Abstracts of papers presented to the AMS, Vol. 7 (2), #86T-85-41:238.

[16] Herrmann, Robert A. (1988) Physics is legislated by a cosmogony, Speculations in Science and Technology, 11(1):17-24.

[17] Patton, C. M and J. A. Wheeler, (1975). Is physics legislated by cosmogony? in Quantum Gravity: an Oxford Symposium, Ishan, C., Penrose, R., and D. Sciama, eds, Clarendon, Oxford.

[18] Herrmann, Robert A. (1993). A special isomorphism between superstructures, Kobe J. Math. 10(2):125-129.

[19] Robinson, A., (1961). Non-standard analysis, Nederl. Akad. Weimsch Proc. Ser. A64 and Indiag. Math. 23:432 - 440.

Important Reference

The above reference [2], from which the mathematical portions of the monograph are taken, contains all of the fundamental concepts associated with the methods and processes used within the discipline of nonstandard analysis as they would apply to all aspects of the MA-model and D-world model.



2. Two General Discussions

[Note: This first discussion is a portion of a general audience non-technical and elementary talk I give on this subject. The numbers that appear in the double square brackets refer to the quotations from [17] as they appear on pages 3, 4.] Let me point out that as long as scientists use mathematics to obtain their theories and written symbols, diagrams, photographs and the like to communicate their concepts then the MA-model can't be eliminated. It will always be there lurking in the background.

Now to answer the question "How was our universe created?" Let's start by considering a single geometric point a few feet in front of you. A geometric point in this sense is a position in our universe and, for the present, has no other meaning. Now, I'm able to magnify this point for you by using a mathematical microscope with a power that's greater than any power that can ever be achieved by human means.

Suddenly, you see the point open up, like the iris of your eyes. What's revealed to you is a background universe, a substratum, or whatever you might like to call it. Now, the Natural-world is the world we can scientifically perceive, and this Natural-world point is still in your field of view with a small portion of the background universe surrounding it. You can't make out much detail, but there's definitely something there. The detail you see is sharper and clearer near to the single Natural-world point. Then clarity slowly fads as you proceed further from that one solitary Natural-world position within our universe. You can find no clear outer edge within your field of view. This background universe forms a portion of what I called the nonstandard physical world - the NSP-world. [Note: The entire collection of all possible standard world points coupled with all of the NSP-world points that are infinitesimally near to them is called the finite or bounded portion of the NSP-world.] The term the NSP-world is also used for other applications. However, for our purposes I'll discuss a portion of this NSP-world, the MA-model.

Our universe is inside and "just as near to" the NSP-world as I have described it. One might say that this background universe is scientifically omnipresent. Now the MA-model portion of this background universe specifically states that there will never be a human language that can give a completely detailed description for the mechanisms that may have produced our universe, but there do exist such mechanisms within this background universe. Such mechanisms exist but, no matter how hard we try, the human mind can't comprehend all of the details. Well then, what can be known about how universes, such as ours, can be created by NSP-world mechanisms? We can know general properties.

To begin with, there exists within the NSP-world a set of "things." These "things" need not be considered as being within our Natural universe. But why do I call them "things"? I do know a lot about these "things." But, unfortunately, if I were to give you a basic description for their contents, I would only use various mathematical terms from the fields of mathematical logic and set-theory. The most difficult task I've faced is to give some "physical" meaning to these "things." These objects could easily fall into the category of those objects within the NSP-world that have NO human language physical descriptions at all. I know that these "things" behave like informational "superballs."

After literally years of reflection, it was determined that these "things" can be described as containing all of the building plans, the laws of "Nature" and even step-by-step images of how an ideal universe will appear as it develops. They also contain a great deal of information which would be incomprehensible to the human mind. It's interesting to note that, after I came to these conclusions, I came across a quotation from one of the greatest scientists of our time. Hermann Weyl is quoted as saying the following:

Is it conceivable that immaterial factors having the nature of images, ideas, 'building plans' also intervene in the evolution of the world as a whole?
We have "things" in the background universe that are ready, when the conditions are just right, to aid in the production of universes, ours included. And different "things" aid in the production of different universes. Actually, these "things," which I have called ultimate ultrawords, don't work alone. Each ultraword is just one piece of an entire process. All pieces of the "puzzle" must be put together before a universe is produced. Now the existence of these ultrawords may not seem startling, but a lot more is yet to come.

What are the conditions that must exist in the NSP-world - conditions needed to trigger the creation of a universe such as ours? At present, there is no human scientific language that can detail these conditions, no human understanding of what these conditions are or were, only that such conditions exist.

I remind you that this cosmogony comes from a mathematical model, a model that can't be eliminated from modern science. This cosmogony satisfies all of the basic theoretical requirements of the scientific method. The universe in which we dwell, our solar system, the Earth, or a virus are Natural systems. Natural systems are studied by the physical scientist in a piecemeal fashion. They apply distinct procedures that seem to describe the moment-to-moment behavior of each distinct Natural system. Now within the NSP-world there's one special process called "*S," its a hidden process, an intrinsic (hidden) ultranatural process. The S-process is one of the entities know as an ultralogic. When the conditions are just right, this force-like process takes one of these ultimate ultrawords and produces a universe. Indeed, it combines together, controls and coordinates all of the distinctly different Natural systems that comprise a universe. The S-process applied to an ultimate ultraword produces each Natural system and also yields the moment-to-moment alterations in the behavior of each and every Natural system [[2, 3, 4, 6]]. But, how does it do this? We can't know many details, but a few simple mechanisms do present themselves.

Within the NSP-world there are objects, of a single type, (that I term ultimate subparticles) that, from the NSP-world viewpoint, can be "easily" combined together to produce every material object, electrons, protons, our earth, and everything else that might be termed material as well as immaterial fields, if such exist [[2, 4, 5, 6]]. This combining process cannot be reproduced by Natural means within any laboratory within our universe.

What about the development of our universe? That is how it changes with respect to time. This force-like process does produce a "beginning" for our universe and, as mentioned, in a remarkable step-by-step manner, it produces in the proper "time" ordered sequence all of the material changes from the very beginning until the universe arrives at a stage such as that which we observe about us. It produces all the Natural events that constitute the moment-to-moment changes that alter the appearance of a universe. This remarkable force-like process applied to an ultimate ultraword yields a solution to the general grand unification problem.

I say that this process is "remarkable" but I haven't explained why. I give you one example. From human perception, we often characterize certain Natural changes in a Natural system as chaotic or random. This means that there seems to be no pattern for such changes - that is no pattern that can be comprehended by the human mind. Indeed, no human predictions can be made as to how individual objects will behave from one moment to another. From our viewpoint, there are no harmonious or regular laws that can produce such individual changes. It can be shown that from the NSP-world viewpoint the opposite is true. This seemingly irregular behavior within our universe is actually only what we can perceive of what is, in reality, an extremely regular process. How is this possible?

Well, it turns out that as a universe develops, as it changes, there are millions of other events taking place within the NSP-world that we can't perceive scientifically. These ultranatural events sustain and hold our universe together, so to speak. It's the ultranatural events that are combined together with the Natural events we perceive that actually comprise a complete change. Thus it's simply a matter of perception. I wish there were words in any language that fully describe this wondrous "combining" together process. There is a technical term, "ultrauniform," that can be used. But this gives no indication of what is actually occurring. The steps in this combining together process are so minuscule, so small, so refined that the human mind can't fully appreciate nor comprehend the S-process. It doesn't correspond to anything that can at present be perceived or imagined by us.

Another startling aspect of the S-process is that every Natural event is connected, within the NSP-world, with every other Natural event. No Natural changes, from the NSP-world viewpoint, are independent from one another. A Natural event taking place in one galaxy is related by the S-process, to events taking place in every other galaxy. But, unfortunately, we can't know, except in general terms, the actual composition of an ultimate ultraword nor describe in any human language the necessary ultranatural events that sustain this process.

Now, what I've described, as best as I can, is the creation of an "ideal" universe. But what happens when there exist creatures within a universe that can alter its ideal development? Well, these creatures can only alter the Natural events, they can't alter the ultranatural events. No matter what these creatures do this "glue" that holds the universe together still remains [[8]]. I repeat, we can have no knowledge as to what these ultranatural events are. They cannot be described in a human language, ever. All we know is that they exist.

From the human perspective, there can be "sudden changes" in the behavior of a Natural system at any time during its development [[7]]. But these changes are not truly "sudden" from the NSP-world viewpoint. Indeed, we can magnify a point in time, as we did with a point in space, and investigate what happens at the moment of change. Again wondrous events occur that are difficult to describe in a human language. Technically, the changes occur in an ultrauniform manner. This type of change, from the NSP-world viewpoint, can be compared with the mathematical concept of an "uniformly continuous" change. But ultrauniform is "infinitely" more uniform, so to speak, than the usual standard concept of an uniformly continuous change.


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

If we look at the MA-model as a whole, is there a way to describe it in its entirety? Yes, there is and this may be the most remarkable aspect of this research. The MA-model can be characterized as behaving in its entirety like a super, super, super to an infinite degree, computer or mind. The processes are similar to how an almost inconceivably powerful computer or mind would behave.



In the next files titled "The Mathematics," are reproduced with additional comments some of the actual mathematics that, when interpreted, describes the properties of the MA-model that correspond to the nonmathematical descriptions.

This portion of this discussion will be somewhat more technical than the previous portion. I will refer to section 3 as the motivation behind this mathematical model is discussed. The expression "Natural world" refers to the collection of all entities that are categorized as Natural systems.

Relative to the behavior of a Natural system, a general scientific approach is taken and it is assumed that scientists are interested in various types of descriptions for Natural system behavior. It is not difficult to show that all forms of scientific description can be reduced to strings of symbols. Developmental paradigms are simply time related descriptions for the ideal behavior of any Natural system viewed simply as strings of symbols. Although some of these collections of descriptions might be generated by a specific theory, a general approach that a developmental paradigm describes a sequence of Natural events is taken. The expression Natural event means an objectively real and physical occurrence that is categorized as "natural" by the physical scientist. This approach does not include any requirement that such a sequence be generated by some accepted and humanly comprehensible theory. On the other hand, theory generated sequences are not excluded.

On the first six pages of the next file are reproductions of pages taken from the paper "Nonstandard consequence operators" [Herrmann, R. A. Kobe J. Math. 4(1987): 1-14]. They detail how the basic somewhat unusual model E is constructed. The terminology used is that of the abstract mathematical structure. For example, certain subtle consequence operators are interpreted as ultralogics, while an ultraword is an unreadable sentence. The theorems on these pages show some of the behavior of nonstandard consequence operators. The most significant discussion on these first six pages is the construction and embedding of the set E. Although the results on these first six pages show some of the behavior of nonstandard consequence operators they are not specifically needed to comprehend what follows. Further, the "time ordering" concept considered throughout what follows can be replaced with the notion of the "universal event number."

Since an ultralogic is based upon the selection of some nontrivial logical process, there is a need to select a logical system that is common to all known logical systems used throughout scientific discourse. After some difficulty, a system, denoted by S and discussed at the beginning of section 7.3 (p. 24) titled ultrawords, was selected. I point out that the ordering of the sections of the included mathematics is not the same ordering in which the processes were originally discovered and used. Intuitively, throughout the modeling of these linguistic concepts, a specific "frozen segment" is considered as a description for a particular Natural event that occurs at a particular moment of time. From a scientific communication point of view, this description is all that can be scientifically known about such an event and is substituted for it.

Developmental paradigms, the deductive logic S and the like are embedded within a special but well-know mathematical structure called a superstructure by means of a fixed encoding and are further embedded into a nonstandard structure. The next step in the modeling process is to show that for a developmental paradigm written in a standard language there exists a new object, called an ultraword, that when the ultralogic (an intrinsic ultranatural process (i.e. an IUN-process)) *S is applied to this ultraword the entire developmental paradigm (or the corresponding Natural event sequence) is logically produced. Theorem 7.3.1 (p. 26) establishes that for each developmental paradigm such an ultraword exists. Defining the Natural world as the collection of all Natural event sequences that correspond to the behavior of all systems that are categorized as Natural systems, it follows that ultrawords cannot be entities within the Natural world under the MA-model interpretation. Also the force-like operator *S cannot be applied within the Natural world. Rather then simply accepting that ultrawords are "things" that cannot be further described, elsewhere additional intuitive meaning for this concept is discussed.

There is a natural question that arises that is relative to the collection of all events generated by the force-like process *S. Are only the developmental paradigm events obtained as a result of the process *S applied to an ultraword w? A conjecture was no. Theorem 7.3.2 (p. 26) shows that other "descriptions" for other types of events also occur when *S is applied to a specific w. A later investigation, Theorem 10.1.1 (p. 38), shows the general composition of these new descriptions or events. But more importantly, Theorems 7.3.2 (p. 26) and 10.1.1 (p. 38) show that these new events must occur. Further, they cannot occur within the Natural world and cannot be described by any natural language. These events have been interpreted as events (called ultranatural events (i.e. UN-events)) that are needed to uphold and sustain a Natural system's development.

The above discussed results do not answered Wheeler's basic question. The basic question is answered by Theorem 7.3.4 (p. 27) in an slightly more general mode. Every subnatural system within the Natural system called the universe can be associated with its own special system generating ultraword w(i)'. Corollary 7.3.4.1 (p. 27) says that if we select a set of ultrawords that generate each and every subnatural system that is within our Natural universe, then there (logically) exists an ultimate ultraword w' such that when the force-like process *S is applied to w' each of the subnatural system ultrawords are produced. Hence, all of the original subnatural systems (the Natural event sequences) are produced by application of *S to w'. Thus within the nonstandard physical world, not within the Natural world, there (logically) exists a force-like process the combines all of the Natural systems together in a consistent manner and yields our Natural universe and all of the moment-to-moment alterations that comprise its development. The mathematical existence of the ultimate ultraword w' yields a solution to the "general grand unification problem" as described by Wheeler.

Since the Natural event sequences are not necessarily predicted by a theory, then is it possible that the ideal developmental paradigms and the corresponding Natural event sequences are somehow preselected? Theorem 7.2.1 (p. 23) coupled with the discussion in 10.4 yields the necessary NSP-world IUN-selection processes. It is at this point that one of the most basic and significant properties of the MA-model becomes apparent.

Suppose that you consider a partial denumerable developmental paradigm d(i) where one of its members F(i) is a frozen segment that represents a specific configuration for a Natural system as it appears at a standard time interval [t(i), t(i+1)) Notice that F(i) is a member of T(i) . The (cosmic or standard NSP-world) time t(i) is conceived of as the moment of time in the past when a time fracture occurs and all other members of d(i) represent behavior "after" such a time. There are infinitely many sequences (finite or denumerable) of standard frozen segments or *-frozen segments that can be adjoined to d(i), and that yield other developmental paradigms that describe Natural or ultranatural behavior for the same Natural system but for cosmic times "prior to" t(i). This yields type d or type d' development paradigm.

Theorem 7.2.1 (p. 23) states that there is an external IUN-selection process that yields each of these developmental paradigms. A developmental paradigm that contains no additional frozen segments of any type prior to t(i) along with its associated ultraword represents the maximum scenario. Note that a maximum scenario can yield by application of an ultralogic, *-frozen segments as well as frozen segments. A developmental paradigm that contains only time ordered standard frozen segments before and after t(i) is a minimum scenario. An intermediate scenario contains some or all *-frozen segments representing ultranatural events prior to t(i). "Sudden alterations" are modeled by the minimum and intermediate scenarios and can be used for various purposes such as the Patton and Wheeler concept of "mutability" of Natural Law or behavior. For the minimum and intermediate scenarios, can we investigate what happens during the NSP-world time when Natural law or Natural constants are altered? Can we "open up the time fracture," so to speak, and look inside?

As discussed in section 7.5 (p. 29), applications of Theorem 7.5.1 (p. 29) yield a startling view of how these alterations are being made within the NSP-world. Furthermore, the force-like process *S that produces all of the event sequences does so in a remarkable manner as described by Corollary 7.4.1.2 (p. 28).

As to various NSP-world objects that can mediate any MA-model alterations, produce informational transmissions, and be considered as the composition of the vacuum, these are automatically generated by this mathematical model as shown by Theorem 9.3.1 (p. 34) where the descriptions are interpreted as descriptions for objects.

As to how the assumed "Laws of Nature" that seem to exist today came into being, the discussion in section 10.2 (p. 40) shows that these can also be assumed to have been produced by *S applied to an ultraword. This answers another basic Wheeler question. The MA-model cosmogony yields the various Natural laws that exist today. Further, an ultraword such as w" gives an external unification to the collection of all written physical theories.

The last section in this paper deals with the "substance out of which the (Natural) universe is made." It shows that using the method outlined in Theorem 9.3.1 (p. 34) the basic properties of subparticles can be obtained. Ultrasubparticles are obtained with the aid of Theorem 11.1.1 (p. 46). Other properties of the infinitesimal and infinite hyperreal numbers coupled with a very simple hyperfinite translation (affine operation) lead to intermediate subparticles that can be finitely combined together to produce every basic entity within our Natural universe.

[Note (1): Relative to cyclic, multi-universe, plasma or any cosmology that claims that our universe has no Natural time beginning or no Natural time ending, such universes still have a "beginning" from the MA-model viewpoint. No result in this paper is dependent upon a universe existing for only a finite period of time. Each Natural system still has a beginning and ending with respect to an identified cycle. There is for each Natural system within an i-th cycle an ultraword w(j)^i. Then there is a cycle ultimate ultraword (w^i)'. The multi-cycle-universe generating ultimate ultraword w' still exists and the force-like operator *S still applies. That is that w' generates each cycle and the contents of each cycle. Also note that if a Natural system j is open-ended in that it continues to alter its appearance for all of time and has either no beginning or no ending in the intuitive sense, then this also can be modeled by considering a denumerable sequence of basic time intervals [a(i),b(i)). Again there is an ultraword w(j)^i for each [a(i),b(i)) and, hence, an ultraword (w(j))' that generates each of the w(j)^i. Thus (w(j))' generates the entire Natural system's behavior. Finally, there is an ultimate ultraword w' that generates each (w(j))' and, hence, each Natural system. 3 MAR 1996]

[Note (2): For an ultimate MA-model conclusion that shows how an MA-model generated universe can be made to vary due to moment-to-moment perturbations, see section 11.2 that starts on page 55. The ultimate MA-model conclusion also models specifically the "quantum" and participatory requirements. 24 DEC 1996]

In the next files, the references, when they have been reproduced, appear at the end of each "chapter."

Note

In the mathematics section 3, objects are chosen that seem to yield the simplest possible entities. This is done to minimize controversy and to allow most conclusions to be established in a convincing and straightforward manner. The philosophy of science employed is the exact same philosophy of science used in theoretical cosmology and quantum logic investigations. In almost all cases, the physical-like interpretations correlate directly to the mathematical structure. The first section of part 3 is a reproduction of a Kobe Math. J. paper. The second section of part 3 is a reproduction of portions of the book Ultralogics and More (see reference [2] of part 1) where all the details for the actual construction of the nonstandard model can be found. It should be self-evident that the results contained within this monograph are only the must basic and that further in-depth investigations should be pursued.

Intuitively, I am confident that all of the theorems are correct. If any "proof" is not convincingly established, then this should be easy to rectify. Finally, I do not contend that this is THE solution to this problem and questions. Although these results are speculative in character, they are no more hypothetical than the Everett-Wheeler-Graham many-worlds interpretation or Hartle-Hawking quantum gravity model. I do contend that since the method was devised in 1979 that these results are the FIRST such solutions obtained scientifically. The term "scientifically" refers to the sciences of concrete mathematical modeling, mathematical logic and interpreting mathematical structures physically.


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