The Hyperfinite and the GID and GGU-models

Robert A. Herrmann, Ph.D
12 APR 2018

Many claim that the most significant aspect of human thought is finite choice. If you check any book that contains a "formal" proof in logic, you will discovery that in order to write a "proof" one needs to choose finitely many strings of symbols from a set of formal expressions that is infinite. It is remarkable that the human mind can actually find a suitable selection. Choosing a finite collection of items is certainly a rather simply human task. It is the process of choosing from a finite set of possibilities that governs an individual's everyday experiences. The intuitive notion of what the term "finite" signifies, such as "counting" and other properties, can be mathematically modeled. The actual finite processes we use are rather simple. A higher-intelligence can duplicate the same "finite" processes just as easily and the "finite" processes the higher-intelligence uses are also characterized, in the same manner, as rather simple from its viewpoint. But a higher-intelligence can apply these "simple finite" processes not just to those processes we consider as finite but also to processes that we would consider as infinite. This is why this special higher-intelligence notion is called the "hyperfinite." Relative to choice, this type of higher-intelligence choice is termed "hyperfinite choice."

It is also a remarkable fact that if the finitely chosen items have an encoding that will allow for them to be ordered, that the human mind can usually accomplish such a task. If you have any finite choice-set of descriptions such as Q = {d(100),d(0), d(32),d(3)} that are numerically coded, then for Q there is a mathematical operator that puts them into an order corresponding to the order 0 < 3 < 32 < 100. Thus, human experience is once again mathematically modeled. Although placing a finite collection of entities in order may take some time on our part, it still is a rather simply mental process to do if you know how the numerical codes are obtained. A higher-intelligence can also duplicate this same process not just for sets we consider as finite but also for certain sets of entities that one would consider as infinite. This higher-intelligence process is often called "ordered hyperfinite choice."

A Somewhat More Detailed Explanation

In "Nonstandard Technical Addition" to the article that presents the "simple" illustration of the construction of a physical globe-of-the-world, one has the ordered result of a higher-form of book page-turning. Each page of the book contains "descriptions d" for the systems that comprise a *slice of a universe and then each such collection comprises a "*slice." These 3-D slices, the Universe-Wide Frozen-Frames (UWFF), yield a sequentially developing universe and can be composed of only physical-systems, "physical-like" systems, other types, or can be empty. In the display (I) these systems are denoted by the "d"s with the * notation as follows:

d = *d(1), *d(2), *d(3), . . . , *d(λ)

This symbolism represents a "hyperfinite" sequence of *descriptions. If not empty, these *descriptions can contain "words" taken only from the general language L, or general language words and distinct members of the higher-language *L. They can also be composed only of distinct words taken only from the higher-language *L and that do not appear in L. The interpreted "finite" part of the hyperfinite follows our experience with the "finite." But, when more fully analyzed, the "stuff" representation by the . . . notation is rather unusual and indicates why we cannot actually perform this form of ordered deduction.

Suppose that one has a collection of simple statements that characterize the term "finite." Behavior involving finitely many objects is very familiar to us. A basic fact we often use is that (1) "any subset of a finite set is finite." In the technical Complete GGU-model, four to six different collections of terms are used. The terms "subset" and "finite" are called "standard terms." The term "hyperfinite" is a member of the set of terms denoting certain special objects. These are called the "internal" objects. When one translates the symbols denoting the standard objects into * symbols, then the mathematical objects so translated are the internal objects. Such a translation process applied to (1) yields "Every internal subset of an hyperfinite set is hyperfinite." Such an internal object is denoted by a symbol that is a member of the symbols used for internal objects. I note that there are internal sets that don't carry the * notation. Relative to the translated statements, it is obvious that this level of higher-intelligence behavior follows the exact same patterns as our behavior. But, are there "new" higher-intelligent behaviors that these translated statements do not, by themselves, actually reveal?

Under more refined analysis, it turns out that there is no known mathematical way that allows us to significantly describe exactly "how" the higher-intelligence actually performs the "step-by-step" *deduction that yields this sequence.

There are many non-internal subsets of this sequence. In order to describe them and their behavior, two other sets of terms are employed. These are called the "external" objects and their relations to the standard and internal objects are obtained by using a language that includes both of these languages and many more additional terms. This is the "metalanguage." The metalanguage notion is from Mathematical Logic. The metalanguage is the most general that is used to describe relations between entities defined in terms of these other languages.

I chose to model physically restricted aspects of such sequences, aspects that we can physical perform. That is, we can perform only finitely many actions. The apparent reason why we loose comprehension and cannot meaningfully apply our general standard language to such internal sequences is that the sequence includes both a temporal and "atemporal" procedure. Human step-by-step processes are temporal. But then it switches to an atemporal sequence of empty or hidden yet not empty *UFWW associated in various ways with ours. The higher-intelligence has designed, for our universe, the sequence of descriptive UWFFs to correspond to many of our "discovered" predictive laws for physical behavior. That is, the "d"s satisfy these laws. The actual notions that are modeled are, in their simplest form, displayed by the above d example. Thus, we are allowed to use discovered linguistically expressed "physical laws" to predict from a description like d(233) physical aspects of the description d(234).

Recall that the complete negative integer "ordering," . . . -3, -2, -1 is like a reversed infinite sequence. In this case, we learn to view the ordering from right-to-left, starting with the -1. On the other hand, we view the ordering of the natural numbers from left-to-right and this corresponds to the order as the numbers increase in value. Then the display d above is viewed as an ordering from left-to-right.

Recall, that on this website I have a paper that indicates how we can imagine the "infinite" set of natural numbers, with or without 0, under they usual ordered presentation. ALSO, all that the additional properties that we can describe for sequence d are predicted. When we investigate the . . . above, its discovered that the step-by-step process actually includes all such *d for each member of the set of natural numbers, ordered as are the natural numbers and here viewed, starting from 1, left-to-right.

Then, for d, another ordered infinite set is "attached" to the completely ordered natural number part of the "*d"s. This "attachment" is within the . . . part of the notation. These attached "*d"s are denoted by what are called the "hypernatural" numbers, the λ, that are distinct from the natural numbers. And, they need to be considered as atemporal, when measurable developmental time is considered for the natural number portion. They are members of a set *N that contains the natural numbers N. These model the "infinite numbers" of Newton and Leibniz and are, indeed, called "infinite natural numbers." I note that the complete left-to-right order being displayed is formed by "ordered" higher-intelligence deduction. And, the higher-intelligence instructions follow all of the same properties as just described.

The difficulty is that the ordered view for the infinite numbers that correspond to the atemporal portion of the sequence, which form a portion of the entire sequence as here being presented, is as separately viewed "by us" from right-to-left. Where as the "beginning" temporal portion that contains the fully physical UWFF descriptions is viewed by us from left-to-right. Further, we mathematically manipulate symbols relative to our comprehension of such human behavior as moving symbols from left-to-right or right-to-left. We even define rules assuming that we know the difference in the "hand" notions. I know of no aspect of the mathematical model that allows us to further describe "how" the higher-intelligence "jumps," so to speak, from the "end," so to speak, of an infinite sequence, to the "beginning," so to speak, of the infinite atemporal portion and then stops at description *d(λ). This is a major example of what I call the lack of knowledge model. The model states that although there are no members of L that describe a particular situation, there can be members of *L that are not members of L that do have descriptive meaning to an higher-intelligence. Theologically. *L may be the Third Heaven language mention by Paul.

Thus, the best approach to interpret an hyperfinite ordered sequence as produced by the Complete GGU-modeled higher-intelligence is to simple state that the higher-intelligence applies its "simple" process of its form of ordered finite deduction and we can have no detailed knowledge of exactly how the higher-intelligence actually produces the results.