Necessary Intuitive Mathematics is No Longer Being Taught.
Partially Measuring the Strength of God's Intelligence and Other Attributes,
What the Term Infinite Means.
And God's Ultimate Attributes are Immeasurable via Set Theory.

Robert A. Herrmann Ph. D.
6 FEB 2011. Last revision 29 NOV 2019.

I. What was Previously Known.

(Throughout this article any comparative attribute can be substituted for intelligence.) What do I mean when I write that the intelligence displayed by a higher-intelligence is "infinitely greater or stronger than" any similar intelligence displayed by any biological entity within a physical universe? For attributes, in general, I use one way to describe this concept. For a higher-intelligence, I use two ways. One is general and the second is specifically defined. I know of no other way to express the following concepts exact by definition and illustration. Necessarily, for me to communicate to you, I need to introduce what may be new terminology. The major requirement for any discipline is that one learns and becomes accustomed to the proper terminology. I have no alternative but to present the necessary terminology. But, the general non-mathematics meanings for many of these terms is often similar to their ordinary meanings.

Over its rather long history, there have been periods where the subject we today term as "mathematics" emphasized displays of deductive reasoning. In he middle 1600s, Newton developed his "Calculus" notions as "a tool" to describe physical behavior. He employed his concept of the "infinitely small" to do so. During the 1800s, the subject of "analysis" came into being and such mathematics became a more formal subject. Once again, it was developed as an application of exquisite displays of deductive reasoning, hopefully, using well defined procedures. In general, the terminology and symbolism was not developed simply to model physical behavior. Then, when appropriate, certain terminology and symbols could be so associated. This is termed as the standard terminology and concepts. However, today, surveys indicate that, for most students, much of the mathematics to which they are exposed appears otherwise. Unless you are a "math major," the strongest interest in the subject is relative to it being used to "model" the behavior of the physical world about us.

Relative to the notion of a "model" for behavior, today, the Complete General Grand Unification (GGU)-model can be constructed so that the standard foundations actually model observable human behavior. However, the deductive results are predicted via mathematical procedures that, at times, may be somewhat difficult to comprehend since the mathematics is of a very special nature. It is called NONstandard analysis. To have any in-depth knowledge as to the mathematics used, mathematics that is used to construct the mathematical Complete GGU-model and its predicted results, one needs knowledge as to the behavior of the underlying standard mathematical structure. Then the differing nonstandard behavior that is being modeled is often "intuitively" understood by comparison.

In 1949, I began my studies of Euclidian geometry at the Baltimore Polytechnic Institute. The first axioms in the book I used will not be found in a list of axioms as presented today. Consider Geometric Axiom 2. A geometric figure may be freely moved in space without any change in form or size. The terms "form" and "size" are not defined. The ideas are to be built from the increasing knowledge obtained as different "forms" are considered. In the paragraph that follows these first four axioms we have, "By the second axiom, the knowledge we have of one geometric object may be transferred to another like object, however distant in space." In the proof of Theorem II on congruent triangles, two triangles appear on the page having the appearance of the stated hypotheses. Step one states, "Place the (triangle symbol)ABC upon the (triangle symbol)DEF so that the side AC coincides with its equal DF, and AB falls on the same of DF as DE." You don't actually redraw the figure, you do it mentally. This begins our training so that after a year of plane geometry we can go on to Euclidean solid geometry and mentally move things about in space.

Do these four axioms appear in the list of "Axioms" one finds presented today? I cannot find them so listed. What are listed are the Euclidean "construction postulates." "1. Through and connecting any two points a straight line can be drawn."

"Can be drawn" certainly appears to be a "modeling" notion. After gaining intuition as to this "space" motion notion, one is able to more easily comprehend, via physical illustrations, the concepts of "translations" and "rotations" as they are axiomatically or algebraically presented. After further geometric notions are introduced, it might be noticed that its possible to take two seemly meaningless words like "elin" and "nitts" and write similar statements about these things, where they are merely names for members of sets. "(1) Every elin is a collection of nitts. (2) There exists at least two nitts. (3) If x and y are different nitts, then there exists one and only one elin that contains both x an y. Then (4) and (5) follow" [1a, p. 43]. You then can start proving things about these "things." But, how do we select statements written in terms of these "words" and really know that we can rationally establish the "Theorems"? In the axioms stated, try substituting the word "line" for "elin" and "point" for "nitt." These actually come from abstracting the parallelogram with its diagonals. This demonstrate one aspect of the concept termed "mathematical abstraction" since the elin and nitts can taken on different applied meanings, not geometry, IF one wants to use the results for other applied purposes.

For this article, what does the word "intuition" signify? Intuition is the mental awareness as to the meaning of words and illustrations, an awareness that need not be fully captured by the words or illustrations themselves. Then this is modeled via symbols and axioms. General abstraction is perceiving a common feature among many illustrations for a concept and then presenting a symbolic axiomatization for this feature. Rules for symbolic manipulation and accepted deductive principles then lead to new rational behavior exhibited by objects that satisfy the axioms. However, for many, many years, using mathematics for some "practical" application was not the driving force behind mathematical exposition.

Consider the following illustration. Start with

(a) = 1 = 1.000 . . . , where the ". . ." means mentally continue repeating the 0s. A 1961 algebra text used at Kenwood Senior High School Baltimore Country Maryland introduces basic "Set Theory" terminology for its basic descriptions. Infinite sets are introduced on page 1 via the conceptual ". . ." notation relative to the known, one hopes, concept of counting. “The set consisting of all counting numbers; that is, 1, 2, 3, 4, 5, . . . .” On page 2, we find "Since its impossible to list all the members of an infinite set, we list several members of the set and then write three dots. The dots indicate that the pattern established by the first few elements of the set continues indefinitely." Then we are told "there is no end to the number of elements in an infinite set." The "numbers never stop." "They go on forever." That is, we cannot "write" them all down, but maybe we can mentally conceive of them going on "forever." Of course, this depends upon our concept of time. Hence, it is most likely, that Adam and Eve were given additional comprehension relative to the infinite and its relation to time. This may have been necessary so that they could better comprehend what they, and all of us, lost due to their disobedience.

(I present a method in [1b], where you can, indeed, mentally conceive of this basic infinite, if you believe that you can do so. Of course, if you don't believe that this is possible, then you won't be able to do so.)

Now consider

(b) = 1 + 1/4 = 1.25000. . . . Next consider

(c) = 1 + 1/4 + 1/9 = 1. 36111 . . . . Then

(d) = 1 + 1/4 + 1/9 + 1/16 = 1.4236111. . .

Looking at the pattern of fractions (rational numbers), the next one that should be adjoined is 1/25. (We are adding 1 divided by the square of the next natural number.) Whether in decimal form or by "clearing fractions," its rather obvious that, as one mentally continues the sums, one gets a rational number. As one continues this processes, it might be concluded that the sums are, ever so slowly, getting nearer and nearer to a number that is approximately 1.644931288. Notice that from (a), we then obtain (b), and then we obtain (c) by "doing something." We are producing something in a "step-by-step" manner. Step (1) is (a). Step (2) is (b). Step (3) is (c). Step (4) is (d). We have a succession of steps that in the production of these numbers intuitively follows the natural number ordering. Notice that the difference between successive numbers appears to be getting "smaller and smaller." What we seem to be doing is extending such ideas to an intuitive "infinite" notion without actually stating the term.

It was establish in the 1800s, that the actual number that the sums are approximating better and better cannot be written by us as a fraction or in any decimal form whatsoever although it is claimed that it has a decimal form that can be imagined. We just don't know each of the digits to use nor the exact place where they should appear. Computers can approximate it for us to millions of places. But, they must stop because we do not have an unlimited supply of energy.

Notice that, in this case anyway, the usual ordering (symbolized by the <) of the counting numbers as learned in elementary school, yields the ordering, (a) < (b) < (c) < (d) < etc., for these rational numbers. The terminology "natural numbers" and "rationals" for the fractions is given in a 1961 Kenwood senior high school algebra book. The algebra book discusses the irrational numbers, which are contrasted with rational numbers via a discussion of what √2 might mean. Then it states, on page 68, that the set of real numbers is "the union of the set of all irrational and rational numbers." Basic set-theory operators are introduced prior to this statement. The book does not discuss all the basic real number properties, however, but does list, on pages 106-108, its eleven general "field" properties that a student should be able to apply.

The example of the (a), b), (c), (d), . . . can be stated more formally in terms of the symbol <. A property of the "real numbers" is that there is a number, call it X, to which these sums are getting "closer and closer." ("Closer and closer" is yet another concept that, in this case, one "simply sees" by subtracting.) A 1964 Calculus text I used while instructing at Kenwood Senior High School states the property of convergence using terms that later appear in the subject "general set-theoretic topology." The terms "open" and "neighborhood" are used relative to the "interval." In topology, general properties one intuitively learns from the interval notion are generalized. For this discussion, one property states that there is no other real number Y such that Y < X and these sums are getting closer and closer to Y. Also, there is no real number Z such that X < Z and the sums are getting closer and closer to Z. That is, the X is unique. The sums are said to "converge" to the unique X. But, what type of set do these sums yield. There yield the same type of set that is formed by the set of numbers 1/2, 2/3, 3/4, 4/5 , . . . and, in this case, these also get "closer and closer" to a number; the rather well know number 1 and 1 has the same property as the X.

For further analytic geometry applications and intuitive comprehension, the numbered line and "directed numbers" are introduced on pages 50 and 51 of the algebra text. It is not necessary that I, at present, discuss the "left hand" directed line notion. But this idea is employed later for the major application to the GGU-model where nonstandard analysis adjoins additional properties to this "left hand" notion.

In this article, after the terms are properly defined, I discuss with respect to the mathematical notion of "infinity" a fact similar to these two illustrations. The (a) - (d) and all those that follow only approximate the X. The fact is that God's attributes can be modeled, where each expression is produced by a "succession process" and is "infinitely stronger" than the previous attribute. But, there is no mathematically expressible X for an ultimate "infinitely stronger" attribute. The mathematics yields a "partial" model for the strength for God's attributes. If one assumes the existence of an in-depth indescribable, an immeasurable, ultimate strength for each of God attributes, then this intuitively would yield each of the mathematically modeled infinitely greater strengths.

Further, the (intuitive) dictionary definition the word "sequence'" is "(1) The coming of one thing after another, succession. (2) The order in which this occurs.'" However, in much of mathematics, the definition is a restriction of this intuitive notion. In order to retain the intuitive notion throughout my writings for the GD, GGU and GID models, I may use this term in a generalization form. A sequence for me need not be just a "function" (illustrated below) defined on the natural numbers. In many cases, it "produces" things. An "ordered'" set D is given and this set need not be the set of counting (or natural) numbers. That is, D may have a simple order "defined on it," like the one for the integers (. . . -3 < -2 < -1 < 0 < 1 < 2 < 3 . . . ). (Notice that this symbolic form yields an ordering that corresponds to our moving from out left to our right. Would this make sense to entities on planet OH who only have one arm?) The standard ordered set is always stated and understood if this generalization is employ. The successive order of what is produced is reflected by the order on the domain, not by any order that may be perceived for the set of produced results. (I do not use the term "net'" for this generalized notion.)

I actually have hopes that you will somewhat understand the meaning of the display

*f(α) < *f(α +1) < . . . • . . . . < f(-2) < f(-1) < f(0) < f(1) < f(b-1), f(b)

as it applies to one of the GGU-model universe generating scenarios. The order "<" displayed here is "defined" so as to have the same properties as those of the integers displayed and those of the "integer-like" numbers denoted by α, α + 1, . . . . This is a model for an higher-intelligence step-by-step development of a universe from a beginning "α" sequential step but, from our comprehension, it appears to have NO past "beginning" physical starting point.

The Calculus book I used from 1964 - 1968 to instruct seniors at Kenwood Senior High School presents to students, on page 171, the "completeness" property for the real numbers. After defining the "the inner and outer (Jordan content type) measures" from which the "upper and lower sums" are obtained this property is used to define the "upper and lower integrals." If these are numbers equal, then the result is the definite integral. This is done so that the objects for which the basic integral is defined are somewhat general in character. However, during 36 years of instruction, I instructed 17 different college level mathematics courses. This included the usual service courses of Calculus I, II, III, IV, Vector Analysis, Differential Equations, and many sections of Mathematics for Engineers and Physics. Except as used for one minor result in the "sequential convergence" portion of many of today's Calculus books, this completeness property and its application to these concepts are not taught in any of the mathematics courses required in order to obtain any of the engineering or physical science degrees at the United States Naval Academy. Since these degrees are of the same or better quality as obtained elsewhere, I assume the same holds throughout United States.

Knowledge of standard mathematically stated properties is necessary in order, by comparison, to have basic knowledge as to nonstandard properties. Indeed, without some knowledge of the standard properties, you cannot know, even slightly, how the above displayed higher-intelligence step-by-step process "beginning with" *f(α) differs from anything that we have experienced. But, many standard properties are no longer presented as part of a student's basic mathematics education.

In order to give comprehension to this display and to the general GD and GGU-model measures for God's attributes, it's necessary to present some "rather simple" notions that I'm sure you can follow after very careful reading. Two "orders" are used in what follows. The same symbol < or ≤ is used for both. You can tell which is which by considering the type of objects that appear on the left and right. The ≤ for sets or ≤ for other stuff.

To define the (ordinary) finite, just return to what "counting" means to a young child. We need to select from a collection of apples, not all of them, but a few so that each child can have one apple. So, we count-out the "required" number. Such counting is the intuitive bases for the mathematics definition. Such counting is not defined except by illustration. We count out one-at-a-time as we select, "one, two, three, four, five" apples and stop. Thus, we have set-up a relation between the symbols 1,2,3,4,5 and the apples. (The 0 can be included to mean, "no selection or no count is made" or "there is nothing to count." This gives {0,1,2,3,4,5}.) What has not been done?

(1) Two or more apples have not been selected and each one given the same numerical name. Hence, in general, two or more objects have not been assigned the same numerical name.

(2) Two or more of these numerical names have not been assigned to the same apple. Hence, in general, two of more numerical names have not been assigned to the same object.

Any such relation that satisfies (1) has various general names; a function, an operator, a transformation, a map, a mapping, a functional etc. But, here, let's call it a function. In French, functions are called "applications" because they can be thought of as being applied to something. This application idea also comes into play when the term "operator" is used. A function "operators" on something and yields something. On page 4 of the algebra book, a special type of function is introduced. It is the "one-to-one (1-1) correspondence." It has properties (1) AND (2).

To construct intuitively a finite set of symbols, write down the symbol { . Then write symbols to the right followed by }. Or, write no additional symbol on the right of { except for the }. The intuitively constructed set { } is called the "empty set." A finite nonempty set of counting numbers is represented by a collection of symbols like {1,2,3,4,5}. I've used the term represented since numbers can be expressed by other symbols as well. Since specific axioms may not be presented in this article, then these symbols, the numbers they represent and their order should be intuitively understood. They can intuitively represent steps in the application of some processes. For the illustrated "selection of apples" function, the stuff inside the { and } (in the specific case the 1,2,3,4,5) is called the domain and the set of objects to which the domain numbers correspond (the apples in the specific case illustrated) is termed the range. Notice that five different apples are selected. I doubt one would pay for five apples and yet the bag in which they are contained only has three.

Often the entire set of objects that contain the objects to which the domain members correspond (all the apples) is called the co-domain. We say a function is (defined) on something. The something is the domain. The function is into if the co-domain may not be the entire range. (I have a few times dropped the "in" from "into" without complaint.) But we state it as "onto" something if the range is the entire co-domain. (There are other terms for this notion.)

Now and then the term from is used for "on" but "from" has a slightly different meaning, although, in general, its okay. But, to remove confusion the "on" is often replaced by the following, where the term "map" is used as a type of "what is being done" term. "The function f maps the domain into the co-domain" or "The f maps the domain into the co-domain." There are variations for these terms. If X is the domain and Y the co-domain, then f maps X into Y. The general symbol for this is f:X → Y, where one states that "f maps X into Y" or "f is a mapping of (on) X into Y." Most of the time the symbolism is the only thing used.

If a function has property (2), then it has various names depending upon its use. One of the easiest to comprehend is the term one-to-one function (the correspondence idea). Another good choice is the term injection, which is easier to write. What I mean by this, is that when constructing a sentence the term injection fits better since it includes the notion of the "function" and "one-to-one" as part of the term. For example, rather stating that such and such is a one-to-one function, simple state that it's an injection. Injection satisfies both (1) and (2). Hence, an injection is "on" a set and "into" another and, of course, it is always "onto" its range.

Here's another symbolically illustrated injection for sets of natural numbers, where → is used to indicate what goes with what and we include the 0 since the domain and range are natural number symbols.

0 → 0, 1 → 1, 2 → 10, 3 → 6, 4 → 4, 5 → 123.

But, this arrow notion is not too easy to follow. So, if this left-to-right → ordering is understood, then abbreviate "number → number" as "(number,number)," where the first number intuitively moving left-to-right in the ( , ) symbol corresponds to the second number in the symbol ( , ). (An ( , ) is called an ordered pair.) Thus, one has the abbreviation

f = {(0,0),(1,1), (2,10), (3,6), (4,4), (5,123)}.

Notice that the one symbol "f" is used for the entire collection of ordered pairs. This ordered pair stuff and there properties appear on page 237 of that high school algebra book. If you have an ordered pair (a,b), then "a" is the first coordinate and "b" is the second coordinate. By-the-way, since the 1961 high school algebra students did receive enough instruction in set-theory, one can define more formally such ordered pairs by letting (a,b) = {{a}, {ab}}. (This is also defined by some as {a,{a,b}}.) This set has the necessary properties. This set contains but two objects. Then the set of all first coordinates can be easily defined for a given set f of such ordered pairs. It is sometimes called a "projection." It is written in short hand form as the set of all x such that x is a natural number and there exists a natural number y such that (x,y) is a member of (symbolically ∈) f. (If one wants, you can further more formally express this in terms of the properties of {{x}, {xy}}. But, when doing finite stuff in mathematics, "simple" notions need not be described. They are considered as commonly understood.)

Anyway, using coordinate language, you can easily find the domain and range. The domain is the set of all first coordinates and the range is the set of all second coordinates. Hence, the function f is on {0,1,2,3,4,5} and onto {123,4,1,10,6,0}. (The way we write a set need not display an order.)

If I had written {(0,0), (0,1), (1,7), (2,10), (3,6), (4,4), (5,123)}, then the (0,0) and (0,1) show that the correspondence does not satisfy (1). If I had written {(0,0), (1,0), (1,7), (2,10), (3,6), (4,4), (5,123)}, then the (0,0), and (1,0) show that such a correspondence does not satisfy (2). Here is another important notion used for f that includes the "applied" idea. Let x denote any member of f's domain. Then f(x) denotes the unique "value" one gets when f is applied to x. Thus, f(2) = 10. and f(5) = 123. The f(x) one gets is often called the "image."

Why all of these different notations? Well, a specific one may fit better into an expression that uses a natural language. When I was twelve-years old and began my study of the Calculus and encountered functions such as f(x) = 5x3 + 3x2 + 10x. I viewed this as various processes that generate numbers from numbers. I would even write f as f( ) = 5( )3 + 3( )2 + 10( ) to indicate the operations. Of course, a function need not correspond to any such operations as I later found out.

Notice that if I reverse each ordered pair in f, then I get an injection on {0,1,10,6,4,123} onto {0,1,2,3,4,5.} Denote this "inverse" by f or f-1. Indeed, given any injection f on X onto Y, then f is an injection f on Y onto X.

The integers form a subset of the real numbers. The real numbers have the "completeness" property that although it was studied in the 1960s by high school students is almost totally unknown today by our college graduates. Using this property I'll try, but as you will note not in a completely successful manner, to "explain" that strange display using the *f and f notation.

A (nonempty) set D of real numbers has a "lower bound" z if z is less than of equal to (symbolically ≤) each ∈ D. Hence, the real numbers greater than 0 and less than 10 have -1 as a lower bound. They also have -1/2,- 1/2234656, and even 0 as lower bounds. The completeness property implies that it you consider all of the lower bounds there is actually one and only one that is "greater than" all of the others. In this case, it is 0. This unique number is called various things, but the term "greatest lower bound" (g.l.b or "inf" for infimum) is rather descriptive and often used. I note that such numbers can be a member of the set used to determine the number. Just take the set of all real numbers greater than AND equal to 0 and less than 10. Then 0 itself is a lower bound and the g.l.b.

As mentioned, the 1964 - 1968, high school Calculus I taught required this concept in order to define the integral. Indeed, on page 172, the Archimedian Principle is stated and established. Both of these concepts are necessary, in order to have any understanding as to how the nonstandard real numbers differ from the standard. This principle states that if given arbitrary positive real numbers a and b, than there is a natural number n such that a < nb. This principle does not hold for the entire set of all nonstandard real numbers, relative to the standard natural numbers.

Consider next the integer numbered line and the "directed numbers." This is the "infinite" like geometric object the associates each point (location) on the line with a real number and intuitively considered the "left-hand" and "right-hand" directions.

. . . _____________________________________ . . .
. . . -4   -3   -2   -1   0   1   2   3   4 . . .
⇐       ⇒

I guess the three dots and arrows indicate that we have an infinity of stuff to the left and well as to the right of 0. One might even state that the line is "infinitely long." I wonder, is it possible to view this real line ordering from the left of the . . . in the ⇒ direction?

A property of the real numbers states that given any negative number n, then there is a negative integer n - 1 less than it. So this implies that the set of negative integers does not have a real number lower bound. The f(x) stuff in the above display for the *f and f has many of the same properties as the real number order. The < represents "ordered steps" in the production of the *f(x) and f(x) objects. Indeed, there is a integer number identifier for each of the f(x) that represents the "step name" for application of f. There is also a type of "number" α for the *f. For universe development, in

*f(α) < *f(α +1) < . . . • . . . < f(-2) < f(-1) < f(0) < f(1) < f(b-1), f(b)

the . . .< -3 < -2 < -1 < 0 denote past observer-time. If you don't look left at the *f(α) < *f(α +1) <, then this probably has the intuitive "standard" meaning. (This is the important suppressed form, where a method for the internal construction of each universe-wide frozen-frame is not considered. Otherwise, it remains a *f type symbol and the change over is from a set of hyper-integers (the α) to an unbounded set of integers.)

But, the actual behavior of the corresponding α < α +1 < . . . integer type numbers does not follow the standard properties. Indeed, not every bounded subset of the

α < α +1 < . . . • . . . < -2 < -1 < 0 < 1 < b-1, b

has a g.l.b. In particular, notice that α is a lower bound for the set of all negative integers. But as far as we can comprehend the matter, the set of negative integers that forms a subset of this "new set of integer-like numbers" does not have a g.l.b.

Consider a change in viewing direction

α < α +1 < . . . • . . . < -2 < -1 < 0 < 1 < b-1, b
→    → . . . •                                      

I can give a little extra insight into this matter if one can, indeed, consider the . . . < -2 < -1 < 0 < 1 < b-1, b , portion as viewed in the → direction, but it does not help with the "step-by-step" generation of a universe because something happens at the • step.

As an example, every member of the entire set of all α's is "less than" any negative integer. This can be expressed as a step-by-step notion for this set. But, how can I express the step-by-step notion for what would be the "remaining" portion of the . . . that seems to "start" at the • and "move" in the → direction? I intuitively know it should follow a step-by-step pattern, but I cannot explain in words or diagram form how it changes from the α's to the standard integers, how the *f(α)'s change to the f(x), where x is a negative integer. This is where we have some idea of the nonstandard notion by slightly comparing it to the better understood standard notion. (In [1b], on imagining the infinite, I give a little mental image as to a special "view" of part of this situation, a "view" that we cannot have for the method used.) So, apply this view to universe generation. This is

*f(α) < *f(α +1) < . . . • . . . < f(-2) < f(-1) < f(0) < f(1) < f(b-1), f(b)
→    → . . . •                                                  

So, I accept that this models the step-by-step generation of an "infinite" past observe-time universe from an actual non-physical event represented by *f(α). I also accept that this models what the LORD declares.

"For my thoughts are not your thoughts, neither are my ways your ways." "As the heavens are higher than the earth, so are my ways higher than your ways and my thoughts than your thoughts" (Isaiah 55:8-9).

So far, the idea of the "infinite" has been the intuitive notion of "forever" or "does not stop" or "continues on," the ". . ." idea. But, this is not much good for a mathematical presentation. Let's abstract these notions and present a corresponding mathematical stated property that, from a behavioral viewpoint, mirrors the intuitive notion. Unless other co-domains or domains are considered, it is understood in what follows, that the co-domain is the natural numbers (or symbols for them) and the domain a set of natural numbers, which intuitively corresponds to the set of symbols N' = {0,1,2,3,4,5,6, . . .} (Note that the natural numbers are actually any entity that satisfies the Peano Axioms. These axioms can be almost exactly modeled by modern "first-order" set-theory.)

Notice that counting, as here demonstrated, uses a special set of natural numbers. These are all the numbers that lie between 1 and some fixed natural number n. We use the intuitive "order" < to construct such sets. Thus {1,2,3,4,5} is such a set from 1 to 5, for the apple counting and the "size" is n = 5. Call each of these sets a segment. This segment is usually denoted by [1,n] and in words this is the set of all natural numbers greater than or equal to 1 and less than or equal to n. Now, from all of this, the "simplest" counting number definition for a finite set of things is describable. (Note: The basic segment can also be defined for the natural numbers that include 0 as [0,n]. This is not done in this article.)

(3) (Definition.) A set of things B is finite if there exists an injection on a segment onto B or the set of things is empty (notation ∅); that is, there is nothing to count. (Note: The first part of this definition is equivalent to stating that there is an injection on B onto a segment.) One can associate ∅ with a zero count, when counting physical things other than symbols. (Further, this definition can be generalized in that if there exists any function on a segment onto B, then there exists an injection from a, possibly different, segment onto B.) The properties of such injections lead to our understanding as to what it means to state that one finite set "has more members" than another finite set. This idea is then extended to the infinite set concept. (Note: When a mathematician gives a definition, it may not be stated in this "if and only if" form. But, this is always to be understood.)

I admit that I can only "count" a finite set. In, say, some ever lasting universe, one may conceive of "unbounded counting" in that given any natural number n > 0, no matter what it is, the set of natural numbers [1,n] can be used to count. We are told that mathematicians actually consider the "number of elements" in a set as its most fundamental property. Then this size language and "number of elements" is carried over to the infinite notion [5, p. 102-103].

Thus, the apples selected form a finite set as well as the set (of natural numbers or symbols for them) {123,1,4,10,6,0}, has "size" n = 6. In counting, we neither consider any order for the range nor what the symbols are. (In what follows, I will not consider any other definition for the finite nor the infinite as defined in (4) below.)

Suppose we write down all the statements that state "A set is finite if and only is such and such holds." Each of these is a characteristic for a finite set. Then since we use classical logic in mathematics, we have that "A set is infinite if and only if such and such does not hold." So, any one of these negated finite set characteristics is a characteristic for an infinite set.

Today, any set of objects that I can mentally count I consider as finite. Under the injection notion a perceived "order" does not alter this size concept. Let's consider the set {3,1,2,}. Now construct the function {(1,3),(2,1),(3,2)}. This function makes {3,1,2} a finite set. Note that for intuitive set notions the set {3,1,2} and ones like {1,2,3}, {2,3,1} are all equal sets since they each contain the exact same objects and if finite have the same size.

(4) (The negation of (3).) A nonempty set B is infinite if there does not exist an injection on a segment onto B or there does not exist an injection on B onto a segment. Hence, intuitively, I cannot actually count such a set. Of course, one can ask how do we show that something does not exist? There are ways to do this, but using all of the basic axioms of set-theory this characteristic is equivalent to one that can be easily applied.

(5) By using of the natural number notion of "induction," and maybe another notion, it can be argued that the set of all natural numbers N = {1,2,3,4,5, . . .} is infinite [3, p. 67-68]. If one accepts this proof, there does not exist an injection on a segment onto N. Note that given any segment, say A = {1,2,3, . . . ,n} = [1,n], there is an injection into N. The injection is {(1,1),(2,2),(3,3),. . . ,(n,n)} Note: As indicated previously, there is a tendency to consider two sets of natural numbers that depend upon their use - the one containing 0, N', and the one not containing the 0, N. (Having nothing better to do, I suppose, a few mathematicians argue against accepting this result since they "see" no way to even imagine the non-finite. This is no longer the case. They state that it is meaningless since, for them, the complete N does not exist.)

(6) Intuitively, the injection notion is used as a substitute for the counting notion. BUT, certainly one should expect that this should yield different properties for finite and infinite sets. This is why intuition may fail when infinite sets are considered.

For the size notion, what do we mean when we state that a finite set B is "larger than, greater than, or has more elements than" the finite set A and how is this notion extended to infinite sets?

II The Intuitive and Formal Meaning for a Set Being "Larger Than" Another

Let's see how we state that a finite set is larger than another finite set. Is A = {3,1,2}=[1,3] "smaller (in number)" than the set B = {7,26,55,1,0}? The function {(1,26),2,0),(3,7)} is an injection from A into B. This is just like choosing 3 members of B. Does there exist an injection on A onto B?

Suppose that f:A → B is such an injection. We use a process accepted throughout mathematics known as the finite axiom of choice. (It can be proved if one accepts certain properties about the natural numbers.) Mostly a mere finitely expressed collection of symbols is sufficient to state that a "visible" property holds. This is considered as entirely intuitive. Thus, the fact that it is an injection is "obvious." Other notions are also intuitively accepted if I can extend my imagination to larger finite sets for which I don't have a large enough piece of paper to write them down. By-the-way, if a set A is a subset of B, (symbolically A ⊂ B) and they are not equal, then A is called a proper subset of B.

First, for each x in A, I choice to remove f(x) from B. Then intuitively, after I have removed the distinct f(1), f(2), f(3), do I have any thing remaining? Well, what I am doing is removing symbols and every time I have done this to B I always have symbols remaining no matter what distinct members f(1), f(2), f(3) of B I choose. This operationally uses the finite choice notion. Of course actual observations are being used. The correspondence is between real written symbols. This is a "physical" correspondence. It is usually called a "concrete" model or approach.

Dedekind gave what appeared as an obvious negation to this finite set statement for a definition of an infinite set.

(7) (Dedekind) A set A is infinite if and only if there is an injection from A onto a proper subset B of A. (If one assumes or establishes that N is infinite, then this holds. This is a rather remarkable generalization that was not, at the time, formally shown to be equivalent to (4).)

But, the finite set statement requires the use of the finite axiom of choice. So, noticing this, today using (4), (7) is established by application of the "general axiom of choice," which, being independent from the basic axioms of set theory, is acceptable to me. The induction property for N is also used. So, we do not need to show something does not exist, but rather we can use the more direct approach and show explicitly that an injection does exist.

(8) Definition of the setorder. For any nonempty sets A and B, the symbols |A| < |B| mean that there is an injection on A into B, but no injection on A onto B. And |A| =|B| means there is an injection on A onto B. The two concepts are combined into the symbols |A| ≤ |B|. For a nonempty finite set A, |A| is the size, where A injectively corresponds to some [1,n]. This definition for ≤ is also used if A or B is infinite. It is, of course, a generalization for our counting notion.

A set is unbounded in the "usual" physical sense if for each segment there is an injection or there is a time or space dependent injection into a set of distinct physical entities. These entities can be mere spatial regions.

Notice that the set definition for < and ≤ also generalizes the counting notion even for finite sets since one need not use the counting numbers {1,2,3} as A. But, by the very definition of what constitutes a finite set, the injection and the non-injection notion can be referred to the original counting idea via the "composition" of two or more functions.

(9) The composition of functions. Let A, B, C be three nonempty sets. Let's denote the range of function f defined on A by f[A]. Suppose that f:A → B and g:B → C. Then the composition function gf:A → C is defined as follows: Let x be in A, then apply g to f(x) ∈ f[A]. The result is denoted by g(f(x)). This yields members of C. The domain of gf is A and the co-domain of gf is C. BUT, there is a slight difference in how we apply g. This is where we need the "in" notion. Although g has B as its domain, for the composition function gf, g is only applied to the range of f, to members of f[A]. In general, if h is a function on a set H, and D is any nonempty subset of H, then the function defined by h(x) only for members of D is denoted by h|D. Hence, for this composition the actual function used is not g but rather g|f[A]. This is hardly ever mentioned and must always be understood.

If f and g are injections, then so is gf. Also note that if h is any injection on H → D, for any D, then h restricted to any nonempty subset A of H, h|A, is also an injection from A into D. I wonder if |A| < |B| and |B| < |C|, then is |A| < |C|? Using other notions such as the if |A| < |B| and |B| < |A|, then |A| = |B| one shows that it does follow that |A| < |C|. [1, p 257-258]. This is an important ordinary property for an "order." It called the transitive property.

Under our definition of finite, we can use the composition idea to go from a segment to obtain a finite set and composition to obtain another set of the same "size." The set order < satisfies the intuitive rules we assign to counting and size for finite sets. Unfortunately, as I'll show, if the sets are infinite sets, then set order < does NOT follow the same rules that hold for finite sets. We should expect this. That's actually a good thing since it helps differentiate between the two concepts. One gains intuition by simply learning and applying the new rules.

Remember, the injection takes the place of the counting notion for both nonempty finite and infinite sets. For nonempty finite sets, it corresponds directly to our human notion of counting.

For physical space, the notion of infinite time or space can be described not as unbounded but rather by the onto injection definition. We don't need to go into an extensive discussion of "different levels" for the set-theoretically defined infinity notion in order to understand the physical use of the term "infinite," where one level may be as good as another. Although this all seems reasonable, some unusual things occur.

We might think from our experiences that the set of rational numbers Q (the "fractions") would be "bigger" in size than N for Q certainly contains rational numbers not in N, say 3/2, But, else, it can be shown that there exists an explicit injection on N onto Q [3, p. 82-83]. Hence, |N| = |Q|. Further, take any two rational numbers x and y such that x < y and [x,y] the set of all rational numbers between x and y inclusive, then something I cannot easily diagram occurs. |N| = |[x,y]| = |Q|. This is one of many unusual things that occur when definition (7) is applied. Below it's shown why this "strange" behavior occurs.

Two more examples, where finite counting does not follow the same rules of behavior for this generalized notion, are enough. As noted above, let N' be the set of natural numbers N with 0 included. The function f(x) = x + 1 is an injection on N' onto N. To see this, one actually needs a definition for the set of natural numbers. But, let's see what can be intuitively done if you allow for the integers.

First, assume that 0 is in the range of f. Then there is a y in N' such that 0 = f(y) = y + 1. Does such a natural number y exist? If it does, then 0 = y + 1. Using integer number algebra, we do know that y < y +1 = 0. But, using a certain property for the natural numbers, the obvious is shown. The number 0 is the "smallest" natural number in N'. This last statement says that y is "smaller" then 0. This is a contradiction. So, 0 is not in the range of f.

Now consider any z in N. Then z > 0. Using integers, z - 1 > 0 - 1 = -1. Hence, z -1 > -1. Thus, integer z -1 ≥ 0. Consequently, z - 1 is in N' and f(z-1) = (z - 1) + 1 = z. So, the range of f is N. Now let x and y be in N' and assume that f(x) = f(y) = x + 1 = y + 1. Then x = y. Thus, f is an injection.

Hence, |N'| = |N|. But, intuitively, |{0,1}| = |{1,2}|< |{1,2,3}| for these finite sets. So, for these finite sets of natural numbers, including just one more number in a set makes it larger than the set prior to inserting the number. Indeed, this holds for any finite set. But, for N, including one more number did not alter its | . . | size. This shows that the set ≤ does not follow the same rule for infinite sets as it does for finite sets. (Indeed, if E is the set of all even numbers and O the set of all odd numbers, by injections, it can be shown that |E| = |O| = |N|. Thus, in this case, adjoining a particular distinct infinite set does not change the behavior of | . . |.) You can begin to understand why using the mathematical definition that mirrors the original intuitive notations for the finite that we seem to understand beaks-down when the infinite is concerned. One will simply need to accept these unusual properties if one wishes to apply modern mathematics to the subject.

Let A = {0,1}. Then for the set of all ordered pairs A X A = {(0,0),(1,1),(0.1),(1,0)} (page 364 of that algebra book), we have that 2 = |A| < |A X A| = 4 = |A| x |A|. This shows that what we expected to happen with this finite set did happen. Each member w ≥ 3 in N can be written uniquely in the following way. First, by definition, 2 raised to the "0", 20, equals 1. Then by factoring all the 2s from w, you get a factor 2x and a factor that is an odd number (2y + 1), where y is some member between 1 and w inclusive. Thus, (a) w = 2x(2y + 1) ≥ 3, where x ∈ N' and y ∈ N. Note that if w = 2s(2t +1) = 2x(2y +1), then from the Fundamental Theorem of Arithmetic, you get 2x = 2s. Hence, x = s. Thus, 2t + 1 = 2y + 1 implies that t = y. So, the form 2x(2y + 1) is unique.

Let T be the set of all natural numbers of the form 2x, where x ∈ N' and O the set of all natural numbers 2y +1, where y ∈ N.

For any nonempty sets B and C, let B X C denote the set of all ordered pairs with first coordinate a member of B and second coordinate a member of C. In set generating notation, this is written as B X C = {(x,y)| x ∈ B and y ∈ C}. Let N'' be the set of all natural numbers greater or equal to 3 and n ∈ N''. Then since there is a unique x ∈ N' and unique y ∈ N such that n = 2x(2y + 1), the correspondence n → (2x,2y +1) is an injection on N'' onto T X O. Hence, |N''| = |T X O|. Now, consider the injection (x,y) → (2x,2y +1) on N' X N onto T X O. Then this yields that |N''| = |N' X N|. By (relative) "induction" and under the definition of addition, for each x ∈ N such that x ≥ 3, there is a unique y ∈ N such that x = y + 2. Further, for each y ∈ N, the unique y + 2 ∈ N''. Hence, for each x ∈ N'' and the unique y ∈ N, the map x → y is an injection on N'' onto N. Thus |N''| = |N|. Then the injection (x,y) → (x + 1, y) on N' X N onto N X N yields that |N' X N| = |N X N|. Consequently, |N''| = |N| = |N X N|. Thus, infinite sets do not follow the same rules as those for finite sets, when ordered pairs are considered. Why does this strange behavior happening?

This happens because mathematicians can define some rather unusual injections that do not capture the intuitive counting concept. Equation (a) does not seem to be how one would conceive of "counting" anything. Unfortunately, the phrase a countable set often means the empty set, any finite set or any set A such that |A| = |N|. It does not mean that we can actually "count" the members.

Without going into a technical definition, can a "real number," which is not a rational number (i.e. is irrational), at least, be imagined? Consider the following type of rule to construct something that seems like it should be a number.

r = .01011011101111 . . .

Notice that I have written, going from left-to-right a 0. Then I have followed each such 0 by repeated 1s, where I have repeated them enough times to correspond to the total number of 0s I previously wrote. So, for this representation for r, the next symbols would be 011111. Now I "image" that I can continue to do this, where the number "n" of 1s varies over the entire set of natural numbers.

In very basic mathematics this often unspoken imagination part is rather important. It's again the . . . part. We know that each time I stop at a particular m, then what I have without the . . . is actually a rational number. Indeed, a rational number with an "m + 1" collection of 1s that differs in value from the collection with "m" 1s by a rather small amount. And, as I continue to increase the natural numbers I use for the number of 1s, the difference gets smaller and smaller. So, whatever r is, a set of increasing rational numbers approximates it rather closely from a rational number viewpoint. Well, r is an example of an imaged real number that is irrational. Why? Recall that each natural number when expressed in decimal form must start, at some point in the expression, to repeat a fixed finite collection of numbers indefinitely. For example, consider 3.12345000000. . . , where the 0 repeats, or say 7.13452323232323. . . . For this article's notation, I still need to use the . . . idea.

Cantor gave an actual definition for the real numbers using only the rational numbers to do so. Others have also used different techniques. Further, one may also define them via axioms. The set of such real numbers includes the rational numbers. Cantor showed, in 1892, that there is no injection on N' or N onto the set of all real numbers R, but since N and N' are subsets of R (and not equal to R), there does exist an injection on N or N' into R. Cantor did this by introducing a new idea into mathematics. If you assume that such an injection exists, then his accepted method shows how to generate a real number that cannot be in the range of such a function. This is somewhat the same as my example for r above. The set R is said to be an example of an uncountable set. So, the notions of what is or is not "countable," even in the above infinite sense, is still used.

Let [1,k], k >1, be any segment and X any non-empty subset of [1,k] and X is not equal to [1,k]. By using the induction property for N, it is shown that there does not exist an injection on [1,k] onto X [3, p. 69-67]. (This is why |{1,2}| < |{1,2,3}|.) Since 0 is in N, then N is not empty. Assume that there is an m in N and an injection f on [1,m] onto N. Notice that m >1, since if not, then f(1) = p in N and {p} = N. But p + 1 is in N. Hence, N is not = {p}. Consider f, an injection on N onto [1,m]. Since [1,m+1] a subset of N. Then f|[1,m+1] is an injection on [1,m+1] onto [1,m]. This contradicts the second sentence in this paragraph, where k = m + 1. Thus N is also an infinite set using (4). (Yes! Of course, this assumes that N exists.)

Suppose that Y is a set such that N is a subset of Y and N and Y are not equal. Suppose that {(a,b)} is an injection with domain [1,n] onto Y. Then consider the subset {(x,y)} of {(a,b)}, where the range members, the y, are restricted to N. (This means that, for the ordered pairs, only those pairs that have members in N are used to form the (x,y).) Reversing the ordered pairs then, due to the injection property, the set {(y,x)} is also an injection on N onto a nonempty subset D of [1,n]. The obvious can be established that any subset of a finite set is finite. Hence, D is finite. By composition maps, there is an injection on some [1,n] onto N. But, this contradicts the fact that N is infinite. Hence, any set that contains N is an infinite set. Thus, N' and R are infinite sets and |N'| = |N| < |R|.

Let's look again at the numbered line.

. . . _____________________________________ . . .
. . . -4   -3   -2   -1   0   1   2   3   4 . . .

I guess one can say that this line is "infinitely long." Let π denote the constant ratio of the circumference of a circle to its diameter. If you had basic trigonometry, you probably learned about the tangent (tan) function and what angle measures called "radians" means. Let (-π/2,π/2) be the set of all real numbers between -π/2 and π/2 not including -π/2 nor π/2. (Don't think of this as an ordered pair in this case.) Since for (-π/2,π/2) only the end points are missing from the real number segment [-π/2,π/2], and (-π/2, π/2) is a subset of [-π/2,π/2], if one assigns a length to (-π/2,π/2), then this length would be π. But, using radians measures, the tan is an injection on (-π/2,π/2) onto the real numbers. So, the "tan" injection turns a "non-ending" object that converges to a finite object into an object that is infinite and intuitively converges to . . . ?

In 1978, I introduced the stronger or greater or better than ordering for the strengths of attributes that can be qualified by the use of the very, string of symbols. As an example, "very, very, intelligent" is stronger than "very, intelligent," which is stronger than intelligent. This ordering can be produced by logical deduction as well. Let VW be the set of all such words (i.e. a basic word with various numbers of "very," strings on the left). We can order members of VW, by counting the number of "very," strings of symbols on the left. This is the better than or stronger than or greater than order for VW. So, an entity that is "very, very, intelligent" has "greater" intelligence than one that is but "very, intelligent" since 2 > 1. (For the mathematical model the "," is needed in all cases.) This is the order used for the attributive GD-world model. The order > is the usual order for the natural numbers.

(10) Suppose that it is proposed that there are "infinitely many" universes that contain "infinitely many" biological entities that know how to count as we do. Let U{U} denote the collection of all of these universes. A physical entity P is a member of U{U} if and only if there is some universe in U{U} that has P as a member. Physically, if the notion of the physically infinite is that of the natural or rational numbers, then it can be shown that |N| = |U{U}|. The set U{U} seems to be one of the scenarios used in an attempt to eliminate God as the Creator. (This also applies to all known eternal atheistic cosmologies using a finite or finitely many universes.)

In the model I use, it is shown that there exists an object called a nonstandard word or ultraword W, since it behaves in some but not all respects like an ordinary word in our language. The W is composed of a word like "intelligent" and a lot of "very," strings on the left side. (You can define intelligence any way you wish.) But, being a nonstandard object, W has properties that ordinary words do not have. In all cases, W is "infinitely" stronger than any member of VW, where the "number" of "very"s does correspond to an injection onto an infinite set. The W can be obtained in various ways. Each member of VW has a finite number of "very,"s on the left. But the word W has hyperfinitely many "very," strings. Without going into the rather complex mathematics required to study the hyperfinite, how can be we characterize this hyperfinite notion? It won't be easy since

α < α +1 < . . . < -2 < -1 < 0 < 1 < b-1, b

is a hyperfinite set. But, fortunately, it is not necessary to actually have any additional knowledge about this concept for me to establish that the strength of God's infinite attributes can only be but partially described mathematically, I won't further describe the hyperfinite in this article.

It has been shown that the comparative "measure" ≤B I use to measure the strength of God's comparable attributes is "bounded" in one sense, but in general is considered as unbounded within set theory. This occurs when other models for the set *N' (the hyper-natural numbers) that correlates to this measure are used. As one should expect, since *N' is a nonstandard set it has properties that are not exactly the same as those of N'. The symbol I have used about a set A, the |A|, actually stands for a type of comparative measure for the size of an infinite set. That is, relative to any model *N' used in my investigations to express strengths of God's modeled attributes, there is another model that contains **N', where as measured by | . . |, God's attributes are stronger for the **N' model than for the *N' model. (This technical paper establishes this and a great deal more.) These results are all obtained by using rational mathematical reasoning. Thus, I do NOT differentiate between < levels of the infinite since, for any such nonstandard H, |H| depends upon the model used. The phrase infinitely greater, and the like, is a generic phrase relative to the common notion of the infinite and these order properties.

Theologically, these measures still apply but they are partial in that the mathematics used does not yield a strongest measure - an ultimate measure. (There is a "type of" ultimate measure if one assumes that "class theory" is consistent.) I simply state that God's intelligence is infinitely stronger or greater than that of any entity within a physical universe OR any combined collection of entities in infinitely many (the U{U}) universes. This "infinitely greater" notion can be simply compared with the infinite set N' or N, where the "infinite" notion is the easiest to comprehend. Adding an ultimate bound for these measures is rationally acceptable. But, the notion is exterior to the set theory being used.

Using the injection concept, it is rather easy to show that there is a comparative "infinite" notation. Indeed, the next result shows that for, at least, one property the finite and infinite correspond. Take a finite set A = {1,2,3,4}. Now I can list the set of all subsets, where ∅ is the empty set. They are the sixteen-member set P(A) = {∅,{1},{2},{3},{4}, {1,2},{1,3},{1,4},{2,3},{2,4},{3,4},{1,2,3},{1,2,4},{1,3,4},{2,3,4}, {1,2,3,4}}. Now notice that there is an injection of {1,2,3,4} into this set. Simply consider f(x) = {x} or (x,{x}) as x varies over {1,2,3,4}. But if our finite counting makes sense, that is the f function, then there should not be such an injection from A onto P(A). Why don't we prove this in general.

Suppose that A is any nonempty set finite or infinite. Set theory allows use to consider the set of all subsets P(A) of a set A. Suppose there is an injection f:A → P(A). Then set-theory states that we can use f to define subsets of A. Such a correct definition is that B is the set of all of members x of A such that x is not a member of the set f(x) (x ∉ f(x)). The assumption, at the moment, is that f is a injection. Thus, there is a b ∈ A such that f(b) = B. Now where is b? Suppose that b ∈ B. Then we also have from the definition that b ∉ f(b) = B; a contradiction. So now supposed that b ∉ B. Then by definition b ∈ B = f(b); another contradiction. How do we escape from this? We simply state that no such injection exists. But, under the set < definition this means that for infinite N', |N'| < |P(N')| and |N'| ≠ |P(N')|. Now substitute for N' the P(N'), and you get the set P(P(N')) and that |P(N')| < |P(P(N'))| and |P(N')| ≠ |P(P(N'))| along with the property that |N'| < |P(N')| < |P(P(N'))|. Thus, a given set and the set of all of its subsets, whether finite or infinite, has this common property.

III A Significant Specific Case.

Let a time interval be measured by a specifically defined sequence of distinct rational numbers (Q) between a and b, where a < b. Denote this interval by [a,b] when a and b are included. The sequence itself is a injection on N onto [a,b] and, thus, |[a,b]| = |N| = |Q|. (Note: N' is also used for sequences.) Given a finite set of hypotheses, how many deductions can we obtain during this interval using a specifically defined set of rules? I accept that only a finite number of deductions can be done by any biological entity within our present environment that understands how to apply the rules.

For a maximum suggested cosmology, assume that in each member of the collection U{U} of universes that comprises an eternal universe, there is a super-agent that can deduce n conclusions for any n in N during the same time period [a,b]. Now each distinct deduction is encoded in the GGU-model by a natural number. Let E be the set of all coded deductions produced by the super-agents in U{U}. There are specifically described rules for such deductions. Mathematical relations within the mathematical theory used for the GGU-model represent these rules. In particular, the sets of rules are modeled by collections of objects called the rules of inference. Let PR denote the set of rules of inference used within in each member of U{U} and used by the collection of all super-agents. Suppose each U in U{U} is the result of consistent processes, then PR is assumed to be a consistent set of such rules of inference. It turns out that for U{U}, |PR| = |N|. Further, for the entire collection E of all super-agent deductions |E| = |N|. (This is established using set-theory.)

The mathematics used for the GGU-model predicts that there exists an hyperfinite set of rules of inference PR' that contains the PR. The modeling process shows that |PR| <|PR'| = |*PR| and this implies that |PR| < |*PR| = |*N|.

(What follows is an important discussion and I hope, I really hope, it makes sense to someone other than myself.)

Now consider what type of intelligent activities are being measured by the GID-model interpretation for the GGU-model. Assume that for any natural number n > 1 and m > 1, a super-agent can deduce from a finite set of hypotheses H, such that |H| = m, n conclusions, where n and m are in N. (The number of conclusions depends upon the rules of reference used and there are always such rules that allow one to deduce m conclusions.) Further, such agents can obtain such conclusions during any finite time interval of the form [a,b]. Thus, for a collection U{U} of all universes, the collection of all the super-agent deductions E has the property that |E| ≤ |N'|. (Certainly, some distinct super-agent deductions can be the same conclusion. On the other hand, there can be rather distinct deductions as well.) The same holds for the collection of all hypotheses U{H} and rules of inference U{PH} they use. That is, |U{H}| ≤ |N'|, and |U{PR}| ≤ |N'|.

Theologically, compared to the entire collection of super-agents, God. at least, can use infinitely many hypotheses and infinitely many rules of inference, and deduce infinitely many conclusions during an "infinitesimally" long time interval. The "infinitely many" in this last sentence corresponds to intervals [1,ν], [1,μ], and [1,γ], where ν, μ, γ are members called infinite numbers and relative to the set order < each measures a set that is "greater in size" than the "size of" N. In the above mentioned paper these infinite sets can be compared to all the infinite sets used with standard physical science via the | . . | notion. It is shown that these intervals are very "large" when so compared to each or combinations of the infinite sets employed for standard physical science mathematical models. Moreover, in our physical cosmologies, we are forced to state that anything deduced by God can appear to occur instantaneously. These "deduction" signatures are the exact signatures used to model the higher-intelligence processes as interpreted in the General Intelligent Design (GID) model.
The most well known type of intelligence used within the discipline called "Intelligent Design" (ID) is the non-mathematical notion of "purpose." Using "purpose" yields a non-scientific model and can only be applied to a miscue number of physical entities. The term used for the ID that uses this approach is "Restricted Intelligent Design" (RID).

Note that the New Testament Greek term translated as "forever, for evermore, everlasting" can also carry the idea of being "immeasurable." I need not list those Biblical verses that describe God's behavior in terms of mental behavior so we can better understand it. One merely substitutes the Biblical God for the term higher-intelligence. Hence, we have a completely rational and specific case that verifies the statement that "God's intelligence is infinitely greater than the combined intelligence of all entities within His created universe(s)." Again I mention, that as shown in the technical note, the term "infinitely" should be considered as generic in character from the set theory viewpoint. It should be considered as "unlimited" or "unbounded" in an ultimate sense.

The modeling techniques I use cannot measure this unlimited notion. They can only given partial measures. Hence, whenever I state that *L represents a higher-language, among other similar statements about higher-intelligence deduction, this does not imply that God's intelligence is limited. It means that although *L represents a higher-language it is not the only higher-language. Indeed, *L can be a subset of other higher-languages of much greater comparative size. However, it is logically acceptable to hypothesize that ultimate bounds exist and such ultimate bounds imply that the |. . .| and attributive < can be used as a partial measures.

God has described attributes that are not comparable, such as foreknowledge. It can be shown that there are also higher-attributes that are not comparable and to which the < measure is not applicable. Thus, theologically, the higher-languages, those entities constructed from it and the higher-attributes are not all of God. Indeed, they should not be used as a complete model for God. So, the generic phrase is definitely the appropriate one to use. I never restrict the strength of God's attributes. His attributes cannot be restricted just to those that His created can express.

Repeating, the new result is that for the set theory used and relative to the interpretation, the "infinitely" part of the phrase "infinitely greater than" as used within my models is not bounded. Thus, the term "infinitely" in the phrase "infinitely greater than" applied to the Biblical God should be generic in character for various reasons. This result gives strict Biblical meaning to the Greek "aion" often translated as "forever" and the meaning unbounded. This also means what Plato meant by its use - "immeasurable." If one has faith in the consistency of class theory, then it is rational to assume that a type of ultimate bound exists. Each member of the class of all such set-theoretic measures yields a partial representation for this ultimate bound.

[1] Abian, A., The Theory of Sets and Transfinite Numbers, W.A. Saunders Co. Philadelphia, (1965).

[1a] Herrmann, R. A., Science declares our universe IS intelligently designed, Xulon Press, Fairfax VA, and other addresses, (2002).

[1b] How to imagine the infinite

[2] Stroyan, K. D. and J. M. Bayod, Foundations of Infinitesimal Stochastic Analysis, North Holland New York, (1986).

[3] Wilder, R. The Foundations of Mathematics, John Wiley & Son, New York, (1969).

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