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.
(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.
Consider the following illustration. Start with
(a) = 1 = 1.000 . . . , where the ". . ." means mentally continue repeating the 0s. Now considerLooking at the pattern of fractions (rational numbers), the next one that should added 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. But at some point, the sum will actually be slightly larger than this number.(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. . .
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.
Finally, notice that for the usual ordering of the natural numbers as learned in elementary school, one hopes, (a) < (b) < (c) < (d) < etc. A property of the "real numbers," which are constructed from the rational numbers, is that there is a number, call it X, to which these sums are getting closer and closer. The property states that there is no other real number Y such that Y < X and this 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.
In this article, after the terms are properly defined, I discuss with respect to the mathematical notion of "infinity" a fact similar to this summation illustration. The (a) - (d) and all those that follow only approximate the X. The fact is that God's attributes can be modeled by a similar ordering, where each expression in the order 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 indescribable, an immeasurable, ultimate strength for each God attribute, then this intuitively would yield each of the mathematically modeled infinitely greater strengths.
In order to show how these measures work, it's necessary to present some "rather simple" notions that I'm sure you can follow after careful reading. Two "orders" are used in what follows. The same symbol < is used for both. You can tell which is which by considering the type of objects that appear on the left and right of the < or < symbol.
To define the (ordinary) finite, just consider what "counting" means. 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 names; a function, an operator, a transformation, a map, a mapping, a functional etc. In general topology, it is called a map or mapping. But, here, let's call it a function. In French, functions are called "applications" because they can be throughout 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. Below, I give an example of this.
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 no specific axioms are presented in this article, then these symbols, the numbers they represent and their order, should be intuitively understood. For the illustrated 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.
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. "The function f maps the domain into the co-domain" or "The f maps the domain into the co-domain." There are variations of 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. 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." 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 it is always "onto" its range.
Here's another injection for symbols that represent the 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.
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 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
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) = 5^x + 3x^2 + 10x. I viewed this as various processes that generate numbers from numbers. I would even write f as f( ) = 5^( ) + 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(<-). Indeed, given any injection f on X onto Y, then there is an injection f(<-) on Y onto X.
If you have an ordered pair (a,b), then "a" is the first coordinate and "b" is the second coordinate. 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.) 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, . . .}
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 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) A set of things B is finite if there exists an injection on a segment onto B or the set of things is empty (i.e. there is nothing to count). One can associate the empty set with zero (nothing to 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. )
I admit that I can only "count" a finite set. In some 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. (This scenario is further discussed below.)
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} where n = 6. We are not considering any order for the range nor what the symbols are. (In what follows, I will not consider any other definition for the finite or the infinite as defined in (4) below.)
(4) A nonempty set B is infinite if there does not exist an injection on a segment onto B. Hence, intuitively, I cannot actually count such a set. (There is more than one definition of the infinite. This is the easiest to understand.)(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]. Hence, 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: There is a tendency to consider two sets of natural numbers depending upon their use - the one containing 0, N', and the one not containing the 0, N.
(6) Intuitively, the injection notion is used as a substitute for the counting notion. BUT, this yields different properties for the finite and infinite.This is why intuition may fail when infinite sets are considered.
What do we mean when we state that a finite set B is "larger than, or greater than" the finite set A and how is this notion extended to infinite sets?
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.
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.) This is considered as entirely intuitive. Other notions are also intuitively accepted if I can extend my imagination to larger finite sets that I cannot write down.
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. So, there is no such injection on A onto B. Notice that actual observations are being used. The correspondence is between real written symbols. This is a "physical" correspondence. It is usually called a "concrete" model.
(7) Definition of {<,=} order. 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|. This definition is also used if A or B is infinite. It is 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 definition for {<,=} also generalizes the counting notion even for finite sets since one need not use the counting numbers A = {1,2,3} as A. But, by the very definition of what constitutes a finite set, the injection, the non-injection notion can be referred to the original counting idea via the composition of two or more functions.
(8) The composition of functions. Let A, B, C be three nonempty sets. Let's denote the range of function h defined on A by h[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). This 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. 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 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|, whether |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].
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 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 < does NOT follow these 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 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. Not 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 does not alter its | . . | size. This shows that < 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 distinct infinite set does not change the behavior of | . . |.)
Let A = {0,1}. Then for the set of all ordered pairs A X A = {(0,0),(1,1),(0.1),(1,0)}, 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 in N can be written uniquely in the following way. First, by definition, 2 raised to the "0", 2^0, equals 1. Then by factoring all the 2s from w, you get a factor 2^x and a factor that is an odd number (2y + 1), where y is some member between 0 and w inclusive. Thus, (a) w = 2^x(2y + 1), where x and y are in N'. Note that if w = 2^s(2t +1) = 2^x(2y +1), then from how the factor 2^s and 2^x are defined, you get 2^x = 2^s. Hence, x = s. Thus, 2t + 1 = 2y + 1 implies that t = y. So, the form 2^x(2y + 1) is unique.
Let N' X N' denote is the set of all ordered pairs with first coordinate a member of N' and second coordinate a member of N'. In set generating notation N' X N' = {(x,y)| x is in N' and y is in N'}. Consequently, (a) yields a unique (x,y) determined form for any member of N. Equation (a) yields an injection on N' X N' onto N. Hence, |N' X N'| = |N|= |N'| or |N' X N'| = |N'|. Thus, infinite sets do not follow the same multiplication rules for the size of sets, when such a set of ordered pairs is defined, as do finite sets. 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|.
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.
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 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" group of 1s that differs in value from the one 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" 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 an infinite set.
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|.
Here is another example of how the mathematics of the infinite differers from the finite. In high school, most students learn about the "numbered line." This is an "infinite" like geometric object the associates each point (location) on the line with a real number.
I guess one can say that this line is "infinitely long." Let pi denote the constant ratio of the circumference of a circle to its diameter. If you had basic trigonometry, you probably learned about the tangent function and what angle measures called "radians" means. Let (-pi/2,pi/2) be the set of all real numbers between -pi/2 and pi/2 not including -pi/2 nor pi/2. (Don't think of this as an ordered pair in this case.) Since for (-pi/2,pi/2) only the end points are missing from the real number segment [-pi/2,pi/2], and (-pi/2,pi/2) is a subset of [-pi/2,pi/2], if one assignees a length to (-pi/2,pi/2), then this length would be pi. But, using radians, the tan(x) is an injection on (-pi/2,pi/2) onto the real numbers. So, the "tan" function turns a finite long object into an object that is infinitely long.
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. 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.
(9) 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. That is, a physical entity P is a member of U{U}} if and only if there is some universe in {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 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 like an ordinary word in our language, where W is composed of the word like "intelligent" and a lot of "very," strings on the left side. (You can define intelligence any way you wish.) But W has properties that ordinary words do not have. In all cases, W is "infinitely" stronger than any member of VW. 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?
First, there is a new set of numbers *N' that behaves with respect to the ordering < and in most others ways like the set N'. Further, the N' numbers are also in *N'. In the usual way, a segment for *N' is defined by H in *N'. Then a basic segment, for this article, is the set of all x in *N' such that they lie between 1 and H, inclusive, or in notational form {x| x is in *N and 1 < x < H} = [1,H] = H'. This type of set is called hyperfinite. Note that if H = 0, then [0,H] contains no numbers. It is empty. If it's not empty, then no matter what the H is, [1,H] behaves like a finite set in many ways but, in general, it is not finite. (Note: If there is a special type of injection from a hyperfinite H' onto another special type of set G, then G is also called hyperfinite.) But there are many H in *N' that are not in N'. For two sets A, B, we write A - B as the set you get be removing all of the members in B that are also in A.
The finite and infinite concepts can be compared in one sense relative to injections. The finite number of "very." strings corresponds, conceptually, to each member of N' via the segment [1,n], where each x in [1,n]. is a member of N' that lies between 1 and n. The "n" is the actual number of objects in [1,n]. If [1,n] = the empty set = {}, then this means no "very," string appears on the left of the basic word like "intelligence." (What is being done here is that a mathematical model for some linguistic notions is being constructed.) For W, there is an H in *N' - N' that yields the "number" of "very," strings in W. Hence, either H or [1,H] can be used as a measure for this number. But, as noted below, this is only a comparison of the finite [1,n] and the "non-finite" hyperfinite [1,H].
Using the ordering on *N', for each H in *N' - N', there is an ultraword W(H) with H "very," strings. But there always exists another G in *N' - N' such that H < G and another ultraword W(G) that has G "very," strings. Although, W(H) is stronger than any comparable human attribute, W(G) is stronger than W(H). This yields a type of sequence of these ever increasing stronger measures for the same comparable Divine attribute. In the mathematics used, there is no K in *N' - N' such that every member of *N' - N' is < K. For a single *N', the model used for | . . | does not satisfy the same order notion as H < G. In this model, |[1,H]| = |1,G|.
In the models I use, it turns out that for any H in *N' - N', where N' is a subset of *N', at last, |N'| < |H'|= |[1,H]| = |*N'|. Of course, each segment S determine by N' has the property that |S| < |N'|. Hence, |S| < |H'|. Moreover, from (8), |U| = |U{U}| = |N'| < |H'|. Since, for injections, < does not follow the same rules for finite sets and infinite sets, where very clever injections need to be defined, then it is up you to decide whether the injection notion corresponds to the notion of something being greater than something else. But, by "interpretation" the intuitive notion of "greater than" can be applied. So, how might we describe this for the GID-model (i.e. the GID interpretation)?
It has been shown that the measure <(B) that I use for measures of 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 *N' are investigated. That is, relative to any model *N' used in my investigations to express strengths of God's modeled attributes, there is another model **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 the H', |H'| depends upon the model used. The phrase infinitely greater, and the like, is a generic phrase.
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 such an 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.
[Within modern set theory and using the function notion, there are sets C that are, intuitively, considered by some to represent "different" levels of the infinite notion since the ordering for any set of these sets is the |. . .| order. In the sense of this order there is no greatest cardinal [1, p. 369, Problem 9.] Hence, if you restrict your conclusions to modern set theory and wish to model the infinite notion associated with God, then a cardinal number is certainly a good choice. But, such an infinite notion is again only partial in character within set theory.]
Let a time interval be measured by a specifically defined sequence of distinct rational numbers between a and b, a < b. Denote this interval by [a,b], where 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} 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 D 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 rules of inference used within in each member of {U} and used by the collection of all super-agents. Suppose each U in {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 D of all super-agent deductions |D| = |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|. How do I compare theologically the infinite hyperfinite with the finite?
(What follows is an important discussion and I hope, I really hope, it makes sense to someone other than myself.)
Next, I 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 an 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 D has the property that |D| < |N'|. 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,nu], [1,mu], and [1,gamma], where nu, mu, gamma are members of *N - 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 by 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 now 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 such ultimate bounds exist and these 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 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 partially representation for this ultimate bound.
[1] Abian, A., The Theory of Sets and Transfinite Numbers, W.A. Saunders Co. Philadelphia, (1965).
[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).