The Hyperfinite, Ultraword and Ultralogic Concepts.

Robert A. Herrmann, Ph.D.

(I) The Hyperfinite and the GID-model

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

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

A More Technical Explanation

Suppose that one has a collection of simple statements that characterize the term "finite." A basic theorem about finite sets is that (1) "any subset of a finite set is finite." In the technical GGU-model, four to six different collections of terms are used. The terms "subset" and "finite" are called "standard terms." The term "hyperfinite" is an "internal" term and can take the place of the term finite. There is also an internal term associated with the standard term subset. This adjective may not appear but from the context it is understood. The term is "internal subset." The above statement (1) is translated into a correct GGU-model statement by making the substitutions. This yields (*1) "any internal subset of a hyperfinite set is hyperfinite." An internal subset of a set is a subset that is also an "internal" object and is denoted by a symbol that is a member of the symbols used for internal objects.

If one were only to use the internal language translations, then it turns out that one could not write an internal statement that is actually stating something different about hyperfinite sets. But, the mathematician can use two other sets of terms, the "external" and different "metaterms." One describes properties relative to these different languages. The metalanguage being the most general that is used to describe relations between entities defined in terms of these other languages.

For example, any infinite (standard) set in the model is also an external set. Then various properties are established. There are subsets of certain hyperfinite sets Z that can be shown to be external subsets of Z, but they are neither standard nor internal. They are neither a member of the collection called that "standard" sets nor the collection call the internal sets. BUT, on the other hand, every standard infinite set X is a subset of a hyperfinite set Y, when the set theory used is described using its metalanguage. Then the set that contains all the members of Y that are not members of X is neither standard nor internal, but rather external. Then there are sets that are neither standard, internal nor external. Indeed, the set, the superstructure, that contains the standard, internal and external sets is such a set.

Viewed from the GGU-model "metaworld," there are differences between the hyperfinite and finite. For the "internal people" that only use internal terms, the infinite set X does not exist. Further from the viewpoint of the "standard people" who use only standard terms, the set Y does not exist. The set X is also an external set. Using only the external language, the set Y also does not exist. There are other external sets that are only external. For example, the set of all infinitesimals, m(0), is external and neither standard nor internal. Unless one is careful, these language variations can lead to contradictions. In nonstandard analysis, one can use six different collections of terms to describe the various entities used.

As a last example, for the GGU-model, one can write a metatheorem, that gives relations between these different types of objects. The (external) infinitesimals m(0) is a subset of the internal hyperreal numbers *R, the standard real numbers R is considered a subset of *R and R and m(0) only have 0 in common. The term "subset" as used in the last statements is the notion used in elementary set-theory and carries no additional restrictions. There is a lot more one could write about these comparisons as viewed from the metaworld. A higher-intelligence, at the least, uses the metaworld viewpoint. The GGU-model employs hyperfinite sequences based upon the hyperintegers. So, one also needs to know how the hyperintegers behave.

(II) Ultrawords, the Hyper-language and the GID-model

The major portions of this article are concerned with properties of the General Intelligent Design (GID) Model and to the most recent refinement. This refinement has been added to the latest articles on this Website and those archived at This article is a strong GID-model interpretation. The logic-system notion is discussed at length in my book "Science Declares Our Universe IS Intelligently Designed." But, rather than employ the mentally applied algorithm AG relative to a specific logic-system, the P deduction process as discussed below, as the basic mathematical entity, the corresponding consequence operator S is employed as its representation. This article has been modified from the original, which was written for the book's *S logic, and is now concerned with the refined P process. (Technically AG is written as the symbol script A.) The refined method only appears on this website and in my articles.

If you don't know what a mathematical model is nor the basic notion of how symbols are applied and interpreted, please see the section on mathematical models in this article. Further, certain undefined terms used here are defined in this glossary. As usual, the "intelligent agent" terminology as employed throughout the GID-model may be but analogue in character in that we may not be able to otherwise comprehend such behavior.

It is important to remember that the GGU-model can also be considered as an "analogue model for behavior" in that it rationally represents behavior and properties, where generalized descriptions F(t) represent actual physical objects. (The "t" is a primitive sequence (time) index.) Indeed, this is all that physical science does. It deals with descriptions as substitutions for physical events E(t). But, directly related to such descriptions is the notion of informational "instructions." These instruction-entities I(F(t)) use propertons (previously called subparticles) and yield an event E(t) that corresponds to the F(t). Thus, it is not the F(t) that generates E(t) but rather the I(F(t)). Each instruction-entity is distinct since, at the least, it carries a different primitive sequence, t, identifier. An instruction can be considered as a member of the set of all generalized descriptions. There are various choices as to the objects to which the ultralogics as discussed below apply, such as *S or *P. Each can be used for the GID-model interpretation. But, one may be more appropriate for the maximum secular GGU-model. In either case, they are applied to "ultrawords." Ultrawords are "significant" members of the predicted hyper-language (higher-language) *L for the GID-model interpretation.

In some cases, by application of the general paradigm approach, a few members of *L that are not members of the language L can be decoded in such a manner that the addition of one or two symbols renders them meaningful to us. This is how the notion of the ultra-properton is predicted. These cases, however, are rather rare. Based upon the modeled aspects of human behavior via measures of intelligence, the GID-model interpretation does state, in general, that the rationally predicted behavior of an higher-intelligence implies that such an entity does comprehend the descriptive content of "infinitely" many members of *L that are not members of L. Modern physical science claims that humankind can predict and comprehend the behavior of the physical-systems we describe. Applying this to *L indicates that each member of a *developmental paradigm has meaningful content for a higher-intelligence. Hence, rationally, this applies to each of the ultranatural events.

Recall that for the GGU-model, ultralogics are termed as "intrinsic ultranatural processes," or IUN-processes within a background or substratum region in which physical universes are embedded. They are also sometimes termed as force-like processes. Using this terminology, they are considered as producing physical-systems, and producing, staining, guiding or controlling physical-system behavior. Their relation to intelligent design can be considered as an extraneous interpretation that is ignored as is done in quantum logic. Further, for the GID-model, ultralogics have the same GGU-model "meanings" except their relation to intelligent design is not ignored and these IUN-processes are coupled with the intelligent agent characteristics that are considered a general signature for the intelligent design aspects. The term "signature" is used for certain characteristics displayed by consequence operators or equivalent logic-systems that are directly related to mental processes applied by an intelligent agent.

When I write a linguistic expression, I almost always first compose it mentally in the form of my "mental voice." This "mental voice" model is usually extended to the behavior of a higher-intelligence, where the term "mind" may be the most appropriate term that generally describes such behavior. For simplicity, first consider an ordinary language L constructed from a nonempty finite set of symbols. For our use, also include a spacing symbol |||. Technically, a (Markov) "word" is a finite combination of these symbols written left-to-right. (I note that the actual foundations of much mathematics are human experiences.) Thus aaabcaa is a word distinct from aacbaaa.

There is an intuitive operation defined for "word theory," which is called by various names. Let's call it "juxtaposition." Often, it is not represented by a symbol. (Mathematically, if the empty word is not included this is called a semigroup operator and satisfies a basic algebraic structure. Including the empty word yields algebraic monoid structure.) Thus, aaabcaa is considered as formed by juxtaposition in various but finitely many ways. For example, juxtaposing aaab with caa, and aa with abcaa yields aaabcaa. There are words like - it|||is|||cold. - which are used to replace "itiscold." so that one does not confuse the meaning. I introduced the notion of "meaningful" into word theory and few seem to know this. Certain combinations of such symbols will change their meaning when the spacing symbol is put in different places. Then a method is introduced that allows one to suppress the finitely many different left-to-right juxtaposing operations that yield the same combination. Also, one can replace the alphabet symbols with "images" or"diagrams." This yields a generalized language.

The methods introduced allow one to examine each member of a word as well as how the words are constructed by juxtaposition. One can construct certain "words" that correspond to the sequential "appearance" or descriptions for a physical-system as it develops. The general language L being used is coded in a special way. This coded language L is composed of mathematically defined objects. (This language can also be replaced by a corresponding "propositional" language P where distinct symbols correspond to the words. Propositional logic is the foundation for the construction of a computer microprocessor.) Relative to their construction, ultrawords have the same mathematically representable properties as such strings of ordinary linguistic entities.

Words have various algebraic properties and one can "count" the number of positions a word will require for its actual expression. However, a special symbol is needed if the members of L are expressed as meaningful statements. For example, a word w might be "A photon is emitted and then absorbed by the electron." The "and" in this statement should not be confused with the "and" used to logically separate two words. This is no problem if only propositional symbols are used as replacements for such words. However, in certain cases the actual content of a word is investigated. Hence, to avoid confusion for the logical "and" a special symbol for the logical "and" is used, where the symbol ∧ appears nowhere in any of the words used.

However, using the propositional language logic is not necessary. The language is mainly employed in my book "The Theory of Ultralogics." Almost all aspects of the GID and GGU-model use the logic-system approach applied to a general language. All of this is then embedded into a special mathematical structure. As originally introduced, a very simple informal system that is a sub-system for propositional logic is employed and is represented by consequence operator S. However, the major logic that is employed for the GID and GGU-models is the AG algorithm defined on members of L.

Let ||| be a spacing symbol. Let F(1), F(2), . . . F(n) represent "n" members of a developmental paradigm. Each F(k) is an abbreviation for a general description. They are necessary distinct via a sequential coding symbol. Now consider the actual word w(n) = F(1)|||If|||F(1),|||F(2).|||If|||F(2),|||F(3).||| . . . .|||F(n). This word is mathematically modeled by a logic-system. The logic-system's mentally applied algorithm AG yields the exact same results one gets by applying the most basic form of linguistic deduction "modus ponens" or the "rule of detachment." Results obtained by application of AG are always considered as a form of deduction. When AG is applied to this word one deduces the set of descriptions {F(1),F(2),F(3), . . . ,F(n)}. But, they are deduced in the left-to-right order as here displayed. All such words w(n) as you let n vary over bigger and bigger numerical values are embedded into a mathematical structure. The mathematics predicts the existence of the ultraword w in the extended language *L and not in L. This ultraword can be represented by the notation w= F(1)|||If|||F(1),|||F(2).|||If|||F(2),|||F(3).||| . . ., where the . . . indicates an infinite extension. The ultralogic *P applies to this ultraword w. This yields the "hyper" deduced set of descriptions {F(1),F(2),F(3), . . .} in the same order as the ordering of the numbers 1 < 2 < 3 < . . ..

Intuitively, mathematicians do consider various entities as having a "size," even when they are "infinite" entities. They also intuitively use such comparative terms as "more" and "greater" for such entities. Robinson states that there is an "L'-sentence whose length exceeds l" (Robinson, A., On languages which are based upon non-standard arithmetic. Nagoya Math. J. 22(1963, p. 91)). The l entity is an infinite Robinson number and, mathematically, when compared with other "infinite" sets, it is rather "large." The notion of "length" corresponds to a function that intuitively "counts" the symbols in the form even if they are repeated. Using this approach, at this point, all that is needed is to state that w is composed of more language "symbols" than any entity in any decoded member of L, where there are members of L that are, at the least, conceptually of any finite length. This is one reason that w is call an ultraword.

For the GID-model, an higher-intelligence can work with such a w as easily as we work with a word from L. This can be technically established. [Such "easy" work is hyperfinite work.] Due to the method of construction, the interior construction of any such w can be, at least, partially analyzed. Portions can be decoded, and these portions are images, diagrams or descriptions in an ordered set d and these are contained in another ordered set d'. For the GGU-model, the d corresponds to a physical event sequence. It is here where one can establish that d' contains symbols that construct words in *L, and these symbols and the words they form are not members of L.

[In more detail using the ultralogic notation below, when the ultralogic *P(*R(λ), • ) is applied to F(1) , you get the d' as well as the coded images, diagrams or descriptions in d. This is equivalent to a higher-intelligence applying infinitely times the modus ponens deduction rule. Since w behaves in a general way like an ordinary word, but, in comparison, w is a member of *L, not a member of L and it is longer than any word taken from L. Hence, such a w is a significant ultraword. I often state that such ultrawords as w are composed of "supercompact information" and *P(*R(λ), • ) decompresses w so as to obtain the specific information it represents.]

The general term "ultraword" has been extended, now and then, to include other types of thought-elements that can be represented by members of *L and are not members of L.

Your author has recently refined such w type ultrawords so that, emergent properties, if they exist, are included. One can think of these "ultrawords" as containing all the specific information that leads to instruction-information that will lead to the formation and development of every physical-system within a material universe. However, one does not obtain the physical-systems and their developments unless an ultralogic is applied. Thus, we have that ultrawords are associated with the GID-model notions of intelligent design by a higher-intelligence and, when realized, the patterns produced [by *AG] are designed by a higher-intelligence. The GID-model ultrawords can be considered as pre-designed.

For the GID-model, each such ultraword has associated with it a set of instructions I(λ). The same approach using *P is applied to an instruction-entity ultraword. It is the instruction-entities that actually produce the physical entities at each moment in the development of a universe. Thus, ultrawords can be considered as objects that contain "supercompressed" specific information or informational instructions. Finally, no physical entity displays, in any way, written or coded characteristics. For the secular GGU-model, only the set of instructions I needs to be employed. There are, however, other schemes that can be employed. For the GID-model, both the descriptions and their corresponding instructions can be used. Thus, based upon ones choice, the technical GID-model may be slightly different than the GGU-model. Whichever approach one chooses, instructions can be considered as a pure analogue notion in that they are but auxiliary objects that aid in human comprehension.

On the other hand, some symbolically expressed instructions can be considered as representing actual physical-like processes that behave in a rather simple manner. They detail the actual number of entities that are to be combined to produce more complex physical-systems. These details have been mathematically expressed and can be considered as actual physical-like processes that "force" physical objects to conform to a description, where the processes are guided by a set of linguistic-styled instructions. Indirect evidence as used in atomic physics and early history cosmology can be used to declare that such processes exist in the substratum (i.e. ultranatural) region.

(III) Ultralogics for the GGU-model and
the GID-model's Higher-Intelligence.

A "logic" is a set of rules for rational deduction. For the GID-model case, these rules are applied to a set of "hypotheses" and the results are called (rational) "deductions." For the GGU-model, the rules are equivalent to a substratum process used in the formation of a universe. As described next, this process directly corresponds to the "rationality" of Nature and is considered as having an intelligent agency signature. For the Complete GGU-model, these specific GGU-model rules directly correspond to the basic concept of GID-intelligence. [Additional mathematically related statements appear between the [ and ].]

We are told, from a human mental viewpoint, that "nature is rational." When we linguistically describe behavior of physical entities, we are able to predict their behavior by following certain linguistic rules - the rules for rational deduction. Look at a physics science-community textbook-derivation for the highly discussed galactic redshift. You can, as I have done, specifically trace out the step-by-step logical steps used to obtain the redshift expression. Any such derivation uses a "logic-system," which contains the "rules of inference" R and a set of mentally applied procedures, AG, the "algorithm," in order to obtain the deduced result. Recall certain applied mathematical notation. If you are given the measure, F, for the force of motion, and the mass, M, then you can calculate the acceleration, a, of a moving body as a = F/M. The force and mass can be considered as changeable. This yields an expression in "two variables," a(F,M) =F/M. Hence we have "a" applied to "F" and "M." In some areas of algebra, the symbolism is reversed to maintain a left-to-right operational notation in that F and M are first obtained and then "a" is applied. This is written as (F,M)a.

The general AG algorithm (operator) is applied to the R coupled with a set of hypotheses H, that is AG is applied to (R,H). Since the AG rules are fixed, this can be written as P(R,H) = AG((R,H)) [or ((R,H))AG in one of my articles]. This particular "redshift" physics logic-system R is part of the much larger physics science-community logic-system that may by mostly implicit in character until one explicitly argues for a prediction. For the GID and GGU-models, this entire procedure is equivalent to results obtained by successive modus ponens (i.e. the rule of detachment) deductions relative to a specially constructed "word." The R, H and AG are abstracted and mathematically modeled.

For the GID and GGU-models, the R coupled AG mental processes are equivalent to ordered modus ponens deductions applied to the hypotheses H. Hence, P(R(n),H) = AG((R(n),H)), for these models, is a logic applied to H, where the R(n) corresponds to finite sequence of "n," "If A, then B." forms and H is the "first" A in the "first" form. These are members of the language L. Indeed, any finite sequence of such forms is a "word" in L. (The notion of what constitutes a language is extended for the GID and GGU-models to include all digitized sensory human-impressions and specific unobservable behavior and entities.) The characterizing aspects displayed by P(R(n),H) are embedded into a mathematical theory. This yields the representation P(R(n),H) and, further, the mathematical object *P(*R(λ), *H) = *P(*R(λ), H) is predicted. [In the technical papers, the AG is denoted as an underlined script A. Further, mathematically the P alone is used as a representation for the collection, in ordered triple form, composed of R(n), H and value the P(R(n),H) as the R(n) and H vary.]

The "bold" notation indicates that this is an entity within the mathematical structure being employed. Note that the terminology being used is an "interpretation" for these symbolic forms. *P(*R(λ), X) is called an ultralogic, when applied to X, and it is defined on special subsets of *L, the internal subsets, and has predicted properties that can be compared to those of P via the interpreted representation P. When a single hypothesis H is considered, in this formate, it is considered as a singleton set {H}. [Note the actual formate uses the order pair concept and the reason for the * on P is relative to the single symbols representation for P. This yields the highly significant form *(P(R(n),H)) = *P(*R(n),*H.]

The logic-system approach is equivalent to a consequence operator C that takes sets of words in L and yields words in L. The operator C also has special properties but C does not display the R being employed. The set of words it uses is called a set of premises. If one takes a set of premises X ⊂ L, then the set of all deductions Y obtain is the result of this operator. This is denoted by C(X) = Y. It has been shown that, in general, each such C also determines a logic-system from which the C can be obtained. Hence, in general, the deductive results obtained by application of C are equivalent to the deductive results obtained by applying AG to a logic-system.

Originally, the logics considered for the "The Theory of Ultralogics" were of the consequence operator form and *C was called a "superlogic." I had to changed the name to "ultralogic" since the term superlogic was already in use. But, why use the prefix "super" or "ultra"? Are "mental" processes characterized by *AG really "super," or as now termed "ultra," when compared to AG? Why do I term the characteristics for *AG as those of a higher-intelligence?

Note that each member of L is directly related to a corresponding member of L. And, how words are formed using members of L is directly related to an operator defined on L. Each alphabet member of L is directly related to a member of L and the length of a word in L corresponds to the notion of length of a word in L. Thus, for our purposes, L behaves like a language. In terms of the higher-intelligence (HI) interpretation, it turns out that HI can use the *P(*R(λ), •) process on members of a language *L that are much larger than those in L, the set that represents our human language. But, our human language is contained in *L. If you restrict these particular HI activities to L, then you get P(R(n), •). So, mentally, human beings are associated with this HI "starred" process. But, basically, for this interpretation what can the HI do that human beings cannot?

Let X be an interpreted finite set of hypotheses that humans can use, and Y the (interpreted) set of all deductions. Consider a defined standard logic-system process G(R',X) = AG(R',X), where R' is the set of rules of inference and X is a set of premises to which the rules are applied. Then using the same interpreted X the HI *G deduces conclusions *Y. *Y can be the same as the deductions Y, but, although *Y contains Y, *Y is, usually, much, much larger than Y. This is the exact definition for a "stronger" process and indicates why *G represents an HI-designed action.

The reason for an increase in deductive power is that when *G is investigated it is discovered that members of *Y can be the result of applying hyperfinitely many *algorithm steps. This is the case for *P. The hyperfinite includes the finite, but, often, it is highly infinite in character. The G that humankind uses is restricted to finitely many steps. This also implies that to obtain any deduction, we can use only finitely many premises. This need not be the case for application of *G

If X is an infinite set of premises, and in science there are many of these since varying parameters are often used, then each deduction uses only a finite subset of X. But, for general *G, hyperfinitely many premises from an automatically expanded set of premises *X, which contain the X premises, and the predicted "rules of inference" *R, can be used to obtain deductions. Such an hyperfinite set can be infinite, and, in that case, it has a "size" "greater than" any infinite entity ever considered for any application of the infinite to standard physical science. Further, for the GID-model, it is predicted that, in a microsecond, the HI can deduce an infinite hyperfinite set of conclusions by application of *P(*R(λ), •) to H.

Another aspect of the HI processes is relative to the language *L. This language contains certain objects that behavior like ordinary language symbols, words, sentences and has a grammar. But, for these objects, no human being can have any direct knowledge as to what are the words, sentences or even the symbols. The HI can use hypotheses from this unknowable portion of its language and deduce meaningful members for our language L for which humans can have direct knowledge.

There are other rather unusual differences between the HI processes and what humans can do but it requires one to have a basic knowledge of the workings of the nonstandard model. Nevertheless, these intuitive notions, in my view, certainly should indicate, on a basic level, why I use the term ultralogics. BUT, now I come to the truly sensational difference between ultralogics and human mental activity.

One can characterize human mental processes in a general manner. Then it turns out that there are other HI processes called "pure" ultralogics. This type does not carry the "star" on the symbol used to identify it. They have all of the same characteristics as listed above but with a major difference. If you restrict their behavior to the language L, you do not get all of the properties being expressed by the pure ultralogic. Indeed, some described results that one would observe would appear to be unguided and chaotic from the human viewpoint. The ultralogics that model probabilistic behavior are of this type. In this case, this shows that HI has intelligently designed each finite sequence of the specific physical events, where it is often assumed that the behavior is random, AND maintains the probabilistic behavior as well.

For the GGU-model, ultralogics are also termed as intrinsic ultranatural processes, or IUN-processes. From a secular viewpoint, they can be considered as force-like processes. Using this terminology, they can be considered as producing physical-systems, and producing, staining, guiding or controlling physical-system behavior. Their relation to intelligent design by a higher-intelligence can be considered as an extraneous interpretation that is ignored as is done in quantum logic. For the GID-model, ultralogics have the same GGU-model "meanings" except their relation to intelligent design is not ignored and these IUN-processes are coupled with the intelligent agent characteristics that are considered a general signature for the intelligent design aspects. From the GID interpretation, ultralogics are designed by an HI, when applied they exhibit HI actions, and, necessarily, the structural and behavioral patterns produced are intelligently designed.

Last revision 14 FEB 2018.

Click back button, or if you retrieved this file directly from the Internet, then return to top of home page. If you retrieved this file while on my website, then return to top of home page.