The Lack of Knowledge Model

Robert A. Herrmann, Ph.D.
30 MAY 2017, Revised 12 FEB. 2020.


[[If some terminology that appears in the article is new to you, please consult this brief glossary, where definitions appear. The terms used in this article are those employed for this interpretation of the mathematical structure.]] The General Intellect Design (GID) Model is based upon the mathematical modeling of general language linguistic concepts. GID intelligence, as defined, is that displayed by application of the linguistic rules for rational thought. The Complete GGU-model is a cosmogony for the generation of universes and, hence, what constitutes GID-design must be rather general in character and should be considered as conceptual.

Each standard language element is considered as but a finite string of symbols taken from a general language L. This is modeled within the mathematical structure and the symbol L employed for the set so modeled. The set L is interpreted entirely in terms of the concepts employed in order to discuss the properties of L and the rules for rational thought as applied to L. Thus, via interpretation, L is often employed as the notation for the standard language L. (However, the actual language L is mostly not the actually symbolic forms that appear in the standard model. Mathematically, I mostly use a special coding via the informal natural numbers although as Robinson did in 1963 one might consider the symbols themselves as "abstract entities" in the model. The coded objects behave mathematically like the informal language that is coded. The first individual to use a coding to discuss formal languages was Gödel.) When this language is mathematically modeled, the "higher" language *L and "higher" thought processes are predicted. Terms such as "higher, and ultra" are employed since the language L and rules for rational thought as related to language elements have measurable properties and, hence, specific comparisons can be made.

Rational Thought Example.

First, consider any finite sequence of "If" and "then" statements of a special form, where the A, B, C, . . ., X are comprehensible phrases. Such a sequence is called a "word."

W = A, If A, then B. If B, then C. . . . then X.

The step-by-step application of the most basic form of propositional deduction yields the statements A, B, C, . . .,X in that order. For the mathematical GID-model, it is mathematically predicted that there can exist another such sequence, *W, extending W, that contains other such word forms taken from *L and they are not in L. Further, it is predicted that an "higher-intelligence" can "deduce" the entire new sequence of statements including these new predicted ones. In most cases, we can have no knowledge as to the contents of these new higher-language statements. This is the major part of the "Lack of Knowledge Model."

[[Technical Note: Two disjoint mathematical "structures" are employed in Nonstandard Analysis. It is due to a notational simplifying procedure used by me and others that such an INTERPRETATION is easily produced. But, not all textbooks on the subject of Nonstandard Analysis use this notational simplifying procedure.]] Since the mathematical modeling procedures that yield this result are rather complicated, I present, next, how nonstandard analysis methods are applied to a much better known concept. This is the set of natural numbers N, a set of entities humans have used for various purposes for a very long time.

The Nonstandard Natural Numbers.

The constructed set-theoretic model is said to be a "standard" model if the relations correspond to relations associated with what is defined as "standard" mathematics. The theories of natural, rational, real and complex numbers are examples of standard mathematical theories. Not all properties of the "informal" natural numbers are expressible in the required special language. However, the most significant are expressible. As an example, for the slightly restricted set of natural numbers N (for this presentation not including 0), consider that for each x ∈ N there exists a y ∈ N such that x < y. This is formally symbolized as

(I) ∀x((x ∈ N) → (∃y ((y ∈ N)∧ (x < y)))).

Using such linguistic constructions, a formal theory of natural numbers is embedded into a "nonstandard structure," as it might be termed, and such statements are transformed into new statements using a new language. The new statement obtained is

(II) ∀x((x ∈ *N) → (∃y ((y ∈ *N)∧ (x *< y)))).

In order to correspond to the original Robinson notation, the most recent approach to nonstandard analysis, the superstructure approach, often uses for the standard number systems a notation convention. The original set N is, technically, not a subset of *N. But the set that is (σN) behaves exactly like N. By notation convention, one often sees, especially in my writings, the expression N ⊂ *N. Of course, those that have actually worked in nonstandard analysis know about this notational convention for these number systems. Furthermore, the original informal < is not the actual object used in the mathematical structure employed. Once again, relative to such "formal" statements, the standard relation to which it corresponds behaves in the same specially expressed manner as the informal <. That is the embedded < for N is but a restriction of *<, when its properties are expressed in the proper form such as stated in (II). Again, the notational convention employs the symbol < in two contexts. One will often find that the notation *< is not employed and only < appears. Of course, one must know to what set this < relation applies, either *N or its restriction to N. The same notational convention holds for the basic number system operations. But, why is the term "nonstandard" employed?

To comprehend how these set-theoretic object behave with respect to each other, six and maybe seven different technical mathematical languages are employed. Then various features of these languages are consistently interpreted using GID-model and Complete GGU-model terminology. For example, the overall language used to discuss all of the comparative properties is the metalanguage. [[Definition is below.]] It turns out that the set *N does not have all of the same properties as the set N. But this cannot be determined by only investigating the properties of *N as expressed by *-transferable statements such as (I) and (II).

One of the major properties for N is that < is a well ordering. That is, if given any nonempty A ⊂ N, then there exists a first element n ∈ A, which means that, for each y ∈ A, n ≤ y. This is not the case for *N and <. Indeed, for N ⊂ *N, it is first shown that N ≠ *N. Then the set of members of *N that are not members of N is denoted by *N - N = N. It turns out that N does not have a first element. Note that in standard mathematics the ordering < of the rational numbers is not a well ordering, but, when restricted to N, it is a well ordering.

Within the nonstandard structure are members of *N, special subsets of *N and relations that behave exactly like the natural numbers N. However, as illustrated, there are other properties of *N that are distinct from those of the N. This is the usual result. There are additional properties produced via a nonstandard model.

I point out, although it is often not stated, that in mathematical logic there is a conceptual difference between symbols used to represent mathematical entities within a specific model and the abstract entities themselves. The subject studies various aspects of rational linguistic thought. Originally, it was applied only to mathematics itself. Today, it is a subject that studies all forms of rational thought relative to any linguistic form. The metamathematics employs the natural numbers and other "mathematical" devices for the analysis. For the set of mathematical entities, it is understood and almost never mentioned, that the same symbol may be used in two or more contexts and one is suppose to know the differences. For example, "informally" a set that has the properties of the natural numbers is used to construct the mathematical structure. And the "n" notation is used. However, one of the objects "in the structure" is a set that has natural number properties. For that set, the same symbol "n" is used. However, since strings of such symbols are analyzed, then, in some textbooks in mathematical logic, symbolic notations such as n' are used as special "names" for the natural numbers being modeled. (The symbol "n" can also be in the general language being modeled.)

Nonstandard analysis is mostly concerned with "model theory" and the analysis of set-theoretic members of the model. Linguistically displayed informal and even formal statements such as (I) and (II) are exterior to the model itself. However, for the GID-model, linguistic word-forms are being analyzed. In 1963, Robinson applied his nonstandard analysis to the semantics of formal first-order languages. In 1978, Herrmann was the first to apply nonstandard analysis to the general syntax of general languages. In various cases, care must be taken relative to the symbolism used for the members of the model and the word-forms they represent. Mostly, however, this is avoided by the use of a significant coding procedure.

The coded objects have the same behavior as the linguistic objects they represent, but are not the same objects. And, it is important to understand that the GID-model and Complete GGU-model are interpretations for the mathematical structure and the interpretations are distinct from the terms used to technically discuss the mathematical structure.

Do Members of *L - L Have Meaning for Us?

Statements between the [[ and ]] are additional technical statements. The notation I use to model general language linguistic properties is especially designed to retain the original "meaning." For example, a string of symbols such as "energy" is coded and represented by an "equivalence class [f]." (Equivalence classes are a major concept within all of modern mathematics.) However, such an equivalence class may also be denoted as energy. In other cases, I may write energy = b. Then [f] is denoted by b. I mention that how we combine actual word-elements to form other word-elements is modeled by an operator defined on the set of equivalence classes L obtained from the language L. These classes code the way language-elements are combined to produce word-forms.

[[Intuitively, the word "the<:-)" is produced by the combinations of "t h e <:-)" such as this single symbol form or as "th e<:-)", "t he<:-), "th he<:-)" and "the<:-)" taken from L. The equivalence class for this word contains partial sequences of natural numbers, such as f(4) = i(t), f(3) = i(h), f(2) = i(e), f(1) = i(<:-)); g(3) = i(th), g(2) =i(e), g(1) = i(<:-)), k(1) = i(the<:-)), . . . , where "i" is the natural number coding. If one uses the notion of the empty word, then the entire collection of these classes forms the algebraic entity termed as a "monoid."]]

Relative to statements characterizing a continuous energy spectrum and not expressing this notion in a form such as the no spacing form used in Koine Greek, a "spacing symbol |||" is employed. Thus, consider the informal set of statements S = {An|||entity|||has|||energy|||100+1/n'. |n ∈ N}. Notice that this uses the symbol "n," for members of the mathematical set of natural numbers N and the statements use the symbolically converted prime notation. As indicated above, in the GID-model, I have retained the informal "forms" but have altered the font to indicate that this is the coded mathematical structure representation. Thus, upon coding, I might write the corresponding set that is an entity in the nonstandard structure as S = {An|||entity|||has|||energy|||100+1/n'. |n ∈ N}. [[The N is a basic member of standard superstructure and S is actually a set of equivalence classes that exists in the standard superstructure.]]

We might conceptually comprehend members of the set S I employ in their entirety, but, being finite creatures, we can only actually perceive or, say comprehend the increasing large finite subsets of S. (However, the least of the "infinities" can now be "mentally" perceived [1].) When the set S is viewed as S in the nonstandard model, the model predicts that there will be coded statements in *L that have in *L but not in L (i.e. in *L - L) entities that behave like symbols for the n', where these "symbols" represent members of N, the Robinson infinite natural numbers. But, they are not obtained from the original coding. Although Greek symbols are members of L, such as λ, these are used to "represent" some members of the "higher-language" since the interpretation is that of the "metalanguage" used. (See definition below.) As mathematical entities, "numbers" such as 1/λ represent (Robinson) infinitesimals. (The set of infinitesimals is a major aspect of Robinson's solution to the 300 year old problem of Newton and Leibniz.)

The term "represents" does not appear in most textbooks that may not be in mathematical logic and abstract model theory. You will read, "1/λ is an infinitesimal." Mathematics is about relations and, hence, the behavior of "abstract entities." Indeed, the expressions "1 < 2" and "I < II" denote properties about two specific natural numbers, but the symbols used come from two distinct languages. These symbolic facts are almost never discussed within the mathematical literature but are supposed to be "intuitively understood." The actual formal language used for this nonstandard structure is termed "a formal first-order language with constants." It is the "constants" that name (i.e denote) specific entities such as "a ∈ A." You often see "There exists an x ∈ A such that P(x). So, there is a b ∈ A such that P(b). But then b etc." It is very important to understand that throughout mathematics one jumps back-and-forth between the "abstract" mathematical structure and interpretations that employ possibly different languages. AND, often the same terms are used to describe both the structure entities and the interpreted entities.

The above "energy" can be directly "interpreted" as stating that "An entity has energy 100+1/λ." The "standard part" operator applied to the number 100+1/λ = 100. Whether or not within our universe such an entity actually exists is not the point. It is the statement that is predicted. I use such predicted statements to define substratum entities termed as "ultra-propertons."

I point out that, as now "informally" decoded, the sentence "An entity has energy 100 + 1/λ." has meaning if you have knowledge as to the algebraic properties of the infinitesimals. Further, for the (coded) entity An|||entity|||has|||energy|||100+1/λ ∈ *L - L, we don't actually know what interpreted "symbol" might correspond to λ. The basic foundations for the Complete GGU-model are modeled using features of human behavior. In particular, the predicted higher-intelligence attributes are based upon specific observed human behavior that measures our intelligence.

The set of all finite subsets of S is denoted by F(S). The transformed set of "hyperfinite" subsets, *(F(S)), is significant. Being a mathematical model, statements predicted by the model are exceptionally rational in character. The fixed interpretation employed states that the higher-intelligence being modeled, via corresponding mental attributes, *comprehends members of *(F(S)). But members of *(F(S)) are "hyperfinite." This includes all denoted entities for each x ∈ *N such that x < λ. This includes each member of N. This set is denoted here by the interval notation [1,λ] and N ⊂ [1,λ].

Mathematicians actually do "think" of different levels of infinite "quantities," where, for example, Wielder uses the term "size" as meaning a comparative measure for this quantity. (The technical term is "cardinality.") Relative to such quantity statements, one can state that the "size" of this set implies that there is a vast infinite "quantity" of statements that can be *comprehended and that we cannot comprehend. This statement can actually be greatly enhanced when one discovers that there are no known general language word-forms that we can use to describe certain aspects of the set [1,λ]. And, that we can describe all similar features for a similar but finite interval. Thus, via this interpretation, the higher-language *L can have an immense "quantity" of statements that could only be "comprehendible" by a higher-intelligence. Theologically, notice that aspects of such an hyper-language can correspond to that of a Third Heaven language.

Although there is an interplay between different technical languages, it is the interpretation that yields the Complete GGU-model. I repeat this rational conclusion. From the interpretation viewpoint, the above discussion gives an example, among many others, that members of the higher-language *L - L exist that are predicted to be *comprehensible by a higher-intelligence, but they are not comprehensible to any finite entity within our universe. For us, such predicted results form the lack of knowledge model. For this reason, the Complete GGU-model will not be further detailed.

Of considerable significance is that the Complete GGU-model employs the instruction-entities, and predicts physical-like as well as ultranatural events. Descriptions for these entities and events require members of *L - L and, with but a few exceptions, such "descriptions" cannot be comprehended by any entity within our universe.

From the glossary on this website:

METALANGUAGE. In logic and linguistics, a metalanguage is a language used to describe another language, often called the object language. For the GID and GGU-models, the metalanguage is to be understood as all of the language elements employed to annalize the language L. As mentioned in the book, "The Theory of Ultralogics" such a language uses members of L but it is to be considered as written in a different color or font. This is not generally done in my publications, BUT it is assumed to be understood by the reader. Indeed, this entire book, can be considered expressed in the metalanguage. Also note that certain conventions are followed. For example, it may be written as N ⊂ *N, where standard N may be a member of L. But this is a convention, where the actual notation is metalanguage notation σN ⊂ *N. The set σN has, within the nonstandard model, all the same mathematically represented properties as set-theoretically defined N. The GID and GGU-model interpretations are fixed interpretations of the metalanguage analysis.

[1] Examples of How to Imagine the Infinite.

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