What few know about the finite.
- It’s very unfortunate that few individuals, including mathematicians, throughout the world have any knowledge as to some highly significant results in mathematics that occurred 44 years ago. I present what follows as “fact. It is “true” under Aristotle's correspondence theory of “truth.” This fact means that the results can all be presented as acceptably proved theorems. This is what “fact” means in mathematics. It need not correspond to any other idea. These results remain fact until an uncorrectable error is found in the proof of a theorem used.When the term non-finite - “infinite” - is considered what is mostly perceived visually is the potential infinite. This is, a type of not completed “moving finite.” There are those that state the we cannot “imagine” the non-potential infinite. It may be a surprise to most but I can mentally image the non-potential infinite. It, of course, helps to have an eidetic memory. I described how I do this in a place in one of my website articles listed below. But that’s not the purpose for this posting.Mathematics as used today by the mathematician is not what it was just 44 years ago. Unfortunately, you will not find the actual facts taught in our schools. This includes undergraduate; even graduate schools and even within graduate school math. departments. A hint as to what has actually occurred was in 1922-23 with the famous Skolem paradox. It was not until 1969 that this was fully explained.The facts are that the most basic characterizations for the finite and, hence, “mental concepts” that don’t satisfy these characteristics are language dependent. This even applies to the very explicit mathematics definition.How finite is characterized in one standard intuitive scientific language does not satisfy the characteristics in another language and there is no contradiction, no paradox. How many mathematics know this? How many can “prove it.”The GGU-model is based upon the finite, not the non-finite. Yet, I constantly use the term “infinite” (not finite) to discuss its findings. I have previously stated that the terms I use cannot be fully understood using any of the prior to 1969 mathematics. Of course, one can learn the new stuff. After all I have a Ph.D. in this sort of stuff and I taught myself all the necessary material. (My graduate school “trusted” me.) It is true that to apply the methods of nonstandard analysis to various areas one needs to have the necessary knowledge from those areas. I’ve applied it to six distinct areas and initiated its use to model theological notions. Further, prior to the GGU-model, no standard mathematical approach solved the Wheeler problem nor verified the rationality of many Biblical statements. These facts alone should lead more to learn the necessary techniques.I simply mention, again, that I use five different languages to discuss five different views. The mathematics languages one sees and to which the vast majority of individuals are exposed today are the intuitive and standard languages. Your textbook’s basic definitions are in terms of this language. If you only have these notions as your guide, then your guide is leading you astray. Modern (abstract) model theory shows that there are other languages that these standard languages determine that have special properties.I use here only one example for the standard, the internal and meta-languages. Each mathematically characterized standard textbook statement for the finite yields another statement written in a different language called the “internal” language. Whatever holds mathematically in one holds in the other. BUT if comparisons are made, this is not so from the view of a larger language used to analysis. This is the language used in mathematical model theory, the meta-language.It’s only the meta-language that gives the actual “total facts” that these other statements yield “language restricted ‘truth.’ ” Indeed, even two different and distinct levels of “truth” were shown to exist in a 1963 paper by Abraham Robinson. Of course, few know this.More to the point when one looks at the standard characteristics for the finite as transferred to the internal language and as understood using the internal language, say by the internal people, then there are the same “true” statement. BUT, they are not the same when viewed using the more perceptive meta-language. Statements about the natural numbers N transform into internal statements about the object *N. These include those that characterize the finite. But, the meta-language used in model theory and logic shows that the internal people only know about the object *N and “certain” subsets using their language. The standard counting numbers N that we use throughout our daily lives is not part of their language. The internal people use *N and to them there is no difference in its characteristics. Although N doesn’t exist for them and they do not need it for their science, which has the same properties as the standard mathematically represented standard science. (As shown on my website, they would actually have difficult time communication their thought to the standard people.)But, what happens with that “easily” understood notion of the finite? An interval of natural numbers say, 1 to 10 inclusive is our basic notion for a finite set. A counting set. We can write statements about putting such intervals together, and the like. For each member of N, these counting sets exist. We use them all the time. That is, take any n in N, then the interval from 1 to n exists. I can write it down or at least conceive of it mentally. The same things holds for the internal people’s finite, where they can take an m in *N and there is the interval they understand the varies from 1 to m. These are all “finite” from the characterizing statements understood by the interval people. They are the exact same statements the standard people use but they are written in an internal language. But what does the meta-language and deduction reveal?The are many members of *N, say p, such that when the meta-language describes the interval from 1 to p it contains every member of N. But N, from this overview is not finite, as far as the standard everyday view given by the standard scientist. But, from the viewpoint of the internal people the interval from 1 to p is behaves like a finite object. The meta-language states that although it behaves this way for them this is not generally true since it contains a non-finite set and *N matches the meta-language characteristic for the non-finite. Thus, what is even “easily” characterized as finite is language dependent. Thus, what is non-finite is also language dependent.In the meta-language, a new term is introduced. The term hyperfinite is used for an internal interval like 1 to p in *N. It has the same properties as the finite when expressed in the internal language. It includes the standard finite and the non-finite meta-language description as well. The GGU-model finite world is embedded into the nonstandard model and all aspects are restated in terms of the hyperfinite. I have just completed this for the GGU-model for foreknowledge.This is what leads to a hyperfinite deduction, when basic human deduction is modeled by the finite. But, from the meta-world, this is a process of deducing infinitely many conclusions over a small finite time period, a characterization for a higher-intelligence. I have analyzed the “infinite” notion used for the hyperfinite and other sets and it has interesting properties. Remember, that one dose not consider such a notion as a form of counting. By-the-way, I won’t get into a discussion as to what the term “science,” in general, entails. All I know is that whatever it is, the GGU-model predicts all there was, all there is and all there every will be for our physical-universe.As to mentally perceiving the non-potentially finite, I give what satisfies me in this case just below the horizontal line in the section “A Semi-Technical Description” in the article on GGU-model processesDr. Bob