Science Declares Our Universe IS Intelligently Designed
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."
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 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."
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." For example, any infinite (standard) set in the model is also an external set. 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. Indeed, every standard infinite set X is a subset of a hyperfinite set Y. 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 GGU-model mathematical structure itself is an example although it is a set. Its name would be a metalanguage term.
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.
As a last example, for the GGU-model, one can write a metatheorem, that gives relations between these three type of objects. The (external) m(0) is a subset of the internal real number *R, the standard real numbers R and external m(0) only have 0 in common, and m(0) is not a subset of a hyperfinite Y that contains the real numbers R. The term "subset" as used in the last statement 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.