During the period leading to the establishment of QM, there was a mathematical thrust to complete the foundation of Set Theory as developed by Cantor in the latter part of the 19th century. One of the key axioms of this theory was the Axiom of Choice (AOC or Selection Choice Axiom) as identified by Zermelo in 1904. After many years of debate on the need for such an axiom, similar to the debate on Euclid's 5th postulate, Paul Cohen in 1963 finally established that the AOC is itself a matter of choice, a property of a set or subset. As he stated [2], "It all depends on whether one's attitudes or the applications one desires call for Cantorian or non-Cantorian set theory." The resolution of the matter was then indeed similar to the resolution of Euclid's 5th postulate. At that point however, and like the situation for Euclid's postulate, someone had to have an example of what a non-separable set could be in the physical world in order to go further in establishing the theory of such sets. Unlike Riemann who had plenty physical examples of non-Euclidean geometry in front of him, nobody from the physical side appears to have brought up physical examples of non-Cantorian sets to mathematicians, and these examples didn't seem to be in everyday life to see, contrary to the 19th century ones.

What instead appears to have happened was that, due to the "practical efficiency" of the QM formalism, the physicists never told the mathematicians: Hey! We have a problem here that you don't know anything about, please help! To the contrary, the reverse was pushed: Mathematicians were to lead the way, as Ernst Mach pushed earlier, and the inspiration was to come from the ivory tower of Mathematics. (Details on this background can be found at the start of phase 1 as well as the last section of phase 2 of the study.) The problem with such an approach may be that by using a statistical view we may have discarded a number of properties of unseparable sets. One of such properties appears to be "creativity," as Goedel's theorem may be telling us that no deterministic machine based on a Cantorian set can be creative, can bring more "objects" than it initially has in its setup (see below). QM then would not be able to predict that a space manifold is created by the unseparable sets of elements making our reality, nor would be able to identify the various modes of existence for that space manifold. Right now QM simply assumes space as an arena completely divorced from its content. Einstein's theory on the other hand sees an influence of the content on space, but misses the creation of that space from its content. Einstein may have felt that QM was too "incomplete" to help there.

Within the above line the term "incomplete" as used by Einstein appears to be incorrect. The formalism of QM comes from a classical base for the understanding of our reality. Since classical reality with its separable character comes from properties of unseparable sets, QM cannot get to the origin of such character, and thus is an approach which must be contained in another more comprehensive one in the sense that Newton's Mechanics is contained in Einstein's Mechanics, and this not through an extension, but rather through a different outlook of reality. We could not get from Newton Mechanics to Einstein's via an extension. The same appears to be the case with the consideration of a new Quantum Mechanics. But in order to get to that new mechanics we need first to establish some of the physical properties that it must display. I provide in phase 1 a rough layout of the physical goals for the unborn mathematics of monadic sets, in an attempt at following the path of Galileo's work, which pointed to the goals of the unborn calculus. The development of the mathematical details will require a combination of mathematical skill, imagination and insight based on the physical knowledge of where we want our theory to end up. Einstein had Riemannian theory to use, we seem not to have an equivalent here.

DEFINITION: A number can be seen as the result of an axiomatic operation (see above) on the elements of the original set of reals. The sequences of possible operations resulting from the finite set of axioms forming an infinite set, a given number on the real line identifies an infinite set of numbers with the same name, each element coming from a sequence of operations on the reals. But once the operations are made the result cannot be distinguished from another result by a different operation since no specific name or set of aliases can be assigned to that element. It is said to be unseparable from the other result.

The importance of the above definition is that before it was enunciated there was an unstated operation in the axioms on reals which kept them separable: Identifying the result of an operation with one of the elements of the original real line. Replacing the infinite unseparable subsets of the "ultrareal" set with the numbers of the original real line will be called the collapse of the ultrareal set into the real line.

I will surmise that this notion of collapse may be connected in some way to the present QM wave function collapse postulate, but such a conjecture will be able to be handled further only when (or if) an adequate algebra on unseparable sets is available establishing the creation of the classical separated world from the unseparable quantum world through the bounded nuclear fields existence (see phase 1 of the study).

Here are a few comments trying to clarify the definition above, in the best way I can do at this point:
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It is understood that if elements of a given set have (or could have in the platonic sense for most elements of the real line) each a specific "name" (taken here as an extension of the common meaning to a specific set of aliases and may include a description under the form of a property definition, such as a convergent series, or an infinite sequence of digits) then they can be mapped to each other, i.e. a one to one mapping can be done, a property which is equivalent to stating the applicability of the AOC to the set under consideration (thus making it separable) as established in the literature decades ago through the demonstration of the well-ordering theorem [1].

It is also understood that the elements of the unseparable sets cannot be identified separately through the sequence of operations producing them as the name of the result contains no trace of the operations, i. e. no process "memory" is built in the operations defined by the axioms on the reals, and this in contrast with operations such as cartesian products which do leave such a trace by combining the names of the elements used in the product.

Lastly, numbers are understood as entities, elements of sets, they are not the names (in the common meaning of the term), signs or definitions representing these elements. They exist in a platonic sense. In unseparable sets the distinction is paramount since elements may not have a unique name, unlike separable sets where the elements and their names, known or unknown, are basically confounded. Elements may exist without their names known, this does not mean they have no name. Mathematics so far has reflected this synonymity between elements and names by not developing a theory of names for elements of sets, or not emphasizing how elements are distinguished or their names identified. Some may say that elements cannot be chosen (and thus the AOC is denied) because we cannot get to their names. Such a view has nothing to do with unseparable sets, it is merely a naming problem within separable sets which needs to be investigated separately.
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If M (x) = 2^R for all x elements of R, the "ultrareal" set replacing the real line is an algebraic system G = (R, 2^R, M). The traditional real operations are thus supplemented with the ultrareal property of unseparability.

Without AOC the Cantorian cardinality notion for sets cannot be established, and there is no way to say in general whether an infinite set is greater than another infinite set. AOC can only apply to the reals R which is separable within G and the subset 2^R of G (which is also separable since the cartesian product gives a specific name to its results). Then a mapping can be done for these restricted subsets where AOC applies. But since the algebraic system G contains both these sets, it must be greater than the set of reals from the higher cardinality of 2^R. This then should give a lemma which is the key to the following conjecture:

LEMMA: According to the generative definition above, the unseparable set of ultrareal numbers contains more numbers than the original real line.

Finally, I deduce a rephrasing of Goedel's Theorem for the reals, showing it is a consequence of the ordinary reals being separable by the definition of unseparability given earlier:

CONJECTURE: According to the above definition and lemma any axiomatics on the numbers of the real line is bound to be incomplete as there will always be an axiom missing to cover all the relations between the elements of the larger unseparable set making up the ultrareal numbers resulting from the axiomatics.

By their mere existence the ultrareals generate (or "create") new axioms for the original set of reals R. The physical meaning of Goedel's Theorem is then to identify the creative ability of unseparable sets. A set of axioms is always incomplete to cover an infinite classical object because this object is created from unseparable sets which are the true makeup of the quantum. We "collapse" ("reduce") these objects into separable ones and then wonder why we cannot describe them with our "finitistic" methods.

The above is only an attempt at a formal picture of an Element of Reality (EOR) or Monad. The notion of "name" for an element of a set will have to be expanded and made more precise, a notion which needed no such expansion when only separable sets were considered. The conjecture above will have to be demonstrated as true by adapting the Goedel numbers to axiomatic operations without collapse.

Next will have to be a formal method describing how such "directions" imply the dimensionality of the monadic computational process. I suggest here a tentative approach through directed graphs over quaternions. In linear algebra (by definition a finite-dimensional vector space over a field) it is known [3] that quaternions are the largest algebra possible with the property of associativity. Octonions (Cayley algebras) and Lie algebras have a higher dimensionality but are not linear algebras because they are not associative. Physically, I would take the property of associativity as a necessity for monadic relations:

(Au)v = u(Av) = A(uv)     for A being an element of a field

and this because u and v must be undistinguishable. (I call them u and v only so I can speak about them.)

Then, when using axiomatic operations for quaternions, each real number may be the collapse not of an unseparable set resulting from operations on reals but from operations on quaternions. We end up then with a 3 times 3D space for each element of a 3D reals space, with the 3D reals being in effect the collapse of an unseparable set of "hyperreals" for each real.

Now the question comes about how consistent parallel operations (computations) on sets of such hyperreals could restrict them to a grouping between bounded manifold evolutions (nuclear field) and unbounded ones (electromagnetic field) , with the generation of a common space manifold between the various content manifolds, all as physically discussed in phase 1 of the study. A well-known work by W. V. Quine [4] starts to establish non-Cantorian sets. There I do find what he calls an "extended theory of classes" which do give me some interesting points. It appears that I will have to treat the unseparable numbers as classes and then manipulate these classes via strictly logical operations. I am not versed at all in such methods, but I do see that the "manifolds" I am talking about may be themselves classes, and their properties such as boundedness and connectivity may be able to be identified from the algebra on the elements of the classes. However, looking through his book I get the feeling that Quine does not develop the theory very much. He notes that it would remain to address how to handle algebraic and geometrical concepts within the set of extended classes. I need more than that from a mathematician on the matter. It seems obvious that Quine did not have an incentive in producing further developments. It is interesting though to find a similar way about producing non-Cantorian sets to the one I used for the reals, except he gets finite classes instead of the infinite ones I propose... and so he definitely appears incomplete (although I cannot be certain, not being a specialist in this area). I do get the feel he was missing an example of an infinite classes system...

If the question above has a meaningful answer a badly needed check of the formalism will be required at that point, and this through connecting it to the present statistical approach of QM, for example via the Feynman path integral. To reach this goal, at least two simplifications will need to be done (1) take the speed of the monadic sets computation as infinite when concerning the common space manifold maintenance while leaving it as finite for its content, (2) reduce bounded (nuclear) manifolds into unresolved punctual "charged spinning matter" as in the times of Schroedinger. A single "particle" prepared at the start of the "path" integration will then represent an infinite unseparable set of EORs selected by the experimental set-up. When this set is collapsed at a later time after an unknown and unseparable evolution, all the computations that did not match the requirements dictated by the common space manifold maintenance will not appear, having ended sometimes before. The path integral in effect automatically retains a statistics on only the "successful" computations, the computations that can contribute to the future of the space manifold. Such a set of adaptive computations then is the basis for the quantum path integral formalism. In turn this formalism leads to the Principle of Least Action and the other variational principles of Classical Mechanics. To give a better feel on how I envision making the formal connection between the future and present QM formalisms I provide at the end of phase 1 a description of various experimental set-ups analyzed through the monadic relations picture.

When it comes to the connection with Relativity theory, identical conclusions must be obtained by the new formalism for spacetime warps due to the presence of content, except for one situation. Since content manifolds transform into each other, part of the common space manifold must also be found to transform into content manifolds upon accumulation of content beyond certain levels (phase 1 expands on the reasons behind this assertion), thereby resulting in a continuity process between space and matter requiring an equation in the line of the equations governing fluid dynamics with sources and sinks as known in Classical Mechanics. Such a continuity equation would then at last complete General Relativity.

Accomplishing even part of the above would be a serious advance in our ability to handle Reality. And needless to say, if Mathematics now comes to the rescue of Physics, Mathematics can only help itself going further in Set Theory.

Such a semi-formal approach could be then generalized to any structure made out of molecules which could exist in more than one conformational state while containing semi-free electrons, as phase 2 suggests. Space in areas containing such molecular arrangements would be lined with "inertial" space manifolds oriented differently, forcing a new dynamics on the content of normal space, outside the one coming from gravitational forces. Such a dynamics would be observed by present biochemists as "tracks" or "accumulation points/patterns" for proteins and other molecules, and would be at the physical origin of embryo initial oocyte compartmentalization as well as subsequent cell divisions. Right now such evolutions are seen as governed by classical statistical dynamics following the 1950s ideas of Alan Turing. Yet as phase 2 of the study points out, the environment in cells is far from allowing a free statistical dynamics, and such a dynamics anyways cannot explain the precise choreography of what is really going on in cells as well as in-between cells.

I am only trying to speed up the discovery process by connecting already known things which were not connected before, and tying them up with experimental knowledge, present or near future. This is the scientific value of my work, and ought to be known because of that, regardless of the informal ways I am taking, not having the means to do otherwise. Of course my work appears incomplete now as the scientific world has been accustomed to formal approaches for so long. (My text both at the start and end of the thesis deals with this issue.) But this is by no means the first of that sort in history...

Galileo used Euclid's geometry to demonstrate a few properties of moving objects in a gravitational field. Several of his "demonstrations" were extremely complex taking several pages and prior properties themselves taking several pages (the "demonstrations" were still haphazard, witness his finding that a rope hung by two nails described a parabola...), while the calculus equivalent took just a few lines. Descartes was right in criticizing that work, as the "formal" part was mostly powder in the eye of the reader, in retrospect they were there only to point future mathematicians to what the results of calculus should be. The infinitesimals were an animal totally different from Euclidean concepts. Unseparable sets are of that kind versus separable sets.