                              built on purely mathematical terms, in this case based on the "non-standard" analysis framework, alternatively to the "standard" analysis framework (whereby the term "standard" is "just" due to a suitable conditioning in early childhood); the field of non-standard (or Mickey Mouse) numbers (where the field of real numbers is a subset of it) has same cardinality (as defined by Cantor), fulfills same (mathematical) Archimedian principle, but is just non-ordered (according to its mathematical definition), as the field of real numbers. If a (mathematical) Mickey Mouse universe enables a consistent quantum and gravitation theory it's an adequate model, which explains phenomena from both areas, simultaneously. They are images in our mind.

"Something, which is named "hyper-real", is perceived as an utopic status, which is only applicable in science fiction stories. But it's just a mathematical definition of something, which is very similar to "real". A "hyper-real" universe corresponds to an Archimedean, non-ordered field, while a "real" universe corresponds to a Archimedean ordered field. Both fields have same cardinality, which is Cantor's (mathematical) definition to qualify and quantify the different kinds of infinity, e.g. integers, rational numbers, real numbers.

Schrödinger E., "Science and Humanism": 7. The intricacy of the continuum: ... "It seems simple to us, because the idea of the continuum seems simple to us. We have somehow lost sight of the difficulties it implies. That is due to a suitable conditioning in early childhood. Such an idea as 'all the numbers between 0 and 1' or 'all the numbers between 1 and 2' has become quite familar to us. We just think of them geometrically as the distance of any point like P and Q from 0. ... Among the points P and Q there is also the square(2). We are told that such a number as square(2) worries Pythagoras and his school almost to exhaustion. ...There worry was highly creditable. ... The idea of a continuous range, so familar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is really accessible to us." ...

Basically people take as a real "particle entity", what's defined as real number, instead of what's alternatively possible as hyper-real number. This is just due to the fact, that Leibniz (monads, differentials) lost the marketing fight against Newton (particle) concerning the branding and related perception of "differential calculus". This led to so-called Standard Analysis and to the perception, that a "real / physical" particle (required as test particle in mathematical physics) is "identical" with a real number: then, finally perception became "reality" in human (western) mind.

(PoP) Poluyan P. V., "Non-Standard Analysis of Non-classical Motion; do the hyperreal numbers exist in the Quantum-relative universe?"

http://www.oocities.org/quantum_math_poluyan/hy_nu/hy-nu.htm

1. already in standard models "particles" are "transcendental objects", which are mathematically modeled by real numbers. As an option to this we propose non-standard numbers alternatively, i.e. the monads (= ideal points). The common denominator between both arithmetic models is the Archimedean principle and the same cardinality. The Archimedean principle enables a "measurement" of the distance / "lenght" between zero and any real number on the x-axis by a multiple (integer number) of a given finite measure unit ("!).

Let a=r+i a finite non-standard number with r real and i infinitely small. Then i can be the differential of "something"

2. There was an initial hyper-real „particle“ in the neighborhood of the big bang, the „inflation“.

3. The Weyl curvature of that inflaton was infinitely small, but not zero in the neighborhood of the big bang, i.e. Weyl(i)=i

4. The Ricci tensor, measuring the size of the volume reduction at that point in space-time (i.e. at the point i) was infinitely large, i.e. Ricci(i)=1/i.

5. Taking 3. and 4. as initial value conditions for Einstein's gravity PDE, where the PDE systems are defined as variational equations system, i.e. in weak form only, and formulated in the k-form calculus (coordination system independent)

a. to match to current quantum theory mathematical concepts

b. are embedded in a hyperbolic geometry following R. Penrose´s arguments

c. to link non-standard analysis with Dirac function/Distribution theory.

If there is a chance for a well posted problem a duality/symmetry between Weyl and Ricci could be constructed by the additional condition, that the k-form analogue of Weyl(1/i)=1/i and the k-form analogue of Ricci(1/i)=i. This would define a periodically swinging back and forth (between the infinitely small and the infinitely large) quantum gravity model, where the Weyl and the Ricci tensors are changing their roles.

Some related quotes

1. Well-ordering theorem from E. Zermelo (1904): every set  can be well-ordered

2. Theorem of G. Cantor: "for every set L the cardinal number of its power set is richer"

Theorem: the cardinal number of the power set of the natural numbers is the same as the cardinality number of the real numbers

3. Cantor's continuum hypothesis: "every sub set of the real numbers has either the cardinality of the natural numbers or the cardinalidy of the real numbers"

The Zermelo "axiom of choise" (i.e. every set of non-empty sets has a function of choise) was key/necessarily required to answer the CH of G. Cantor positively.

4. L. Kronecker, "God made integers, all else is the work of man"

5. Hoffmann D. W., Die Gödel'schen Unvollständigkeitssätze, Springer Spektrum

For the related Gödel's incompleteness theorems we refer to

Ebbinghaus H.-D., Flum J, Thomas, Mathematical Logic

6. Weyl H., "The Continuum, a critical examination of the foundation of analysis"

H. Weyl: "Preface: It is not the purpose of this work to cover the "firm rock" on which the house of analysis is founded with a fake wooden structure of formalism - a structure which can fool the reader and, ultimately, the author into believing that it is the true foundation. Rather, I shall Show that this house is to a large degree built on sand. I believe that I can replace this shifting foundation with pillars of enduring strength. They will not, however, support everything which today is generally considered to be securely grounded. I give up the rest, since I see no other possibility.

At the center of my reflections stands the conceptual problem posed by the continuum - a problem which ought to bear the name of Pythagoras and which we currently attempt to solve by means of the arithmetical theory of irrational numbers. ..... Concerning the epistermological side of logic, I agree with the conceptions which underlie Husserl. .... Our examination of the continuum problem contributes to critical epistemology's investigations into relations between what is immediately (intuitively) given and the formal (mathematical) concepts through which we seek to construct the given in geometry and physics.  ...(chapter I, concluding remarks): The concept of function has two historical roots. first, this concept was suggested by the "natural dependencies" which prevail in the material world - the dependencies which consist, on the one Hand side, in the fact that conditions and states of real things are variable over time, the paradigmatic independent variable, on the other hand, in the causial connections between action and consequences. ....With the help of a tradition bound up with that complex of notions which even today enjoys absolute primacy in mathematics and which is connected above all with the names of Dedekind and Cantor, I have discovered, traversed, and here set forth my own way out of this circle. Only after having done so did I become acquainted with the ideas of Frege and Russell which point out in exactly the same direction. ....chapter II, §6, ... If the time-points with their relations of "earlier" and "equal" can really furnish the foundation of a pure theory of time, then the intuition of time must suffice to determine whether this correspondence between time-points and real numbers holds or not. If it does not hold, then we should attempt to expand or modifiy our principles of definition in such a way that the desired agreement comes about. ....In confronting these questions we cannot avoid the concept of set (or sequence), no matter how we twist and turn; and the scope of this concept depends on the principles of definition! Now, I think that everything we are demanding here is obvious nonsense: to these questions, the intuition of time provides no answer - just as a man makes no reply to questions which clearly are addressed to him by mistake and, therefore, when addressed to him are unintelligible. ....

Chapter I, §4, No one can desribe an infinite set other than by indicating properties which are characteristic of the elements of the set. And no one can establish a correspondence among infinitely many things without indicating a rule, i.e. a relation, which connects the corresponding objects with one another. The Notion that infinite set is a "gathering" brought together by infinitely many individual arbitrary acts of selection, assembled and then surveyed as a whole by consciousness, is nonsensical; "inexhaustibility" is essential to the infinite...But as things now stand we must point out that, in spite of Dedekind, Cantor, and Weierstrass, the great task which has been facing us since the Pythagorean discovery of the irrationals remains today as unfinished as ever; that is, the continuity given to us immediately by intuition (in the flow of time and motion) has yet to be grasped mathematically as a totality of discrete "stages" in accordance with that part of its content which can be conceptualized in an "exact" way. More or less arbitrarily axiomatized systems (be they ever so "elegant" and "fruitful") cannot help us here. we must try to attain a solution which is based on objective insight. At  this point, we would do well to explore somewhat further the consequences for the foundations of analysis and set theory of our view concerning the concepts of set and function. .....

chapter II, §6, The system which, for the moment, we shall call "hyperanalysis" arises if, starting from the level attained in §3 of this chapter, we lay a new foundation for pure number theory, a foundation in which we admit the real numbers as a new basic category alongside the naturals. ... This new system certainly does not coincide with our version of analysis. On the contrary, in hyperanalysis there are, e.g. more sets of real numbers than in analysis. For hyperanalysis admits sets in whose defintion "there is" appears in Connection with"a real number". Thus, hyperanalysis contains neither Cauchy's convergence principle nor, in General, our theorems about continuous functions. ..."

7. A common basis to syncronize Kant´s philosophy with mathematics was given by

Riemann B., "On the Hypothesis which lie at the Bases of Geometry"

http://www.ita.uni-heidelberg.de/research/bartelmann/Lectures/relativity/riemannGeometrie.pdf

8. In his work, "Principles of Nature and of Grace Founded on Reason", G. W. Leibniz summed up the problem "why there is somethning and not nothing?" (cited from the book of S. Blackborn, "Philosophy"):

"Nothing takes place without sufficient reason, that is to say that nothing happens without it being possible for one who has enough knowledge of things to give a reason sufficient to determine why it is thus and not otherwise. This principle having been laid down, the forst question we are entitled to ask will be: why is there something rather than nothing? For "nothing" is simpler and easier than "something". Further, supposing that things must exist, it must be possible to give a reason why they must exist just as they do and not otherwise.

Now this sufficient reason of the existence of the universe cannot be found in the series of contingent things, that is to say, of bodies and of their representation in souls ... Thus the sufficient reason, which needs no further reason, must be outside this series of contingent things, and must lie in a substance which is the cause of this series, or which is a necessary being, bearing the reason of ist existence within itself; we should still not have a sufficient reason, with which we could stop.And this final reason of things is called God."

9. Some comments from A. Einstein, Grundzüge der Relativtitästheorie, WTB, Bd. 58, 1956:

... (48a) ...Die MAXWELLschen Gleichungen bestimmen das elektrische Feld, wenn die Verteilung der elektrischen Ladungen und Ströme und Ladungen bekannt ist. De Gesetzte aber, nach denen sich Ströme und Ladungen verhalten, sind uns nicht bekannt. Wir wissen wohl, dass die Elektrizitäten in Elementarkörperchen (Elektronen, positiven Kernen) bestehen, aber wir begreifen es nicht vom theoretischen Standpunkte aus. Wir kennen die energetischen Faktoren nicht, welche die Anordnung der Elektrizität in Körperchen von bestimmter Grösse und Ladung bewirken, und alle Versuche, die Theorie nach dieser Seite hin und zu vervollständigen, sind bisher gescheitert. Wir kennen daher, falls wir überhaupt die MAXWELLschen Gleichungen zugrunde legen dürfen, den Energietensor für elektromagnetische Felder nur ausserhalb der Elementarteilchen.

... 49) ... Wir wissen heute, dass die Materie aus elektrischen Elementarteilchen aufgebaut ist, sind aber nicht im Besitze der Feldgesetze, auf welchen die Konstitution jener Elementarteilchen beruht. Wir sind daher genötigt, uns bei der Behandlung der mechanischen Probleme einer ungenauen Beschreibung der Materie zu bedienen,  welche der von der klassischen Mechanik verwendeten entspricht. Die Dichte der ponderablen Substanz und der hydrodynamischen Druckkräfte (Flächenkräfte) sind die Grundbegriffe, auf die eine derartige Beschreibung sich stützt.

Die Gleichung (90) beschreibt die Bewegung (motion) des materiellen Punktes unter der alleinigen Einwirkung der Trägheit (inertia) und Gravitation. ...(90a) ... Die Christoffel-Symbole (in dieser Gleichung) spielen die Rolle der Feldstärke des Gravitationsfeldes. Diese Grössen haben nicht Tensorcharakter.  ...Die Einheit von Trägheit und Gravitation wirkt sich formal dadurch aus, dass wohl der gesamte (Summen-) Term von (90) Tensorcharakter hat, nicht aber die beiden Glieder einzeln genommen, von denen man in Analogie zu den NEWTON-schen Gleichungen das erste als Ausdruck der Trägheit, das zweite als Ausdruck der Gravitationskraft zu betrachten hätte.

...unter dem Gesichtspunkte einer tieferen Analyse ist der Energietensor der Materie nur als ein vorläufiges, wenig tiefgreifendes Darstellungsmittel für die Materie anzusehen. In Wahrheit besteht ja die Materie aus elektrischen Elementarteilchen und ist selbst Teil, ja als der Hauptteil des elektromagnetischen Feldes anzusehen. Nur der Umstand, dass die wahren Gesetze des elektromagnetischen Feldes für sehr intensive Felder noch nicht hinreichend bekannt sind, zwingt uns vorläufig dazu, die wahre Struktur dieses Tensors bei der Darstellung der Theorie unbestimmt zu lassen.

... (121) .... Die Materie besteht aus elektrischen Elementarteilchen. Diese können auf der Basis der MAXWELLschen Theorie nicht singularitätsfrei als elektromagnetische Felder aufgefasst werden; man braucht in MAXWELLs Theorie nicht enthaltene energetische Terme, um der Tatsache gerecht zu werden, dass das einzelne Elementarteilchen trotz der abstossenden Wirkung seiner gleichnamig geladenen Teile aufeinander Bestand hat. Poincare hat daher, um dieser Tatsache gerecht zu irgendwie und vorläufig gerecht zu werden, im Innern dieser Teilchen einen Unterdruck angenommen, welcher die elektrostatische Abstossung kompensieren soll. Es kann nun nicht behauptet werden, dass dieser Druck ausserhalb der Elementarteilchen verschwinde. Diesem Umstand werden wir in unserer phänomenologischen Darstellung dadurch gerecht, dass wir der Materie ein Druckglied beifügen. Dieses ist aber nicht mit dem Druck der Hydrodynamik zu verwechseln, der ja nur zur energetischen Darstellung dynamischer Verhältnisse innerhalb der Materie dienen soll.

10. Additional comments on those topics are given e.g. given by E.Schrödinger, "Space-Time Structure", (12.29) ff.:

"Physical interpretation of the GRT can not be done in a general coordinate system. It requires locally a SRT coordinate system (10.13) ff), which does not necessarily require (3,1)-spacelike/timelike space-time structure: K. Gödel gave an example of a new type of cosmological solutions of Einstein´s field equations of gravitation."

11. H. Bergson, "Materie und Gedächtnis", Dieses Buch bejaht die Realität des Geistes und die Realität der Materie und versucht die Beziehung zwischen beiden klarzulegen an dem speziellen Beispiel des Gedächtnisses. Es ist also ausgesprochen dualistisch. Aber andererseits betrachtet es Körper und Geist auf eine solche Art, dass es viel zur Milderung wenn nicht Hebung der theoretischen Schwierigkeiten beizutragen hofft, die immer aus dem Dualismus erwachsen sind und die daran schuld sind, dass er, den durch das unmittelbare Bewusstsein nahelegt und der gesunde Menschenverstand annimmt, bei den Philosophen in sehr geringem Ansehen steht."           