The Problems of Modern Cosmology, the Fine Tuning of Constants and Parameters, and the Notion of Time

1. Introduction

The three main parts of theoretical physics that form the central body of modern cosmology are general relativity, the Standard Model of particle physics and the underlying structure of quantum field theory, as well as thermodynamics and statistical physics.

One could say that the central theory at the heart of cosmology is the so-called Standard Hot Big Bang theory, which implies that the observable universe has evolved from a very dense and hot (energetic) initial phase, which underwent cosmological expansion to produce the actual universe. The Einstein Field Equations (EFE) governs cosmic evolution, which are the central equations of general relativity.

As is well known, this theory certainly has four important observational confirmations: the expansion of the universe, the existence and properties of the cosmic microwave background radiation (the CMBR), the relative abundances of light elements including isotopes in the observable universe, and the number counts of distant radio sources compared to nearer ones.

However, this cosmological theory also raises at least 10 questions or problems that remain unsolved, or which are still debated to this day. Among these are the flatness, the horizon and the smoothness problems. The flatness problem consists in the fact that the spatial sections on large distance scales of the observable universe are very close to flatness, while the EFE imply that the flat solutions are unstable and are the most improbable models.

The horizon problem implies that very distant regions in the actual universe have almost the same CMBR temperature. Yet, the Big Bang did not provide sufficient time for such regions to establish causal contact and thermalize in order to reach almost the same temperature.

The smoothness problem corresponds to the fact a precise and adequate amplitude of density fluctuations is needed in order to act as seeds for structure formation (to form galaxies, stars, …): if the fluctuations are too small, then not enough cosmic structure forms, and if they are too large, then too much over dense structures would form, such as black holes. One would need a correct mechanism in order to produce in the early universe adequate primordial density fluctuations.

Further problems correspond to the initial singularity problem and the nature of a quantum theory of gravity, the matter/antimatter asymmetry problem, and the topological defect production (such as magnetic monopoles) after Grand Unified Theories (GUT) phase transitions (symmetry breaking) in the very early universe.

In addition, the general initial conditions problem and the general fine tuning problem of constants of physics and of cosmological parameters are still debated, as well as the dark matter and dark energy problems. Cosmic evolution strongly depends on the general initial conditions existing in the initial phase of the universe—for instance, if the geometry of space–time was too chaotic or inhomogeneous at early stages, then cosmic expansion would not have produced our flat and relatively smooth universe. The fine-tuning problem of constants of physics and parameters of cosmology entails that even very tiny variations of these numbers could imply that the observable universe today would be very different. It seems that, with respect to these numbers, the universe is structurally unstable and that their values turn out to be critical—such numbers correspond to the speed of light in vacuum, the electric charge of the electron, the ratio of the electron to proton mass and the fine structure constant. Yet, there exists today no well-established theory or explanation of the fine-tuned values of these quantities and of their ratios: we know their values through measurement, not through computation from a well-defined theoretical framework.

In modern cosmology, one strategy to try to tackle some of these issues corresponds to the so-called inflationary paradigm, which is often invoked in order to address the flatness, the smoothness and the topological defect problems. Cosmological inflation means that the universe in its very early stages underwent a very short period of rapid, accelerated cosmological expansion. This would entail that the actual universe has emerged from a much smaller region, enabling all of its parts to thermalize and solve the horizon problem. Moreover, due to this phase of accelerated expansion, the eventual topological defects would get diluted such that their local density would be virtually zero. Furthermore, an inflationary phase of expansion would turn the instability of the spatially flat models into a stable future attractor solution, rendering the flat solution the most probable one, thereby solving the flatness problem. Also, results found by Garry Gibbons and Stephen Hawking provide an adequate primordial density fluctuation spectrum as being generated by quantum fluctuations in the so-called de Sitter space–time during the inflationary phase, thus providing a solution to the smoothness problem.

However, inflationary cosmology not only does not provide a solution to the other problems of the Standard Hot Big Bang model, but also introduces new problems of its own.

First, in the original model proposed by Alan Guth in 1981 (called “old inflation”), the driver of the inflationary accelerated cosmic expansion is the potential energy density of the so-called Higgs field. However, it turned out that the universe that emerges from such an inflationary phase would not correspond to the observed one, unless some parameters were fine tuned. Subsequently, new models have been proposed having a different shape of the potential energy, such as new inflation, extended inflation, hybrid inflation and assisted inflation. In fact, a proliferation of possible models occurred—indeed, there are more than 100 different models.

Second, if the initial conditions characterising the primordial space–time are too inhomogeneous and chaotic, and if the “potential terms” do not dominate over the kinetic terms and over the spatial gradients of the field, then it is possible that inflation does not even start. The so-called chaotic inflationary model and the eternal inflation model try to overcome these issues linked to particular initial conditions: for some regions of space–time, inflation can start and generate a suitable macroscopic domain resembling the observable universe, whereas in other regions inflation does not start or produces a domain, which is different from our universe. An associated major problem of these inflationary scenarios is that the identity of the quantum field, whose potential energy density would be the driver of the accelerated expansion, is still unknown—it is generically called “the inflation field”.

In such a context, ideas such as the multiverse have been developed, essentially suggesting that our macroscopic universe is one amidst a huge (possibly infinite) number of other causally disconnected universes, and that our universe happens to be characterised by the observed fine-tuned constants and parameters, entailing the possibility to develop carbon-based life forms. The possible values of the fundamental constants and parameters would be distributed essentially randomly over the “multiverse landscape”.

At the heart of the questions regarding the nature of the universe in its very early stages, the so-called quantum theory of gravity would be helpful to address the singularity problem. Also, it would help understand the general initial conditions with which the universe started its evolution towards a remarkably homogeneous and isotropic stage at late times, far from the initial Big Bang singularity. It is remarkable that although general relativity and quantum physics are each on their own very well-tested theories till now, when one tries to unify them in a single theoretical framework, a great number of difficulties emerge such that there is actually no certainty on which proposal is moving in the right direction. Jay Armas has published 37 interviews of leading researchers in the field of quantum gravity in his book “Conversations on quantum gravity” [1], all of the experts have different approaches. It turns out that there is no general consensus regarding which approach is the correct one, and after some 50 years of research in this field no new fundamental leading principle or idea seems to have been proposed in order to motivate theoretically a given approach to quantum gravity. Current research essentially extends, modifies or simply uses known techniques from quantum physics and general relativity, but no novel idea has yet emerged.

Popular proposals are loop quantum gravity, Euclidean quantum gravity and superstring theory or M-theory. Considering the latter, in a way which is similar to the case of inflationary cosmology, one has a proliferation of possible universe models—of the order of 10^500. This substantially leaves the general initial conditions problem and the fine tuning problem unsolved, apart from stating that the fundamental constants and parameters are distributed randomly over such a “string theory landscape”. This situation is rather peculiar, since the history of physics generally shows a tendency to unification and simplification of the theoretical models—think of the unification between “terrestrial and celestial mechanics” with Newton, thermodynamics and the kinetic model of matter, electromagnetism and optics, mass and energy in special relativity. Instead, in recent times, one has a proliferation of possible explanatory solutions, or a complexification of the models—this is reminiscent of the ancient Ptolemaic universe, where one has cycles and epicycles upon cycles in order to explain the observed motion of the planets on the celestial sphere—the heliocentric Copernican model has provided a significant simplification.

In the subsequent sections, we will consider another possible road towards a solution to this “crisis of modern cosmology”, recently suggested in the work of Lee Smolin and Roberto Mangabeira Unger [2]. They argue in their book “The singular universe and the reality of time” that superstring theory or the multiverse/landscape model could well turn out not to be a correct solution to the crisis of modern cosmology. A new methodology, similar to natural philosophy, could provide new heuristic ways to penetrate deeper into the structure of theoretical physics, thereby outlining new possible paths towards the solution of the problems inherent to modern cosmology. In particular, with respect to the fine tuning problem and the general initial conditions problem, the models studied in Ref. [2] consider that there exists only one universe. New paradigms should be invoked which regard the nature of physical laws and the concept of time, as well as the way mathematics is used as a language to describe scientifically the natural world and to express the inherent physical laws.

In the next section, we will outline what the critical aspects actually are of the current paradigms that are used to describe the world, and subsequently, we will develop ideas suggested in Ref. [2] which entail that mathematics has great power but also immense limitations, providing examples showing how fundamental mathematics has difficulties when dealing with time and with phenomenal particularity.

2. Critical aspects of current theoretical models

In their work [2], Unger and Smolin explain which shortcomings appear to be present within the theoretical and mathematical formulation of models that aim to describe the world. We will list these critical aspects and subsequently analyse some particular features of the problems inherent to the use of mathematics to formulate theoretical models, which ought to provide a better understanding of nature and of the properties of the observable universe, developing examples showing how fundamental mathematics have difficulties when dealing with time and with phenomenal particularity.

• A first aspect of fundamental physics theories corresponds to reductionism, and to a wider extend to methodological individualism in science: one assumes that the properties of a system can be explained using the properties of its constituents or parts. The laws of thermodynamics explain the macroscopic thermal properties of physical systems, and these are interpreted in terms of statistical and mechanistic classical Newtonian physics applied to particles, atoms or molecules. The behaviour of these atoms and molecules depends, however, on their microscopic properties, ultimately explained by the characteristics of elementary particles such as the quark and the electron in the Standard model. The following question naturally arises, namely: where do the elementary properties of these particles, like mass, electric charge and the colour charge of quarks within quantum chromodynamics, come from? The Standard model of particle physics does not explain these parameters. Superstring theory or M-theory tries to offer an explanation going beyond the Standard model. In these theories, elementary particles correspond essentially to vibration modes in a higher dimensional space–time of extended objects such as one-dimensional strings—the whole known particle spectrum should be produced in this way, but a very special kind of topology of compactified higher dimensions, such as the Calabi-Yau manifolds, is needed in order to produce the observed particle species. Compactification entails that the extra space dimensions are “curled up” on a very small scale, since they ought not to be observable at ordinary energy levels, but only at the so-called Planck scale. However, the problem remains to explain why this extra-dimensional space would have exactly this very special topology. How does hyperspace gets its topology? There is at present no convincing compactification mechanism, which produces these topologies, and, most importantly, which guarantees their stability. Why would only three space dimensions be extended and macroscopic, while the other six extra space dimensions of the theory remain curled up on a very small scale? Again, instead of simplification and unification, one seems to have complexification.

• As opposed to reductionism, relationalism is also part of modern theoretical physics, such as relativity theory and Yang-Mills quantum field theory. However, one ought to keep in mind that this approach too has possible problems. First, relationalism implies a fundamental dynamical and changeable character in the network of interconnections, yet relativity and other relational models presuppose immutable laws. Second, relationalism entails the existence of individual entities that form the subject of the relations—as explained in Ref. [2], not every property can be a relation, there ought to be intrinsic properties, which the relations relate.

• According to Unger and Smolin [2], another aspect corresponds to the assumption of the Newtonian paradigm, which entails that any physical system can be described by a fixed set of unchangeable laws and that there is a distinction between laws and initial conditions. While varying the initial conditions, and using a fixed set of laws, one can in principle predict the behaviour of the system in various circumstances. Theoretical physics models usually boil down to a fixed set of differential-type equations, which have infinity of solutions, so, a particular choice of initial conditions or boundary conditions needs to be specified. Also, any set of equations will always be effectively characterised by a certain number of fixed parameters or constants. In the case of cosmology, this duality between laws and initial conditions leads to the general initial conditions problem. An approach due to Hartle and Hawking in Euclidean quantum gravity, called the no-boundary proposal, argues that in the quantum regime at the initial stage of the evolution of the universe, one does not need boundary conditions: there would be essentially no boundary. On the other hand, the approach suggested in Ref. [2] argues that time would be real and would affect everything, including the laws of physics themselves, and that one would not have a fixed immutable set of equations describing the dynamics of the universe. One then has no distinction between initial conditions and laws.

• A further characteristic mentioned in Ref. [2] corresponds to “false universality” present essentially in the Newtonian paradigm. It consists in assuming that locally determined laws of physics are valid for the entire universe, in any circumstance and at any energy level. This assumption is still part of general relativity and even of quantum physics. Moreover, general relativity and quantum physics are supposed to hold also in other “universes” in the multiverse/string landscape scenario. Smolin in Ref. [2] argues that close to black hole singularities, or close to the initial Big Bang singularity, the laws of physics could undergo change—events “without precedents” in this case would occur and would produce genuine novelty inherent to time and true change.

• Another critical aspect of modern theories corresponds to what one could call “universal anachronism”, which assumes that physical laws pertinent only to a small part of the history of the universe hold throughout the whole cosmic history. Physical laws in the very early universe might have been different from the laws we witness in our large and low-energy universe. High-energy environments (or other environments, which are different from the ones we observe in the local universe) might stimulate degrees of freedom, which are otherwise not excited in the dynamics of a low energy region, which would exhibit regular, stable and thus fixed physical laws.

• Smolin and Unger argue that mathematics has great power to describe the natural world, but it is also important to realise that it has immense limitations—they propose a “selective realism of mathematics”. Mathematics would have difficulties when dealing with time and change, and with phenomenal particularity. Mathematics only captures selected parts of the dynamics of natural systems. It is claimed in Ref. [2] that no mathematical structure can be isomorphic to the history of the universe.

• An interesting concept, which is mentioned in recent treatises such as in Ref. [2], but also in Ref. [3], corresponds the so-called “qualia”, that is to the properties of systems that are observed through conscious perception. This refers to an old philosophical problem which has been, and which still is, very much debated. In order to illustrate this point briefly, consider the perception of green colour. As argued by Roderich Tumulka in Ref. [3], no mathematical structure or computation can explain how green colour appears to the conscious observer. Of course, green colour has its wavelength and its refractive properties; it is an electromagnetic wave and one knows most of the mechanisms that generate electro-chemical impulses in the eye, which travel to particular areas of the brain, and cognitive neurosciences provide a lot of information on these processes. But the knowledge of electromagnetism and optics, of neuroscience and cognitive neuroscience is not capable to explain to someone how green colour appears to the conscious observer—the only way would be to directly experience green colour. Someone who knows all of physics and cognitive neuroscience, but which has never experienced green colour, will not be able to deduce how green actually would appear to his consciousness. Max Tegmark has argued that the universe might actually be a purely mathematical entity [4]—the universe not only would be well described by mathematics, it actually is mathematics—but the existence of qualia and of conscious perception certainly poses an important challenge to such an idea. Smolin in Ref. [2] argues that qualia are not just “illusions”, but ought to be considered as novel physical phenomena, which might correspond to events without precedents.

3. The limits of mathematics and time

Is it possible to illustrate why and how mathematics has difficulties when dealing with phenomenal particularity, time and change? One way would be to consider a fundamental part of mathematics, that is, set theory, which forms the basis of most of mathematical concepts, including numbers.

A standard axiomatic theory in pure mathematics corresponds to the so-called Zermelo-Fraenkel set theory, with the addition of the axiom of choice (ZFC set theory). Before entering into set theory, let us consider some more conceptual or philosophical aspects regarding the notion of time and change. The American philosopher Irving Copi has highlighted one of these: consider some entity A. If A is subject to change, it will become a distinct entity B. Thus, we end up with two distinct entities, and the following question arises: how one can speak of the change of one single entity? In other words, is there an identity over change? In order to deal with this paradox of time and change, we will heuristically discuss an alternative set theory, which differs from the standard ZFC model. The essential idea is that sets are “collections” of elements, but, in addition, they also have information content. In a sense, this alternative set theory is analogous to non-Euclidean geometry, because one of the fundamental axioms of ZFC theory is not valid in this new model. Our approach at this stage is non-formal and heuristic, and will need further research in order to develop rigorous formulations of the models.

The starting point is the concept of information and sets. The difference or duality between information and meta-information plays an essential role. In order to illustrate this aspect, consider as a concrete example a book. The title, the author and the whole content of the book represents information. On the other hand, the colour of the cover represents meta-information with respect to the information—it cannot be deduced, in any way, from the information content. The location of the book in space also represents meta-information: the book could be on a table, or on a shelf of some library. Now, as far as sets are concerned, consider a set S containing three books. This set contains the information, which describes in detail the content of the three books. However, considering meta-information, one can construct a set S′ that contains exactly the same three books, but which has different meta-information content: for instance, in S the books could be aligned on the same shelf, while in the set S′ the books could be on the vertices of an equilateral triangle. The elements of the sets S and S′ are the same, but these sets are different. This means that the so-called extensionality axiom of ZFC theory, which states that two sets are equal if their elements are the same, is not valid. In the real physical world, the extensionality axiom is in fact not always true because of the meta-information content characterising location in space, or the relative ordering of elements in space; that is, sets of things can have an information content with respect to its elements as well as meta-information that is essentially relational.

As is explained by George Tourlakis in Ref. [5], sets are considered in ZFC theory regardless of their “inner structure” or of “intention”, of how the sets came about, that is formal or axiomatic mathematics aims at representing reality within an artificial but formal and precise language. In this representation, as is argued in Ref. [5], there is always something lost, partly due to the decisions we make about what features of reality are essential. This artificiality, or abstraction, represents in this case an instance showing that mathematics does not capture phenomenal particularity.

One can try to generalise ZFC theory by violating the extensionality axiom, and by considering sets as defined by their elements, as well as by a meta-information content, in order to attempt to derive a richer mathematical description, which would offer possibilities to include some phenomenal particularity. Moreover, this setting is suitable for our idea about time and change. If a set S is described by its elements, its information content as well as by the meta-information content, then the identity during change corresponds to the elements of the set S. In addition, the set S at some time t and the set S′ at time t’ have the same elements and in this part the extensionality axiom would hold through change, guaranteeing identity. But meta-information can be added or can change, and this entails that change really can occur and that the sets S and S′ are not the same. Since meta-information is independent from the information regarding the elements, change would indeed introduce genuine novelty.

From a heuristic (not formal), but slightly more technical perspective, the following departure from ZFC theory could be formulated. A set S can be described by its elements e(i) and its information content I(S) regarding its elements:

For any such set S, meta-information, Im(S), can always be added yielding a new set

In this context, the empty set plays a crucial role. Consider, for instance, two distinguishable sets S and S′, characterised by different elements:

These two sets can share meta-information, which can be of relational type between the elements e(i) and f(i), so that their intersection contains no elements, but contains shared meta-information. This intersection would correspond to an “empty” set Ф with no elements, but with a meta-information content:

where Im(S,S′) represents the meta-information between S and S′, such as the “distance” between the elements of S and S′, or the difference in “colour” between the elements, or any difference that distinguishes the elements in some suitable configuration space.

This idea of considering an empty set having no elements but carrying meta-information is suitable with respect to the above explained idea of change: a system can change not only because new elements are added, but because new meta-information content is added—that is, while preserving identity (elements), change can occur (meta-information). Indeed, consider the union set between a set S and the empty set Ф, that is S U Ф. No new elements are added, but this union set has an information content given by

since

This last relation means that the intersection between any set S and the empty set Ф cannot be the empty set Ф, unlike in ordinary ZFC set theory, such that S∩Ф differs from Ф, and carries no elements nor information. In this alternative theory, the empty set carries non-trivial meta-information and thus cannot be part of any non-empty set S.

This is paradoxical from ordinary ZFC set theory. Indeed, suppose, by absurdum, that the empty set Ф is not part of any non-empty set S. But then there must exist some x, not belonging to S, and which is element of Ф, which leads to a contradiction. In the alternative theory, non-empty sets S and the empty set Ф cannot be viewed as existing “at the same level”, in a way which is analogous to the difference between the real part and the imaginary part of complex numbers. It is as if S would lie on a “real” axis, while Ф would lie along the imaginary axis of some abstract Wessel plane. With respect to the notion of time, set Ф is never “simultaneous” to set S—it “lies somehow in the future of S”. One can thus see how a simple idea of change easily leads to paradoxes or contradictions in fundamental mathematics if tools from elementary set theory are used.

These is another area in elementary set theory, which would suggest, in similar way, a notion of time, namely the paradoxes of set theory were first discovered Georg Cantor. The use of the power set theorem and of Cantor’s so-called diagonal argument shows that given some non-empty set, it is always possible to construct a larger set, which contains the original set as a subset. Consider a set S containing three elements A, B and C. It is possible to construct the set S′ of all the subsets of S (as explained above, we do not include the empty set in S′, and we also take into account the order between the elements):

which contains S as a subset.

Now, if one considers the “set of all sets”—or the universe set—one has a paradox. Indeed, if the universe set can be specified, then given Ω, it is always possible to construct the set of all the subsets of Ω, which is a larger set containing Ω as a subset. This power set theorem works even in the case of infinite sets, as has been shown by G. Cantor, and gives rise to the possibility to formulate a theory of transfinite numbers. This suggests a notion of time related to the above-mentioned ideas, and entails that change is always possible, and that no set can be considered as a completed, fixed whole. Whenever a possible universe set Ω is given, it is always possible to construct a different set Ω’, not included in Ω. The different subsets of Ω’ contain the original elements listed in Ω but are associated in different ways and in distinct combinations—this represents relational information and corresponds actually to a form of meta-information.

These set theoretic paradoxes, which in a sense represent limits of mathematics and can be dealt with by choosing appropriate axioms or definitions of what a set ought to be. One can distinguish between sets and classes, and introduce axiomatically the notion of proper class, which, by definition cannot be part any other set.

However, according to our viewpoint these limits of mathematics actually point towards a notion of time. No set can be conceived as a completed immutable whole—there is no timeless “block universe set”—the possibility of relational or meta-information always opens up new possibilities.

In the next section, we try establish heuristically links between these theoretical reflections and the physical world—in particular with features of quantum physics.

4. Links with the physical world, quantum physics

In Ref. [6], we discussed the philosophical concept referred to as the nomological machine idea of philosopher Nancy Cartwright. Essentially, the viewpoints highlighted by Smolin and Unger in their work entail that the natural or physical states form the laws, rather than the states being subject to immutable laws.

Moreover, causality is considered as a primitive concept in Ref. [2]. Usually, laws are considered as fundamental, whereas causal relations are instances of the laws. However, according to our approach, laws are particular instances of causal relationships or interconnections. The natural or physical systems can be considered as nomological machines, which in situation-specific circumstances can produce or exhibit regular law-like dynamics, whereas in other circumstances, some unpredictable, uncomputable or changeable dynamics occurs.

In Ref. [6], the idea has been discussed which considers the standard cosmological model as a nomological machine. Another system was considered too, which corresponds to a set of mutually reflecting spheres, which produces in each sphere sequences of iterated reflections within reflections, ad infinitum. An observer can, within each sphere, perceive sequences of nested spheres, which can generate regular, recursive patterns, but also irregular and unpredictable sequences. The recursive patterns would correspond to law-like behaviour, while chaotic sequences would correspond to non-recursive causal interconnections, which can entail the possibility of changing or mutable laws.

One can consider the physical example of a set of four perfectly reflecting spheres a, b, A and B. Light rays will be reflected back and forth between the spheres. This would correspond to a primitive causal connection. In this way, an observer can perceive a sequence of nested spheres within each sphere, which can be infinite in principle. For instance, the observer could perceive in the sphere b either the reflection of b (itself), of a, of B and A. This leads to four “two lettered words,” bb, ba, bB and bA. Within each of these secondary “spheres”, subsequent reflections can be observed: bbb, bba, bbB, bbA or bab, baa, baB, baA or bBb, bBa, bBB, bBA or bAb, bAa, bAB and bAA. If the reflection process proceeds, then the observable sequences reach infinite length and correspond to “infinite words”, like bBaBbbaBAAbaA…

In Ref. [7], D. Mumford, C. Series and D. Wright explore such mathematical objects by using essentially Möbius transformations in the complex Wessel plane and on the Riemann sphere. These are the most general transformations that map circles into circles, and one of the cases corresponds to four circles in the complex plane a, b, A and B, which are mapped one into each other by the iteration of a Möbius transformation. This iteration generates the successive reflections which we mentioned earlier, and which generate infinite sequences of nested disks. The “words” that are generated by the transformations are represented differently in Ref. [7]: the sequence of letters represent the sequences of iterated maps. The transformation named “a” maps the outside of disk A into the inside of disk a, while the transformation “b” maps the outside of disk B into the inside of disk b etc. Two lettered words are in this case aB, ab, ba, bb, aa, bA, Ab, AB, BA, BB, Ba and AA. These two lettered words can also be generated if one would create all the possible “binary” subsets of the set {a, b, A, B}, allowing the same element to be repeated and taking ordering into account. Only four combinations are not present in the reflections studies in [7], namely Aa, aA, Bb and Bb. Thus, the mappings between circles (or spheres) generate a new set of 12 elements (words) that are part of the possible subsets of the set S = {a, b, A, B}.

When united with the set {a, b, A, B}, one obtains a larger set S′ which contains the original set as a subset and contains 16 elements. Sequences of three lettered words can also be generated by iterated maps, aaa, aaB, aab, aba, abb, abA, baB, baa, bab, bba, bbb, bbA, bAb, bAA, etc. All these new words are part of the set of all possible three lettered subsets of S′. If the reflection process is iterated ad infinitum, one obtains infinitely long words that correspond to the infinite successions of nested disks, and whose ultimate intersection corresponds to a limit point—all the possible limit points generate the limit set. As explained in Ref. [7], the number of limit points is actually uncountable. Computable sequences form a countable set, and thus, it follows that most of the sequences are uncomputable.

Turning our attention to quantum physics and to the orthodox Copenhagen interpretation in particular, the following property must hold, namely: any system quantum mechanics applies to must be part of a larger system that contains the system as subsystem. Any system ought to be potentially observable “from the outside”. This leads to a paradox when one would try to apply quantum mechanics to the universe—how could an act of measurement be performed on the universe? This is reminiscent of the paradoxes discussed in the previous sections in the context of elementary set theory: the universe set Ω cannot be conceived as a completed whole. In same way, one can assume that it is not possible to conceive “the universe” as a completed whole. In addition, if one would adopt the fundamental axioms suggested by Unger in Ref. [2], then the reflecting sphere model provides a way to generate in any case a new and larger set, which contains the original set as a subset, thus making sure that quantum mechanics can always be applied to any set or system. In a sense, the universe, having constituents or parts with reflect each other or which interconnect with each other, would “observe itself from the inside”.

Another interesting feature of quantum mechanics, which suggests interesting viewpoints, corresponds to the “No-Cloning Theorem”. This theorem states that it is not possible to clone a general quantum state through unitary processes, see Ref. [8]. If this “quantum xeroxing” would be possible, then one could perform the measurement of two conjugate variables, one on each copy, thereby violating the Heisenberg uncertainty principle.

Now, within our alternative set theory, suppose that it would be possible to clone any set S, such that the set S and its copy are distinguishable in some suitable configuration space: this means that S and its copy share relational meta-information. Then, we have that, since S ∩ S = S:

where Im(S,S) represents the meta-information that distinguishes between S and its copy S.

However, in this case we contradict the fact that, as stated earlier in Section 3, the intersection between two distinguishable sets is the empty set Ф{Im(S,S′)}. Moreover, it not possible that a set S determines by itself its own meta-information content, which is relational in nature. Thus, in some suitable configuration space, it is not possible to observe a set S and an identical copy. One can observe a set S and an identical copy only at different “instants of time”, but not “simultaneously”.

We hope that these heuristic arguments can be developed further through future research, in order to find a way towards a unifying framework, which would contain general relativity as well as quantum physics. If time is real and fundamental, then everything, including the laws of physics and the associated fundamental constants, might be subject to change. There have been already proposals that study possible variations of the constants, such as the fine structure constant, or varying speed of light theories were the speed of light changes through cosmic evolution. All of the properties of the observable universe might be the result of a unique dynamical process, which would ultimately affect everything.

Acknowledgments

I thank very much Prof. Pawel Tambor, with whom I have worked before, for valuable discussions of philosophical nature on these topics.