SINGULARITY  A Fundamental Entity
We must try to understand the beginning of the universe on the basis of science. It may be a task beyond our power, but we should at least make the attempt.
Steven Hawking
So far, the scientific community has sought to uncover the secrets of our material world. In this endeavor we have progressed quite far.
When the twenty century began, it seemed that new and amazing discoveries were being brought forth every day. However, each new discovery raised even more puzzles and questions. In physics, the dilemma between Einstein’s general relativity and quantum mechanics (the mechanics of subatomic particles that may govern the macrocosm as well) has attracted the attention of theoretical physicists. Their efforts have led to a clearer picture of reality, a wondrous landscape unfolding before our eyes.
One of the many mysteries in this new picture is the idea of singularity. According to contemporary physics (and in particular the Big Bang theory), singularity is the sizeless, massless, ultradense point that is the origin of our universe. It is also assumed that a singularity lies at the centre of each black hole. On this site, we will propose a theoretical model in which singularity acts as medium, holding together and connecting the spacetime components of the universe. According to the presented model, singularity remains active and plays a major role throughout the evolution and every instant events of the cosmos.
In this context, while the objective, tangible properties of elements are manifested in spacetime, the subjective qualities of these elements are attributed to the singularity.^{[1]} 2. ^{[2]}
Using the metaphor of a “fabric” to describe the interconnectivity and wholeness of the world is nothing new. Scientific findings provide ample evidence to support our commonsense, experiencebased notion that the world is an interconnected system. Many attempts have been made to prove such wholeness. The great twentiethcentury physicist David Bohm tried to present his “implicate order” as a model. He believed that at some deeper level of reality subatomic particles are not individual entities, but are actually extension of the same fundamental something. Unfortunately, he could not finish his unbroken wholeness theory before his death.
Notes to Zero and Infinity
Ian Stewart the Emeritus Professor of Mathematics at the University of Warwick, England, says
“Nothing is more important than nothing. Nothing is more puzzling than nothing. Nothing is more interesting than nothing.
What lies at the heart of mathematics? You guessed it, nothing”
In the introduction chapter I have mentioned that zero and infinity has to be dealt with in a new fashion. Here I offer my reasoning and speculation. Zero represents nothing and seemingly it is opposite to infinity. On the other hand, zero and infinity share some common features.
Here, I will argue that zero and infinity cannot be considered members of natural number system. I try to show that if we do, inconsistencies appear within the system.
Peano Axioms
A set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. The principles of the Peano arithmetic are as follows,
1. Zero is a number.
2. If a is a number, the successor of a is a number as well.
3. Zero is not the successor of any number.
4. Two numbers of which the successors are equal are themselves equal.
5. (Induction axiom.) If a set s of numbers contains zero and also the successor of every number in s, then every number is in s.
Peano's axioms are the basis for the version of number theory known as Peano arithmetic.
Ideally a formal system has to be complete and without inconsistencies. In mathematical logic a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, otherwise the system is said to be incomplete. A consistent theory is one that does not contain a contradiction.
Kurt Gödel's incompleteness theorems are among the most important finding in modern logic. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F.Therefore no formal system is complete within its boundaries.
Is zero a natural number?
Among Mathematicians, Cantor, Peano, and Burbaki thought that zero is a natural number, While Euler, Kronecker and Sloane thought otherwise.
Let us see why there are disagreements. Does inconsistencies appear by inserting zero in the natural number system?
1/ adding two numbers should create a bigger number, whereas by adding zero to zero or any other number, the said number will remain the same.
2/ the system is undecidable if both s + a and s  a are equal. Since s + 0 and s  0 are equal then piano arithmetic appears undecidable in respect to zero
3/ Logical trap; zero divide by any number is zero. This implies that, every natural number is equal to another which appears contradictory.
4/ any number to the power of zero equals one, whereas zero to power of zero is not equals to 1. This also exhibit inconsistency in the system.
5/ similarly with n factorial such as 5! We do not follow it up to zero because it defies the purpose. 5x4x3x2x1x0. Mathematicians take 0! = 1. Another exception.
Furthermore,
a/ if natural numbers represent physical entities, why one has to correlate nothingness by a number?
b/ unlike any other number, zero cannot take positive or negative sign.
C /unlike any other number, zero is not analytical. half of zero doesn’t have any meaning.
Infinity
The concept of infinity has been a major problem in mathematics. From Aristotle to Galileo, Cantor, Gödel and the others had struggled withthe concept. Actually, it is said that Cantor and Gödel have had nervous breakdowns over it and ended up in mental hospitals.
Some mathematicians believe that infinity is not a number. Yet, the Zermelo–Fraenkel set theory plus the axiom of choice (ZFC), the most common foundation of mathematics contains the axiom of infinity. By Gödel's first theorem, even the extremely powerful standard axiom system of ZermeloFraenkel set theory (denoted as ZF, or, with the axiom of choice, ZFC) which is more than sufficient for the derivation of all ordinary mathematics is incomplete.
The German mathematician, Georg Cantor believed that infinity is part of number system. He even suggested that there are different levels of infinity. However,
1/ Different infinite sets having the same cardinality, suggested by Cantor is being observed by mathematicians although with reservations^{ (1)}. Intuitively a countable set such as natural numbers should have less members than an uncountable set.
2/ If infinity #1 + infinity # 2 is indistinguishable from infinity # 1 minus infinity # 2 then the finding cannot be explained by natural number system.
3/ Again a+a is supposed to be bigger than a. This rule does not apply to infinity either. Adding any number to infinity doesn’t change it. This is another exception within natural number system regarding infinity.Similarly, infinity times infinity is infinity, another exception.
4/ infinity is not successor of any actual number, therefore it defies the second Peano arithmetic.
5/Different levels of infinities mentioned by Cantor is counterintuitive as well.Aleph zero being smaller than aleph one and so on …. up to infinity of infinities raise many questions.
Furthermore
àA/ while any other number is analyzable (for example, one tenth of number 1 is meaningful), just like zero, infinity is not analytical (one tenth of infinity does not relay any meaning).
B/There are unlimited points between zero and one in the number line. As a matter of fact, there are unlimited numbers between every two consecutive integers. This creates a big puzzle in mathematics.
Does infinity exist?
Stephen Simpson, a mathematician and logician at Pennsylvania State University asks “What truly infinite objects exist in the real world?” If spacetime universe is quantized and made of discrete elements, how can we assign infinity to anything at all?
Taking a view originally espoused by Aristotle, Simpson argues that actual infinity doesn’t really exist and so it should not so readily be assumed to exist in the mathematical universe.
The idea is that everything in spacetime is quantized. Even atoms in the universe is finite and measured at about 10^80. So he concludes that there is no room for anything infinite in spacetime universe.
Does zero exist?
Zero represents nothingness. Logically if there is a thing, there should be a nothing as well. Thing without nothing makes no sense. But where can we find nothing?
The fact is zero and infinity are needed and being used in mathematical calculations and they are building blocks of mathematical theories. They simply can’t be ignored. They should be accounted for in our theories.However, zero and infinity are affecting natural numbers and mathematics in bizarre and unusual ways.Is there a solution?
Adding axioms
Gödel's incompleteness theorem implies,
For any statement A unprovable in a particular formal system F, there are, trivially, other formal systems in which S is provable (but we need to take A as an extra axiom)
Kurt Gödel postulates, “Mathematic which works with numbers (discrete domain) cannot be complete. It needs uncalculable domain outside it to make the whole scheme complete.” Similarly Turing halting problem suggests the same.
Continuum Hypothesis
The continuum hypotheses (CH) is one of the most central open problems in mathematics. It is the number one unsolved problem in the Hilbert's list.
It deals with the question of is there a continuity (ultimately smoothness) between 1 and 2 or any other consecutive numbers? Is there discreteness (steps) between numbers?
The continuum hypothesis was advanced by Georg Cantor in 1878, his first paper. But he couldn’t prove this “continuum hypothesis” using the axioms of set theory. Nor could anyone else.
Is there a way out of the gridlock?
Continuity and discreteness
Zero is not analytical. 1/3 of zero doesn’t have any meaning. Similarly, infinity is not calculable. 1/3 of infinity is meaningless.One may conclude that zero an infinity are continuums by nature. Natural numbers on the other hand are countable, therefore they are discrete.
If zero and infinity are not countable, then combining them with countable natural numbers, can prove troublesome.
Let us explore as Gödel suggested what happens if we extract uncalculable domains out of Peano arithmetic and add them as separate axioms.
While natural numbers do represent discrete domains, zero and infinity not being analytical can represent a continuum.
If we extract zero and infinity out of the natural number system and insert them to an added layer then we may have a system where some of the inconsistencies are removed while a more encompassing theory is achieved.
In such a scheme, the discrete domain (countable) can overlay over a continuum layer containing zero and infinity.
The two layers are different form each other (discrete versus continuum). Yet they can interact with each other with a different set of rules.
Explaining the unexplained
Extracting zero and infinity out of the natural number system and inserting them in an underlying continuum layer, opens the door to a new realm. In this scheme we need to look outside the box where contemporary concepts are modified.
In contemporary mathematics dividing by zero is considered undefined therefore forbidden. However, if we allow division by zero and take multiplication sign as coupling with zero one can find solution for the logical trap mentioned above where dividing different numbers by zero makes them to appear equal.
X x 0 = 0, Y x 0 = 0,
Then X = Y
In such a model, one can conclude that in a Cartesian system where X and Y coordinates represent discrete values, the image of any point projected to point zero equals to zero. In point zero images of all points in the field overlap each other. Therefore it demonstrate equality.
Removing limits from Calculus
When calculus originally introduced by Newton and Leibnitz, it was noticed that, the discipline frequently encounters zeros and infinities. This was troublesome. The 19^{th} century German mathematician Karl Weierstrass came to rescue by introducing limits to calculus in order to avoid zero and infinity in calculations.
Although, artificially placing limits in the field helps to deal effectively with discrete elements, it ignores a good portion of the domain (the continuum portion).
Zero is scattered in any field. In calculus, one can choose any point of the field as point zero, the practice is called blowing up the origin. The separate continuum layer reintroduces zero and infinity to calculus.
No need for renormalization
Certain phenomena such as selfenergy of the electron as well as vacuum fluctuations of the electromagnetic field seems to require infinite amount of energy.
To avoid infinities different technics of renormalization has been used to circumvent the divergence.
One way is to cut off (renormalize) the integrals in the calculations at a certain value Λ of the momentum which is large but finite. Again visualizing zero and infinity as a separate domain removes the necessity of artificial renormalization. We just need to builds theories that convey a meaning to them.
Extra dimensions and super space
Many mechanisms—for example, electromagnetic fields—cannot be explained in the context of a fourdimensional universe alone.To explain these mysteries, mainstream physicists chose to theorize another spacelike manifold in addition to ordinary spacetime. This manifold is called superspace. In basic terms, the idea of superspace presumes that the points in spacetime are actually crosssections of bundles which are extended into this proposed superspace. This is to compensate forinconsistencies.
Maybe we don’t need to assume an extra spacelike entity. Maybe a continuum layer can provide answer to our paradoxes.
Hilbert Space
In mathematics the separable up to infinite dimensional Hilbert inner space is essential. In the formal model theInner product spaces generalize Euclidean spaces (in which the inner product is the dot product,) to vector spaces with up to infinite dimensions.
Hilbert space is indispensable tool in the theories of partial differential equations, quantum mechanics, Fourier analysis and ergodic theory, which forms the mathematical underpinning of thermodynamics.
Further on I will use the Hilbert space to develop the proposed model.
Zero Point Energy
Vacuum energy is the zeropoint energy of all the fields in space, which in the Standard Model includes the electromagnetic field, other gauge fields, fermionic fields, and the Higgs field.
It is the energy of the vacuum, which in quantum field theory is defined not as empty space but as the ground state of the fields. In cosmology, the vacuum energy is one possible explanation for the cosmological constant.^{[3]} A related term is zeropoint field, which is the lowest energy state of a particular field.
Normally point zero energy does not affect discrete fundamental forces (gravity, electromagnetism, weak and strong forces) in spacetime universe.
In this scheme, some of the most puzzling question in theoretical physics may find explanations,the cosmological constant puzzle can obtain an explanation. The underlying layer can be source for the mysterious dark energy sipping into fabric of spacetime.
Nonlocality and quantum entanglement, quantum tunnelling, spin of subatomic particles, virtual particles, nature of fields and alike can obtain explanation as well.
I will refer to them in the coming pages
1/ Hermann Weyl a German mathematician, Theoretical physicist and philosopher wrote:

“… classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the fall and original sin of [Cantor's] set theory …."

http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory#cite_note11

2/ http://www.scientificamerican.com/article/infinitylogiclaw/
Revisiting Singularity
"Singularity was brought to our attention after Albert Einstein presented his Field Equations on November 18th, 1915. But Einstein himself was trying to deny it for the rest of his life. In 1939, he tried to show that “Schwarchild singularities do not exist in physical reality.”
—John Earman, Crunches, Whimpers, and Shrieks
Singularity is understood to be the nucleus for the Big Bang event. According to this theory, about 14 billion years ago, the universe started with a huge, rapid expansion of a condensed zero size point. However, this is not the only place we encounter singularity in the field of cosmology.Singularity is also known to be present at the centre of balck holes.
Supposedly, there is a black hole at the center of each galaxy, as well as many others, which are spread throughout the galaxies. Since billions of black holes are predicted to exist in the universe, there should be many more singularities than the one that started the Big Bang. However, are there in fact many singularities, or is this just one, presenting itself at different spots?
In contemporary physics, singularities have been labeled a “catastrophe,” a “problem,” or else something “nonreal” or “nonphysical.” However, the idea of singularity is present in every astrophysical theory, as well as in quantum mechanics and appears in all mathematical equations related to these theories. As mentioned before, several attempts have been made to bypass singularities. Theories such as supersymmetry, supergravity, and superstring were developed primarily to remove singularities and infinities from matter and fields. Theoretical physicists have been trying to ignore this “problem” for almost a century.
Brian Greene, one of the main advocates of superstring theory, declares that physicists, while developing the theory, “knew that there were significant aspects that we would need to work out before we could establish that our second half of the story did not introduce any singularities—that is pernicious and physically unacceptable consequences.”^{[3]} But others, such as John Earman, see it differently. In his book Crunches, Whimpers, and Shrieks, he states,
“(This book is) written in the faith that, if adequately revealed, the problem of spacetime singularities will not remain the orphan of the philosophy of science and that if adopted as a rightful child, it will enrich not only the philosophy of space and time, but other members of the family as well.” ^{[4]}
Roger Penrose believe that “whatever the quantum gravity theory (the socalled theory of everything) turn out to be, it out to provide explanation for the particular structures that apply to spacetime singularities of our actual universe. These having an importance to physics that turn out to be fundamental, going far beyond the issues normally thought to lie within the scope of quantum gravity."^{[5]}
As a gateway to an alternative view of singularity, I would like to refer to Einstein’s remarks:”If we imagine the gravitational field … to be removed, there does not remain a space of type 1 (Minkowski spacetime), but absolutely nothing, and also no ‘topological space.’” Earman argues, “Einstein is surely right that, whatever the technical details of the definition of spacetime singularities, it should follow that physical laws, in so far as they presuppose space and time, are violated or, perhaps more accurately, do not make sense at singularities. This is a good reason for holding that singularities are not part of space time.”^{[6]}
According to the statement, we should forget about spacetime as an adjective for singularity. Rather, we would have to consider singularity a separate entity altogether.