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The Big Bang Theory, first suggested by George Gamov in 1948, is the leading theory to describe the beginning of the universe. The theory explains how the universe emerged from a tremendously dense and hot point about 13.7 billion years ago, called singularity. Precise predictions of background radiation temperature, even before it was actually discovered, as well as recent findings suggesting that our universe was much hotter in the younger ages, support the Big Bang Theory. However, physicists have mainly avoided the conundrums of an infinitely compressed zero-sized starting point.

The problem arises when we go to ultra small scales. We cannot find any meaningful size smaller than the Planck length (10-33 cm) in the objective universe. In one view, the Planck cube (the cube of the Planck distance) is the building block of three-dimensional space. Therefore, the only way that we can pass beyond this size is to exit the objective world and imagine singularity. Our experiment based scientific laws do not govern in this arena; therefore, any attempt to carry over these laws is doomed to assumptions, uncertainty, and confusion. We have dared to explore beyond the boundaries. Let’s now delve even further.

Ordinal Numbers

The question is how the zero-point singularity could have originated the objective space-time universe we know. Ordinal numbers can be suggested as a mathematical formalization of the Big Bang theory. Ordinal numbers are the simplest set of all and are sometimes called the empty set or the null set. The empty set is normally shown as follows,

Ø = { },
where the empty brackets delineate a set. Since there is nothing inside the brackets, it is an empty set. Let us associate Ø with the number 0. Then we can go further and define a set whose only member is Ø:
Ø = { Ø },
We use our logic to come to the deduction that { Ø } is not the same as Ø. The set { Ø } has one member on it, whereas Ø itself has none at all. Here we can associate { Ø } with number one. For number two, we collect the two previous entities to come to the following,
Ø = { Ø, { Ø }},
Then we associate 3 with the collection of all the above sets as,
Ø = { Ø, { Ø },{ Ø, { Ø },}}},
The number 4 is associated with the sets containing the members that we have encountered before, to come to,
Ø = { Ø, { Ø },{ Ø, { Ø }}, { Ø, { Ø },}}},{ Ø, { Ø },{ Ø, { Ø },}}}},
and so on. [1]

The concept of ordinal numbers shows how natural numbers can be derived from nothing. Moreover, each of these numbers have zeros in their core. We use natural numbers to delineate different objective elements in space-time, such as units of matter, space and time. The ordinal number concept can be used as formalism for the evolving objective universe from a zero-size, non-material singularity at the beginning of time. It also can be used as a model for discrete universe with the presence of singularity at each point of it. Obtaining natural numbers from null sets has other physical manifestations as well. In the Standard Model, subatomic particles such as electrons are considered size-less and mass-less objects. However, they are computable and we use natural numbers to denote them in calculations. In quantum theory waves are reduced to objects, which are computable.

In my assumption mentioned in the "Quantum Mechanics," section, Super position of states [2] is not an actuality. However, it exists in informational domain. One can argue that the act of state reduction (changing multiple states to one state) and "objectification" may be driven by conscious observer. Some theories advocate that the objective reality as we know it comes to existence within observer's mind. Mind you that singularity in this model is non-material and can encompass infinite information needed by super-position.

Origin of Matter

Although there is strong evidence to support the main concept of the Big Bang theory, there are different opinions about how it actually happened and progressed. In Gasperini and Veneziaino's pre-Big Bang scenario, [3] the universe started out as cold and essentially infinite in spatial extent. In addition, they suggest that the Big Bang is not the initial event in the creation of our universe; rather, it is a few steps further.

Guth Inflationary Theory

 [4]The inflationary theory is the most popular and accepted version of the Big Bang event. In the inflationary model, the initial stage involved a very rapid and massive expansion of the space. This took just a fraction of a second.

Then the expansion slowed down but continued for about seven billion years. This period is called the deceleration stage where the rate of expansion was slowed down by the gravity of the matter inside the universe. Then as space got bigger and less dense, the gravity got weaker. As a result, the expansion of the universe has been accelerating for the past seven billion years.

Cosmologists consider the cosmological constant the factor behind the accelerating force. This constant is supposed to exert repulsive force upon the space-time. It is believed that dark energy, with its negative pressure, is the source for the cosmological constant. For the expansion to accelerate, we need constant penetration of dark energy. Where does this energy come from?

In an article about dark energy [5] Nasa declares: "More is unknown about dark energy than is known. An explanation for how space acquires energy comes from the quantum theory of matter. In this theory, "empty space" is actually full of temporary ("virtual") particles that continually form and then disappear. But when physicists tried to calculate how much energy this would give empty space, the answer came out wrong - wrong by a lot. The number came out 10120 times too big. That's a 1 with 120 zeros after it. It's hard to get an answer that bad. So the mystery continues."

The estimated amount of dark energy inside space-time is only about 10-9 Joules per cubic meter. Where does the discrepancy come from? A possible explanation is that we are measuring the energy at the boundaries ,within the dusk of the boundaries, where we are heading to the infinite energy of singularity.

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