Author(s): Todor Zh Mollov
In the present paper we establish, that
(i) The universe is infinite,
(ii) If in its own eternal existence the infinite universe obtains some state (some death), then it receives this state (this death) countless many times and the initial moments of these states (these deaths) are unbounded from below and from above in the time axis. Besides
(iii) We do a review of some results, connected with the occurrence and the development of the universe.
Result (i) solves one from the basic opened questions in the cosmology, namely whether the universe is finite or not. Result (ii) implies some previous results: (a) the consciousness and the perfection of the consciousness of the universe have no beginning and no end in the time, (b) the universe is cyclic and (c) if the universe has had a Big Bang, then it has had an infinite number of Big Bangs.
In this paper we consider the following basic questions for the occurrence and the development of the universe
In the paper [5] it is supposed automatically, that the time of the universe does not have a beginning and it is proved, that
“The basic question whether the universe has no beginning in time or has a beginning is not resolved in philosophy, physics and cosmology. This question is considered from a physical, philosophical and mathematical point of view. For the existence of the universe there are two main physical and philosophical models that contradict each other: model without a beginning of the time of the universe so that it undergoes an infinite number of expansions and contractions in the past and in the future and a model in which the universe has a beginning in time. In the first case the universe is called cyclic. The above two models are connected with the concept of infinity, which complicates the question of their truth.
... In particular the existent physical and philosophical models for a cyclic universe and for a universe with a beginning in time do not give still a final solution of the problem. We note also the following. By the theoretical physicists P. Steinhardt and N. Turok” It appears that we now have two disparate possibilities: a universe with a definite beginning and a universe that is made and remade forever. The ultimate arbiter will be Nature” [12]. In this way P. Steinhardt and N. Turok acknowledge in fact, that the physical and the philosophical models either for a beginning of the universe in time or for a negation of its beginning represent conjectures. For the above statement we note explicitly that the Nature cannot be an arbiter for this, what from the last two opposite assertions is true.” [7].
By the Bulgarian physicist A Polikarov the universe is stationary, that is it is unchanged in entirety [10].
In the beginning of the 21-century new cyclic physical models are obtained, connected with a dark energy and a dark matter of the universe and, in this connection new cyclic physical models are created. Such models are the model of P Steinhardt and N Turok [11,12], connected with the theory of the strings as well as the model of L Baum and P Frampton [1,2], which uses the phantom energy of the universe.
The famous English physicist Penrose, received in 2020-year joint Nobel Prize in Physics along with Reinhard Gentzel and Andrea Ghez for the black holes of the universe, creates model, namely a conformal cyclic cosmology [8,9]. In it the existence of the universe is in endless cycles called zones and with the completion of one cycle a Big Bang and a new cycle begin. This model is continued in the joint work of Gurzadyan and Penrose [3].
Since the results of [5] about a cyclic universe and about the infinite repetition of the consciousness and the perfect consciousness of the universe remain unanswered and without discussion, we made a strict mathematical interpretation of the above statements in [6].
There are many physical models in which the universe has a beginning in the time. We will not consider them in detail and we shall mention only the physical model of A. Mithani and A. Vilenkin [4].
Recently physical models for parallel universes are created, for example [13], that is for universes, which have an empty intersection among themselves and for a mega-universe. The concept for a mega-universe joins parallel, old and new universes. These concepts are admissible from a physical view point. However, they contradict to a general concept a universe: the universe is the whole matter and energy which had existed, exists and will exist and all” parallel and other universes”, if they exist, are a part of the whole universe.
We do not use the concepts a parallel universe and a mega-universe and we keep the traditional concept a universe instead of these universes.
By the Big Bang theory, that is by the Big Bang model the universe arises before 13,7-13,8 billion years from an extremely small, dense and hot fireball by a unique Big Bang and the initial inflation continues until now.
In [7] we establish, that the universe is eternal (Proposition 2.1), that is the universe does not have a beginning in the time. If we accept the original Big Bang theory, then we prove in [7], using the axiomatic method, that (i) there exists an infinite number of Big Bangs in the past and in the future (Theorem 2.2) and (ii) the universe is cyclic (Corollary 2.3). Besides we prove directly also in [7], using the axiomatic method, that the universe is cyclic without an using of a Big Bang of the universe (Theorem 3.1). The statement (i) is a negation and a correction of the original Big Bang theory. according to which the universe possesses only one Big Bang.
We note that here, as well as in [5- 7], we do not consider the physical nature and laws of the universe and we consider the universe from a neat mathematical view point.
Proposition 2.1: [7] The universe is eternal and for it there exists an infinite time axis (-∞,+∞).
“Proof. The law of the preservation of the energy holds for every closed material system. Therefore, it is applied for the universe, since the universe as a whole is a closed material system. This fact implies that the universe exists eternally and it will exist eternally. Consequently, there exists an infinite time axis (-∞,+∞).
Proposition 2.1 implies, that the universe has no beginning and that the time has exists eternally along with the universe.
The indicated time axis is called an axis t. In this axis we shall enter distinct intervals either (-∞,t) or (t,+∞) from the existing of the universe, where t is a fixed moment on the axis t.” [7].
Under states of the universe, we understand for example an expansion, a contraction, a cycle, a consciousness, a big bang, different types of death like a Heat Death, a Big Freeze, a Big Gap, a False Vacuum, a Big Rip, a Big Rift and others.
Axiom 2.1. In its own eternal existence the universe obtains some state.
Axiom 2.2. If the universe is finite, then it cannot repeat its states countless many times.
Now we can prove the following result, which solves one from the basic opened questions in the cosmology, namely whether the universe is finite or not.
Theorem 2.2: The universe is infinite.
Proof: By Proposition 2.1 for the universe there exists an infinite time axis (-∞,+∞). Suppose, that the universe is finite.
We will construct by an induction an infinite series of moments on the time axis, such that in each such moment the universe obtains some state.
Let t1 be a present moment of the existence of the universe, that is t1 is a point on the axis t. Since in the interval (-∞,t 1 ) of the axis t the universe has existed eternally, then, by Axiom 2.1, the universe has obtained some state with an initial moment a1,a1 < t1.
We proved by an induction, that there exists an infinite number states of the universe. For the completeness of the induction, we receive an infinite strictly decreasing sequence t1,t2,...,tn,... from moments. The induction is complied. The proved contradicts to Axiom 2.2. Therefore, the universe is infinite. The theorem is proved.
In the following axiom and a theorem, we shall designate with A an arbitrary state of the universe.
Axiom 2.3: In the own eternal existence the universe has obtained a state A.
Analogously to Theorem 2.2, we shall prove the following result.
Theorem 2.3: If in its own eternal existence the infinite universe obtains some state, then it receives this state countless many times and the initial moments of these states are unbounded from below and from above in the time axis.
Proof: We will construct by an induction an infinite and unbounded from below and from above series of moments on the time axis, such that in each such moment the universe obtains the indicated state.
The induction is complied. The theorem is proved.
Since a death of the infinite universe is a state, then we receive from Theorem 2.3, for example, the following result.
Corollary 2.4: If in its own eternal existence the universe obtains some death, then the universe receives this death countless many times and the initial moments of these deaths are unbounded from below and from above in the time axis.
Corollary 2.5: The universe is cyclic.
Proof: Since, by Theorem 2.3, the infinite universe receives countless many times every your state and the initial moments of the indicated states are unbounded from below and from above in the time axis, then we can conclude, with a substitution of state with cyclic, that the universe is cyclic. The corollary is proved.
Besides, Theorem 2.3 implies the following previous results.
Corollary 2.6: The consciousness and the perfection of the consciousness of the universe have no beginning and no end in the time.
Corollary 2.7. If the universe has had a Big Bang, then it has had an infinite number of Big Bangs.
This paper and the works [5-7] imply that the following results are proved.
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2.Baum L, Frampton P (2008) Entropy of Contracting Universe in Cyclic Cosmology. Modern Physics Letters A 23: 33-36.
3.Gurzadyan, VG, Penrose,R (2013) On CCC-predicted concentric low-variance circles in the CMB sky. Eur Phys J Plus 128: 22.
4.Mithani A, Vilenkin A (2012) Did the Universe have a beginning? https://arxiv.org/abs/1204.4658.
5.Mollov T Zh (1965) On the problems of the Forms of Motion in the Universe (Bulgarian), Philosophical Thought, No 2, year XXI. pp: 116-119.
6.Mollov T Zh (2009) Consciousness and Cyclicity of the Universe. International Journal of Pure and Applied Mathematics 52: 687-692.
7.Mollov T (2024) Correction of the Big Bang theory and cyclic universe. SCIREA Journal of Mathematics 9: 1-10.https: //doi.org/10.54647/mathematics 110418.
8.Penrose R (2006) Before the Big Bang: An Outrageous New Perspective and its Implications for Particle Physics. Proceedings of the EPAC 2006, Edinburgh, Scotland pp: 2759-2762.
9.Penrose R (2010) Cycles of Time: An Extraordinary New View of the Universe. https://www.scirp.org/reference/referencespapers?referenceid=1650445.
10.Polikarov AP (1955) On the Infinite Existence of the Universe (Bulgarian), PhD Thesis, Sofia.
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12.Steinhardt P, Turok N (2001) A Cyclic Model of the Universe. Science 296: 1436-1439.
13.Tegmark M (2003) Parallel Universes. Scientific American 288: 40-51.
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