Ilya Prigogine, the author of the scientific work "Order from Chaos" in Chapter 8 states: "Poincare proved that any closed dynamical system eventually returns to an arbitrarily small neighborhood of its initial state. In other words, all states of a dynamical system are repeatable in one way or another”. This means that both space and time are subject to cycles.
Until recently, another statement by Henri Poincaré remained a hypothesis. Poincaré's hypothesis was considered one of the great mathematical mysteries that touch upon the problems of the physical and mathematical foundations of the universe.
Grigory Yakovlevich Perelman.
Translated from the mathematical to the usual statement of the great Henri Poincaré sounds like this: any infinity that has three dimensions and tends to one point becomes like a sphere.
The method of proof, applied by Grigory Perelman, is that for geometric objects one can find the equation of smooth variation. The original surface during the changes will smoothly transition into the sphere. The proof of the hypothesis is that, bypassing the intermediate moments, one can immediately look into infinity, at the very end of evolution, having found a sphere there.
Let's apply this formulation (as already proved by Grigory Yakovlevich) to our physical space.
Curved space.
The expanses of the Universe are endless, and its space is three-dimensional. Over time, it gets harder. But the mathematical infinite set can consist of both an infinite number of kilometers and an infinite number of hours.
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Mathematically, an infinite set can only tend to a point that is not this set. Otherwise, such a point would already be included in this set. Therefore, every member of any infinite set must in some way strive to establish a connection with one single point.
According to Euclid, a point is a formation that has no parts. Regardless of its size. Nobody forbids having a point the size of a galaxy. The main thing is that at this point it is impossible to select individual parts. A point is something whole or a unit, which can be denoted by the letter A.
After the replacement, the text of the hypothesis will look like this: Infinite space from A-1, A-2, A-3…. to A-∞, each point tends to curl up around a single A.
The whole space folds around one point. But, counting does not end there, but leads to an increase in the surface of "point A", layering around it all the following kilometers of space. Layering members of space leads to the concept of time, counting the number of new layers of space.
If we take each spatial layer as a quantum of time and designate it B, then we can see that the countdown from B-1, B-2, B-3 … to B-∞ also turns out to be infinite.
It is infinite and strives to the starting point, strives to become like a sphere!
This conclusion removes the need to reverse time when traveling into the past. It is replaced by fast forward movement in time. Without violating the second law of thermodynamics (about the eternal growth of the entropy of closed systems).
Perelman proved the fundamental possibility of finding the coordinates of any point we need in the space and time of the cyclic Universe, even if only in mathematical theory.
Traveling to the past, in cyclical time, is the same as traveling to the distant future. Ahead are dinosaurs, dark ages and me, who wrote this text yesterday.