COSMOLOGY. DIFFERENCES AND HIERARCHY OF INFINITE SETS BY THE EXAMPLE OF

MULTIDIMENSIONAL SPACES AND SOME RELATIONS OF THESE SPACES, AND A BIT ABOUT THE

UNIVERSE

1. For simplification, we discuss only flat (Euclidean) spaces (S) here.

1.1 The set object is the 0-dimensional space point; the space dimensionality is denoted by n;

infinity of points is denoted as i; thus, full denomination of the infinity of points in the

multidimensional space is nSi.

1.2 The differences of these sets are the levels. The level of a limited unidimensional space, the

straight-line segment, will be the entry level taken as the measurement unit. Full denomination

of its set of points is 1Si1/0, where the figure before the slash is a level, and the figure after the

slash is the number of unlimited (infinite) measurements.

1.3 An infinite straight line can be subdivided into segments; their infinity level will correspond to

the level of the set of points in the segment, as will be shown in par. 2.1. Thus, the level of such

straight lines increases by one unity and is denoted as 1Si2/1.

2.1 A bi-dimensional S is a plane. The plane bounded by both dimensions is a section. Let us

conceive it as a line of closely pressed segments and intercept all of them, from the first to the

last one, by a segment. All its points will also be the points of interception with such segments;

consequently, the level of the set of segments corresponds to the level of the set of points in the

segment and will look like 1Si1/0, whereas the level of the set of points increases by one unity

and will be 2Si2/0.

If someone like “Maxwell’s demon” could have succeeded to take the section into separate

segments and make a straight line of them, the line would have been infinite.

Let us introduce the correspondence sign (no equal sign is applicable here), ↔, and we can

record that 2Si2/0 ↔ 1Si2/1.

2.2 A bi-dimensional S with one infinite dimension is an infinite strip. Similarly to par. 2.1, let us

conceive it as the closely pressed infinite lines intercepted by a segment. It is evident that the

level of Si strip corresponds to the Si level of such line increased by one unity, i.e. 2Si3/1.

2.3 A bi-dimensional S with two infinite dimensions is an infinite plane.

We will mentally subdivide it into strips of any final width and intercept all of them by an infinite

line, subdivided into segments at the strip boundaries. According to par. 2.1, i level of segments

at this line is taken as a unity. Hence, the infinite plane level corresponds to the strip level

increased by one unity and is denoted as 2Si4/2.

3.1 A tri-dimensional S is 3S. 3S bounded by three dimensions is a volume. We can conditionally

subdivide it into the closely pressed sections and intercept all of them by a segment. Their set

corresponds to the set of the segment points, the section level increases by a unity, and the

volume level becomes 3Si3/0.

3.2 A tri-dimensional S with one infinite dimension is an infinite tunnel. Similarly to the above

described, we increase the (2Si3/1) strip level by a unity and have the tunnel level as 3Si4/1.

3.3 A tri-dimensional S with two infinite dimensions is an infinite layer. Similarly to par. 2.3, we

subdivide it into tunnels, etc., and have its level as 3Si5/2.

3.4 An infinite tri-dimensional S. Using the known technique, we subdivide it into layers and have

3Si6/3.

4. Basing on the above said, we can compose a table of infinity levels in multidimensional space

points:

0-dimensional S is 0Si0/0.

A unidimensional S is 1Si/0 – 1Si2/1.

A bi-dimensional S is 2Si2/0 – 2Si3/1 – 2Si4/2.

A tri-dimensional S is 3Si3/0 – 3Si4/1 – 3Si5/2 – 3Si6/3.

A four-dimensional S is 4Si/0 – 4Si5/1 – 4Si6/2 – 4Si6/3 – 4Si8/4.

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n-dimensional S is nSin/0 – nSi(n + 1)/1 – nSi(n + 2)/2 …. nSi(2n – 1) /(n – 1)-nsi2n/n.

5. Increase in level of the infinity of points of a multidimensional space by one unity is not an

arithmetic operation but the indication that the augmentable value can be contained in the

space in any amount and within the bounds of the first-level infinite set.

6. If multidimensional spaces of more than three dimensions do exist really, then the existence

reality of intermediate states, except completely limited and completely infinite states, is even

less probable.

7. Certain properties of multidimensional spaces (nS).

7.1 Taking volume (3S) and plane (2S) as an example, we can conclude that nS cannot stay inside (n

– 1)(S), independently of their sizes.

Consequently, if (n > 3) do exist in the entire infinite multispace (MS) (we can further assume

that they do exist), then this space represents a certain hierarchy, where all n1S can stay only

inside (n > n1)S.

7.2 By examining (1S) line in (2S) plane and …. plane in (3S) volume it becomes evident that the

increase in their dimensions, expansion of less dimensional spaces to more dimensional, can be

unlimited by any dimensions of the latter, because only still more dimensional nS can be present

(see par. 7.1) by their bound overrun.

Hence, if our Universe, during billions of years already, expands from a point to its present-day

dimensions, then most definitely, it might take place in the four-dimensional space at least.

7.3 In any nS any limited number of (n – 1)S can be found (see par. 5).

8. The most interesting feature is the intersections of spaces.

Here, we will examine only the simplest case, i.e. two nS are intersected in (n + 1)S.

Two (1S) lines, intersecting in (2S) plane, form (0S) point.

Two (2S) planes, intersecting in (3S) volume, form (1S) line.

8.1 Consequently, two nS will intersect if, being joined together, they leave the bounds of their

dimension, i.e. it takes place in (n + 1)S at least.

8.2 When two nS are intersected, their intersection point, i.e. the geometric locus of common

points, is (n – 1)S.

9. Hence, two 4S, by intersecting at 5S (or at a more dimensional S), form 3S in their intersection

place.

9.1 Let us assume as follows:

9.1.1 The entire infinite multispace (MS) is filled with certain non-structured, dimensionless “dark”

energy (DE) of uniform concentration in all spaces.

9.1.2 Two bounded 4S, moving with definite speed in 5S, collided and began intercepting.

9.1.3 Generation of a tri-dimensional space out of two four-dimensional ones, and doubling of DE

therein is accompanied by the generation of tri-dimensional matter out of the “excessive” DE.

9.1.4 The entire process is extremely violent and is similar to the explosion of an assumed singular

point in the shape of a very-high-temperature bundle of energy. This bundle must increase very

rapidly, depending on the convergence rate and the surface shapes in the convergence start

point.

9.1.5 In addition, this bundle must rotate very quickly, due to the process non-uniformity in its various

sections, with the same prompt chaotic variation of its rotation axis, until the mass elementary

particles as well as accompanying inertia, gravity and centrifugal force emerge.

It is known that the Universe possesses certain condensation of celestial bodies along the

straight line, its length being millions of light years. Probably, as a result of the rotation axis

stabilization and the rotation deceleration, certain matter condensation was formed in the axis

area, and in continuation, some celestial bodies were formed. The Universe might continue its

rotation even now, and this condensation is the rotation axis.

9.2 Probably, the presence of a certain “dark” energy in the Universe that is now assumed in

astrophysics as expanding the Universe, is the remnant of energy following DE doubling after 3S

has emerged out of two 4S parts. If the concentration of the remaining DE was higher than the

concentration in the entire Multispace, it possibly expanded and will further expand the

Universe space until the level of this concentration becomes equal in the entire MS.

10. I am convinced that real matter in all its types and states can be only tri-dimensional in the tri-

dimensional space (such “tridimensional chauvinism”), and for this reason, generation of

alternative dimensions in the special intercepts should occur differently.

11. If a hypothesis on singular point explosion seems the only potential explanation of all facts that

are insomuch known about the Universe, then everything narrated here and concerned with the

Universe emergence are nearly groundless speculative assumptions (the condensation line of

celestial bodies, the Universe expansion and the presence of some “dark”, not yet revealed

energy, can serve only as a very shaky proof).

11.1 But astronomy and astrophysics are being incessantly progressing, their engineering modalities

improve, and new data appear and will appear all the time that might not correlate with the

generally accepted hypothesis.

The underlying assumptions consist of several components, whose parameters can be selected

and changed, some new ones being added, and some unnecessary ones rejected, thus creating a

hypothesis on the reasons of the Universe emergence, its background. Maybe, the final result

will greatly differ from our observations – but it is not bad at all, the beginning is what is

important.

Semion PEVZNER, engineer

20/28 Tavor St.,

Natzeret Illit 17602, Israel

Contacts:

S. Pevzner (in Russian): 972-53-3348103

972-4-6020742

P. Pevzner (in English): 972-50-7617213

e-mail:

[email protected]