Sunday, May 1, 2011

12. Choosing your concepts of the external physical reality besides internal existential reality


Personal development evolves our consciousness and awareness. But consciousness and physical body are connected. "As within so with  out" says the ancient quote. Personal development would not be significant to many if it did not change our material life too. And it certainly changes our physical body too.  The changes are two ways too: Consciousness changes the body and physical reality, and the changed body and physical reality helps the spirit to proceed with the changes of the consciousness further. It is like the arrangements in a room and the emotions: Positive emotions help us put a liking order to the furniture and house ware in the room that we live, and the arranged furniture etc helps us further to have nice emotions as we live in the room. So it is important to have the right perspective and ideas about physical reality. The more the consciousness evolves, the higher the frequency of the spin of protons, electrons, and neutrons in our body. And a body with higher frequency of the spin of protons, electrons, and neutrons helps the consciousness to evolve further. A body with higher frequency as above entertains easier feelings closer to the metaphorical "gaseous" or "liquid" states too and not only to the "solid" state. 
Here we list 12 principles or laws of physics in the new millennium that enhance  the known in the academic world laws of physics hitherto.
For more about it see
http://thenewuniversalattraction.blogspot.gr/2012_12_01_archive.html





Here is a list of past Physics assumptions that I consider that the 21th century and the new millennium physics has already started and will eventually turn all of them false
1) The inertial mass of bodies ( of constant amount of molecular matter) at low speed (non-relativistic) cannot be decreased below the inertia of the rest mass.
2) All matter starts with protons, neutrons, electrons. In other words, there are not smaller permanent particles (Quantum particles are excluded as they are not permanent)
3) Nothing goes faster than photons
4) All macroscopic electromagnetic interactions are described with the linear equations of Maxwell.
5) All forces acting on laboratory macroscopic objects at low speed (non-relativistic) are of the next 5 types a) Inertial, b) by contact with other material bodies made from protons, neutrons, electrons, c) Newtonian gravitation forces d) Maxwell's electromagnetic forces e) no other type of forces.

 Inspired by the  12 universal laws  from  Milanovich and McCunes book "The Light Shall Set You Free"(1998) I resume some of the principles of my research in physics  but without the mathematical equations and details,   to 12 universal physical laws for the new millennium physics. Six of the laws 1,5,7,8,9,12 deal with the vertical interplay of the levels, and the other six 2,3,4,6,10,11 deal with horizontal interplay at each level.



1. The Law of levelled physical material reality

1.0 What we consider as physical material reality is subject and conditioned to our development in consciousness our hidden beliefs and our scientific operational competence and development. E.g. in ancient Rome the protons. neutrons, and electrons could not  stand a chance to be considered as physical reality or existent at all. What was considered as physical reality was only gross pieces and aggregations of matter made by protons, neutrons, and electrons. Anything else would be shear philosophical or metaphysical. 
1.1 After the 19th century we are universally mature enough to accept that any form of material existence consists of elementary indivisibles. 

1.2 The content of this law is that all mater does not starts from protons, neutrons and electrons only. What we call traditionally as gravitational field, electromagnetic field, etc is accepted to have energy density in space even inertial density in space (there are classical college physics experiments with the momentum and inertia of the electromagnetic field, and similar in Einstein’s physics for the gravitational field). Therefore these  fields have material existence, but they do not consist from protons electrons and neutrons! We would contradict ourselves if we would  not assume that they consists too from permanent indivisible particles. Such particles have to be permanent and extremely small (e.g. 10^(-36) times smaller than the electron) otherwise quantum and nuclear physics would have pinned them down and would know them. (e.g. the neutrino that can cross the earth and not hit a single earth’s atom is 10^(-6) times only smaller than the electron. All particles of nuclear physics and quantum mechanics besides the proton, neutron, electron are of very short life duration and not permanent). This material reality of the fields we call the 4th material density or resolution, or 1st aether, or 2nd order  micro-atomic matterial reality. There exist also in a similar way the 5th material density.  The content of this law is also that the 3rd material density as well as the 4th and 5th material densities consists from particles. Also that all the material physical reality consists of superposition of such material densities.

1.3 As far as I know our experiments go up to the 4th material density and barely only go up to the 5th material density. But they may as well exist 6th or more. Some say it exists till 11 (or 9 if you strat counting levels from the 3rd density) but as we mentioned at the beginning, this depends on the evolution of the civilisation and the consciousness of the human beings in it. An obvious question is why we start counting from 3. This is so, as we are counting the matter that is granulated in to the planets and stars in aggregation as 2nd material density. The next question would be what is the 1st material density. We consider the overall world of galaxies and everywhere that classical  light can go as one celestial body or…..particle assuming that there more such particles creating the 1st material density. 



2. The Law of Vibration
This Law states that everything in the Universe (e.g. protons, neutrons, electrons of the 3rd material density) vibrates.  The spin of protons, electrons, and neutrons has a vibration frequency. The same with an atom that may have many frequencies like a musical chord (which is also the basis of  smelling: Our sensing cells of smelling detect the atom’s chord-vibrations ). This holds also for the basic particles of the 4th and 5th the material density. So each material density has its own vibration characteristic frequency. Depending on the vibration source, the vibration wave may propagate in many different material densities with different speeds. In the 4th material density there is also “sound” as compression wave, but it may have the speed of light.

3. The Law of Action
This law in the 3rd material density is known and analysed as the 3 Newtonian laws:
a) The conservation of momentum a) The law of force and acceleration, c) the law of action reaction.  Later than Newton scientists proved based on the previous 3 Newtonian laws, the law of conservation of energy. Lagrange and Hamilton were also able to derive the momentum and energy conservation from
the  law of stationary action

4. The Law of Correspondence
This Law states that the known classical principles and laws of physics that explain the physical world  at the 3rd density- energy, waves, vibration, and motion - have their corresponding principles in the 4th and 5th material density. "As above, so below" was the ancient quote. As in the 3rd material density there are the 3 material states , gaseous, liquid, and solid, so there are in the 4th and 5th. E.g. the gaseous and liquid 4th density mater can go through the gaseous, liquid and solid 3rd density mater (e.g. the 4th density gaseous electromagnetic waves of a mobile telephone device can go through air, water, and sold walls). But the 4th density solid mater will hit hard the 3rd  solid mater and would not go through. It seems though that it is difficult technology to isolate or produce such 4th or 5th density solid state mater.

5. The Law of Cause and Effect
The contact causalities or particle collision causalities or horizontal causalities that we know in the 3rd material density  (e.g. law of action and reaction) hold also in the 4th and 5th material density.

6. The Law of Compensation and Deeper causalities
Besides the horizontal contact causalities within a material layer hold also the vertical causalities among material layers. The top-down flow (from 5th density to 4th, to 3rd density) is relevant to our (in-out) creative abilities.

7. The Law of Attraction
This law is essentially what Newton started studying in his universal attraction (later called gravitation) and his inverse square law. This law can only be fully revealed and the inverses square law explained only if more than the 3rd material density is included. This is the process of discovering the Unified Field Theory which is nothing else than the equations of the gaseous 4th material density. Nevertheless in the 4th material density exist also liquid and solid states. The 0th key to start understanding the mechanism behind the inverse square law of Newton, in universal attraction or gravitation is the identification: The Newtonian gravitational scalar potential φ is proportional to the aether temperature (or gaseous 4th density matter temperature)
Both the presence of matter and infrared solar radiation contribute to the creation of aether heat that in its turn creates gravity. But the factor of infrared solar radiation seems to be the major. 

8. The Law of Perpetual Transmutation of Energy
The material densities (3rd, 4th, 5th) are in continuous energy exchange through contact, friction, vibration, etc. There is a natural flow of energy  from the 3rd density to the 4th density through friction of the spinning and rotating protons, neutrons and electrons in an atom. This transfers heat from the 3rd material density and is creating heat in the 4th material density. Also any  3rd density material object moving within the gaseous 4th density matter (aether), creates a  drag-force and carries way partially and locally only  the  gaseous 4th density matter (aether). This may resemble the De Broglie’s law , but its mathematical details are different.

9. The Law of Relativity

The content of this law is that the space measurements and time measurements, if attached to a particular material density only are relative not absolute. So the space and time as measured in the familiar 3rd material density only, may not be the same when measured in e.g. the 5th material density only. For a more unified measurement of space and time, many material layers have to be coordinated. E.g. we might measure time with duration that our bodies change and replace all cells, that in earthly human beings for the current level of evolution is about 7 years. And this because in our human existence we also define a coordination between the various material densities (e.g. 3rd material density, the 4th material density , the 5 material density.) This law has also a further meaning. Our level of evolution of the consciousness defines what is external material reality  (which has many layers, e.g. coarse matter by protons, neutrons, electrons, and fine matter like the electromagnetic field or aether) and what is internal reality (which again has layers e.g. like  mind and spiritual awareness and intention). As we evolve some of the internal layers become external, and new deeper reality becomes internal. 



10. The Law of Polarity
10.0 The content of this law is that as in the 3rd material density  there is the emergence of the electric positive, negative and neutral, so is in the 4th and 5th material density. Therefore there are triads of micro-electrons micro-protons, micro-neutrons in the 4th material density and triads of nano-protons, nano-electrons and nano-neutrons in the 5th material density. These triads create these densities as the triad of proton neutron electron creates the familiar 3rd material density.

10.1  In the 3rd material density, it is traditionally described with the system of linear equations of Maxwell’s electricity. What we know today as the Maxwell’s electromagnetic field was called originally by Maxwell himself as the Electro-magnetised  aether.  Aether is a gas formation of the 4th material density.  Nevertheless, if after this law we introduce besides the polarity of electrons, protons, neutrons, the corresponding polarity in the gaseous 4th density (aether) and utilise the non-linear Navier-Stokes equations of the gases (that are derived by the energy and momentum conservation, but now applied after law 3 of action to the 4th material density) we may correct the Maxwell equations of electromagnetism to non-linear equations. This is part of the process of discovering the Unified Field Theory, which is nothing else than the theory of the gaseous 4th material density with its intrinsic polarity. This Unified Field Theory is not what was much discussed as Unified Field theory at the quantum scale of strong, weak and electromagnetic interactions which was never attained by quantum physics.  It is a different laboratory-scale unified field theory that unifies Universal Attraction of Newton, and Electromagnetism of Maxwell. In addition a 3rd new laboratory-scale and macroscopic field is introduced that underground physics sometimes calls anti-gravity, but it is rather a the gravitodynamics (or aethrodynamics) as contrasted to Newtonian gravitostatics, that escape even Einstein’s gravitation. The more complete understanding of the physics of the gaseous 4th material density seems that it may lead to the discovery of the ability to significantly reduce the inertia mass of slow moving (non-relativistic speeds) bodies without reducing their quantity of matter (as number of atoms).

Remark: Once we have discovered the macroscopic Unified Field Theory, the free energy or overunity magnetic devices have full and simple explanation. The same for more overunity devices (e.g. based on Oxyhydrogen or water). Not only they are explained but also understood as forms of renewable energy.   They are of course explained and computed within the energy conservation and exchange of the  of 3rd and 4th material densities simultaneously. Exclusion of the 4th material density and restriction to the 3rd material density keeps them unexplained and inaccessible to Academic Science explanations.
There are persons for whom the aether vision has opened.  They see the aether glow around objects and bodies. They can see also the weak colored light of aether around the aether-centers (charkas) of the human body. But maybe a day will come that most of the people will have aether-vision. All that it takes is to have higher spin frequency of our electrons, protons and neutrons in our physical body so as to have the aether vision opened. In other words higher energy too, and an easier interplay of consciousness and matter in us.
The key to start really understanding the electromagnetism as the dynamics of the electromagnetised aether is to identify the classical scalar and vector potentials of Maxwell’s electromagnetism to fluid parameters of aether. The 1st key is : The scalar electromagnetic potential a0 is proportional to the pressure of the non-neutral aether (or non-neutral gaseous 4th material density)
The 2nd key is:The vector electromagnetic potential A is proportional to the vector of momentum of the non-neutral aether ( or non-neutral gaseous 4th material density).





11. The Law of subjectivity

Human beings have and are developing their bodies not only in the 3rd material density but also in the 4th and  5th.  The classical literature of acupuncture or aetheric energy centres (=the chakras) exists because we have a living and evolving body in the 4th material density. It seems that our cells and  DNA exists and functions (also as field vibrations transmitter and receiver) not only in the 3rd material density (or 1st order material reality) but also in the 4th and 5th ( or 1st and 2nd aether, or 2nd and 3rd order micro-atomic material reality) By changing our beliefs about  the true reality of our living bodies a new order of identity and self emerges for us. The frequency of the spin of the protons, neutrons, electrons in our body (3rd density) as well as the frequency of the spin of the permanent particles of aether or aetherons  (4th density) depends on our level of consciousness and mind, and does affect the way that  matter (3rd density matter) interacts with the classical electromagnetic and gravitational fields, or aether (4th density matter) (And of course with the 2nd aether (or 5th density matter) which for the time being our civilization is not aware as external physical reality, although as human beings we are aware of it as internal reality and experience .)



12. The Law of Creation

The content of this law is that the physical leveled material reality (and e.g. its protons, neutrons, electrons etc) has been created, in the same way that we consider plants and animals as created. Normally (with probably some exceptions) for a material layer to be created, a finer material layer must exist in advance. Thus for the known material layer made from protons, neutrons and electrons, to be created, a finer material layer (like aether or sub-aether) must exist. Our times  seem to be rare times, when (decade of 1990's) a new material layer has been created, through a simultaneous emission from all black holes in all the galaxies of  a particular frequency. (counting from the visible material layer made from protons, neutrons, electrons as 1st, then this new material layer  is the 10th)







Natural name
Other name in the web
Usual name in the web
Other names in books
2nd frequency macro-resolution reality
1st density physical reality
1st dimension  reality
Many-worlds reality
1st frequency macro-resolution reality
2nd density physical reality
2nd dimension  reality
The cosmic-ball or manifold
1st frequency micro-resolution reality
3rd density physical reality
3rd dimension physical reality
Material reality
2nd  frequency micro-resolution reality
4th density physical reality
4th dimension  reality
1st Aetherial reality
3nd  frequency micro-resolution reality
5th density physical reality
5th dimension  reality
2nd Aetherial reality
4th  frequency micro-resolution reality
6th density physical reality
6th dimension  reality
3rd Aetherial reality
5th  frequency micro-resolution reality
7th density physical reality
7th dimension  reality
4th Aetherial reality
6th  frequency micro-resolution reality
8th density physical reality
8th dimension  reality
5th Aetherial reality
7th  frequency micro-resolution reality
9th density physical reality
9th dimension  reality
6th Aetherial reality
8th  frequency micro-resolution reality
10th density physical reality
10th dimension  reality
7th Aetherial reality
9th  frequency micro-resolution reality
11th density physical reality
11th dimension  reality
8th Aetherial reality
10th  frequency micro-resolution reality
12th density physical reality
12th dimension  reality
The new-born reality during the 90s















As I perceive it "miracles" are to remind to us that the creative process is not only in the causal sequence of its events, of more than 80% from causal events withing the same physical density (or resolution) e.g. 3rd, and less than 20% among different densities (in other words rather horizontal creative process, like traditional technology, which is also collective) but can be also, conversely more than 80% from causal events among different physical densities and less than 20% from causal events in the same physical density (in other words vertical creative process), plus the use of personal or group power of will.

AXIOMATIC SYSTEM OF MATERIAL ATOMIC  EUCLIDEAN GEOMETRY (without the infinite) 


We have as initial concept of objects 

a) The High resolution or precision  points  , or invisible points or material atoms 

b) The Low resolution or precision points, or visible points.

The Low precision or visible congruence of visible linear segments AB of two visible points points A, B, and the Low precision or visible congruence  of angles (to be defined below) 

The High precision or invisible congruence of invisible linear segments AB of two invisible points points A, B, and the High precision or visible congruence  of angles (to be defined below) 

c) The visible lines and the invisible  atomic lines

d) The visible planes  and the invisible  atomic planes

c) We may apply the digital set theory on the points of the digital Euclidean geometry

d) And of course we may apply the digital formal logic to make arguments and proofs.

e) Besides the congruence equivalence relations we have the next   initial relations 

Among visible or invisible  elements. 

An invisible point A belongs to a visible point B , denoted by A ε B

A visible (or invisible)  point A    belongs to a visible (or invisible respectively) line L   ,    denoted by A ε L

A visible  (or invisible) point A   belongs to a visible (or invisible respectively) plane P   ,    denoted by A ε P

 A visible (or invisible) line L belongs to a visible (or invisible respectively) Plane P, denoted by  L ε P


A visible (or invisible) point A is between two visible (or invisible respectively) points B, C. 

We design 6 groups of axioms
In the next axioms the term point refers to visible or low precision point and the line an plane to visible line and plane or refers respectively to the invisible material atomic points , lines an planes. 


The incidence of invisible to visible elements is intended to define a homomorphism of the geometric structure of invisible material atoms to visible, relative to the relations, belonging ε, order < and congruence =

I. Incidence

  1. For every two points A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of “contains,” we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: “The lines a and b have the point A in common,” etc.
  2. For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
  3. There exist at least two points on a line. There exist at least three points that do not lie on a line.
  4. For every three points ABC not situated on the same line there exists a plane α that contains all of them. For every plane there exists a point which lies on it. We write ABC = α. We employ also the expressions: “ABC, lie in α”; “A, B, C are points of α”, etc.
  5. For every three points ABC which do not lie in the same line, there exists no more than one plane that contains them all.
  6. If two points AB of a line a lie in a plane α, then every point of a lies in α. In this case we say: “The line a lies in the plane α,” etc.
  7. If two planes α, β have a point A in common, then they have at least a second point B in common.
  8. There exist at least four points not lying in a plane.
  9. There is a special point denoted by O, which is called the center of the space, and a special linear segment OA, which is called the unit of measurement of lengths in the geometry.  
  10. For every invisible point A, there is a visible point B, so that A belongs to B.
  11. For every invisible line a, there is a visible line b , so that a belongs to b.
  12. For every invisible plane a, there is a visible plane b , so that a belongs to b.
  13. The relation of belonging from invisible elements is transferred as holding to a belonging of corresponding visible elements that the invisible elements belong to.
  14. Two invisible points A, B belong to the same visible point C is an equivalence relation among the invisible points.

II. Order

  1. If a point B lies between points A and CB is also between C and A, and there exists a line containing the distinct points A,B,C.
  2. Of any three points situated on a line, there is no more than one which lies between the other two.
  3. Pasch's Axiom: Let ABC be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points ABC. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.
  4. Every line a, has two points ω1 and ω2 so that every other point of the line , lies between ω1 and ω2 . We call them the end points of the line. All end points of lines define a spherical surface with center the point O (center of the space). All end points of lines of a plane define a circle, with center the center 0 of the space. 
  5. The relation of order from invisible elements is transferred as holding to an order of corresponding visible elements that the invisible elements belong to.

III. Congruence

  1. If AB are two points on a line a, and if A′ is a point upon the same or another line a′ , and as long the A'ω is larger than the AB, then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing AB ≅ A′ B′. Every segment is congruent to itself; that is, we always have AB ≅AB.
    We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in at least one way.
  2. If a segment AB is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″; that is, if AB ≅ A′B′ and AB ≅ A″B″, then A′B′ ≅ A″B″.
  3. Let AB and BC be two segments of a line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ having, likewise, no point other than B′ in common. Then, if AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.
  4. Let an angle ∠ (h,k) be given in the plane α and let a line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one ray k′ such that the angle ∠ (hk), or ∠ (kh), is congruent to the angle ∠ (h′k′) and at the same time all interior points of the angle ∠ (h′k′) lie upon the given side of a′. We express this relation by means of the notation ∠ (hk) ≅ ∠ (h′k′).
  5. If the angle ∠ (hk) is congruent to the angle ∠ (h′k′) and to the angle ∠ (h″k″), then the angle ∠ (h′k′) is congruent to the angle ∠ (h″k″); that is to say, if ∠ (hk) ≅ ∠ (h′k′) and ∠ (hk) ≅ ∠ (h″k″), then ∠ (h′k′) ≅ ∠ (h″k″).
  6. If, in the two triangles ABC and A′B′C′ the congruences AB ≅ A′B′, AC ≅ A′C′, ∠BAC ≅ ∠B′A′C′ hold, then the congruence ∠ABC ≅ ∠A′B′C′ holds (and, by a change of notation, it follows that ∠ACB ≅ ∠A′C′B′ also holds).
  7. The relation of congruence from invisible elements is transferred as holding to a congruence of corresponding visible elements that the invisible elements belong to.

IV. Parallels 

  1. Euclid's Axiom ):Let a be any line and A a point not on it. Then there is at least one line b in the plane, determined by a and A, that passes through A and does not intersect a. 

V. Continuity

  1. Axiom of Archimedes. If AB and CD are any segments of two lines , then there exists a number n such that n segments CD constructed contiguously from A, along the ray from through B, will pass beyond the point B, as long as Dω is at least larger than twice the AB.
  2. Axiom of line completeness. An extension of a set of visible points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible. (This axiom states that there is always a fixed and constant number of visible points on any  line.) 

VI. Resolution (or Density)

  1. Axiom of sufficient high  resolution or density Let a line a passing from the center of the space O and  the units of measurements  OA on it,  and let ω(a), Ω(a), demote the finite cardinal number of visible and invisible points that belong to a. And let  ω(n) be the size of the model of the natural numbers constructed on the line a  through congruence and the above axioms. Then it holds that  
            2^ω(n) <= ω(a)
            2^ω(a) <=Ω(a)  



Let is denote by ω(E) the finite cardinal number of all visible points, and by Ω(E) the finite cardinal number of all invisible points. If a line a passes through the center of the space O, and let us denote by ω(n) is   the size of the model of the natural numbers constructed on the line a and the unit of measurements OA,  through congruence and the above axioms.



If two set-theoretical models M1, M2 of the above Holographic Euclidean geometry have the same , ω(n), ω(E),Ω(E) , then they are isomorphic. 

In other words , the above axiomatic system is categorical up to the units of measurements and visible and invisible density

Finally we give here still an alternative axiomatic system, in which we have two levels of precision and invisible points  , and we endowed the invisible points , with a similar geometric structure ,  as the visible points, lines and planes.

The first high precision invisible points are also called material atoms, while the second hither precision invisible points are called ethereal atoms. The second higher precision invisible points , lines and planes  are called ethereal.  The geometric structure of the invisible ethereal   points, lines and planes is again, that of incidence, order, congruence ,parallelism, and resolution.



AXIOMATIC SYSTEM OF ETHEREAL ATOMIC EUCLIDEAN GEOMETRY (without the infinite) 

The distinction of matter from aether is similar to the distinction of matter , and the electromagnetic and gravitational field, which are properties of a second atomic material layer called in history aether. 



We have as initial concept of objects 
a) The Highest resolution or precision  points  , or invisible points or ethereal  atoms 
b) The High resolution or precision  points  , or invisible points or material atoms 

c) The Low resolution or precision points, or visible points.

The Low precision or visible congruence of visible linear segments AB of two visible points points A, B, and the Low precision or visible congruence  of angles (to be defined below) 

The High precision or invisible congruence of invisible atomic linear segments AB of two invisible points points A, B, and the High precision or visible congruence  of angles (to be defined below) 

The Highest precision or invisible congruence of invisible ethereal linear segments AB of two ethereal points points A, B, and the Highest  precision or visible congruence  of ethereal angles (to be defined below) 

c) The visible lines , the invisible  material atomic lines, the invisible  ethereal atomic lines, 

d) The visible planes  ,  the invisible  material atomic planes ,  the invisible  ethereal atomic planes


c) We may apply the digital set theory on the points of the digital Euclidean geometry

d) And of course we may apply the digital formal logic to make arguments and proofs.

e) Besides the congruence equivalence relations we have the next   initial relations 

Among visible or invisible  elements. 

An material invisible point A belongs to a visible point B , denoted by A ε B

An ethereal invisible  point A belongs to a invisible material point B , denoted by A ε B

A visible (or invisible material or ethereal )  point A    belongs to a visible (or invisible material or ethereal respectively) line L   ,    denoted by A ε L

A visible  (or invisible  material or etherealpoint A   belongs to a visible (or invisible material or ethereal respectively) plane P   ,    denoted by A ε P

 A visible (or invisible material or ethereal ) line L belongs to a visible (or invisible  material or ethereal respectively) Plane P, denoted by  L ε P


A visible (or invisible  material or ethereal) point A is between two visible (or invisible  material or ethereal respectively) points B, C. 

We design 6 groups of axioms
In the next axioms the term point refers to visible or low precision point and the line an plane to visible line and plane or refers respectively to the invisible material atomic points , lines an planes or  refers respectively to the invisible ethereal atomic points , lines an planes.


The incidence of invisible material  to visible elements and invisible ethereal to invisible material , is intended to define a homomorphism of the geometric structure of invisible material atoms to visible, and of ethereal to material relative to the relations, belonging ε, order < and congruence =

I. Incidence

  1. For every two points A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of “contains,” we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: “The lines a and b have the point A in common,” etc.
  2. For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
  3. There exist at least two points on a line. There exist at least three points that do not lie on a line.
  4. For every three points ABC not situated on the same line there exists a plane α that contains all of them. For every plane there exists a point which lies on it. We write ABC = α. We employ also the expressions: “ABC, lie in α”; “A, B, C are points of α”, etc.
  5. For every three points ABC which do not lie in the same line, there exists no more than one plane that contains them all.
  6. If two points AB of a line a lie in a plane α, then every point of a lies in α. In this case we say: “The line a lies in the plane α,” etc.
  7. If two planes α, β have a point A in common, then they have at least a second point B in common.
  8. There exist at least four points not lying in a plane.
  9. There is a special point denoted by O, which is called the center of the space, and a special linear segment OA, which is called the unit of measurement of lengths in the geometry.  
  10. For every invisible material point A, there is a visible point B, so that A belongs to B.
  11. For every invisible ethereal point A, there is a invisible material point B, so that A belongs to B.
  12. For every invisible ethereal line a, there is an invisible material line b , so that a belongs to b.
  13. For every invisible material line a, there is an visible  line b , so that a belongs to b.
  14. For every invisible ethereal plane a, there is an invisible material plane b , so that a belongs to b.
  15. For every invisible material plane a, there is a visible  plane b , so that a belongs to b.
  16. The relation of belonging from invisible ethereal elements is transferred as holding to a belonging of corresponding invisible material elements that the ethereal elements belong to.
  17. The relation of belonging from invisible material elements is transferred as holding to a belonging of corresponding visible  elements that the invisible elements belong to.
  18. Two invisible material points A, B belong to the same visible point C is an equivalence relation among the invisible points.
  19. Two invisible ethereal points A, B belong to the same invisible material point C is an equivalence relation among the invisible ethereal points.

II. Order

  1. If a point B lies between points A and CB is also between C and A, and there exists a line containing the distinct points A,B,C.
  2. Of any three points situated on a line, there is no more than one which lies between the other two.
  3. Pasch's Axiom: Let ABC be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points ABC. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.
  4. Every line a, has two points ω1 and ω2 so that every other point of the line , lies between ω1 and ω2 . We call them the end points of the line. All end points of lines define a spherical surface with center the point O (center of the space). All end points of lines of a plane define a circle, with center the center O of the space. 
  5. The relation of order from invisible ethereal elements is transferred as holding to an order of corresponding invisible material elements that the ethereal elements belong to.
  6. The relation of order from invisible material elements is transferred as holding to an order of corresponding visible  elements that the invisible elements belong to.

III. Congruence

  1. If AB are two points on a line a, and if A′ is a point upon the same or another line a′ , and as long the A'ω is larger than the AB, then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing AB ≅ A′ B′. Every segment is congruent to itself; that is, we always have AB ≅AB.
    We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in at least one way.
  2. If a segment AB is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″; that is, if AB ≅ A′B′ and AB ≅ A″B″, then A′B′ ≅ A″B″.
  3. Let AB and BC be two segments of a line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ having, likewise, no point other than B′ in common. Then, if AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.
  4. Let an angle ∠ (h,k) be given in the plane α and let a line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one ray k′ such that the angle ∠ (hk), or ∠ (kh), is congruent to the angle ∠ (h′k′) and at the same time all interior points of the angle ∠ (h′k′) lie upon the given side of a′. We express this relation by means of the notation ∠ (hk) ≅ ∠ (h′k′).
  5. If the angle ∠ (hk) is congruent to the angle ∠ (h′k′) and to the angle ∠ (h″k″), then the angle ∠ (h′k′) is congruent to the angle ∠ (h″k″); that is to say, if ∠ (hk) ≅ ∠ (h′k′) and ∠ (hk) ≅ ∠ (h″k″), then ∠ (h′k′) ≅ ∠ (h″k″).
  6. If, in the two triangles ABC and A′B′C′ the congruences AB ≅ A′B′, AC ≅ A′C′, ∠BAC ≅ ∠B′A′C′ hold, then the congruence ∠ABC ≅ ∠A′B′C′ holds (and, by a change of notation, it follows that ∠ACB ≅ ∠A′C′B′ also holds).
  7. The relation of congruence from invisible elements is transferred as holding to a congruence of corresponding visible elements that the invisible elements belong to.
  8. The relation of congruence from invisible ethereal elements is transferred as holding to a congruence of corresponding invisible material elements that the ethereal elements belong to.
  9. The relation of congruence from invisible material elements is transferred as holding to a congruence of corresponding visible elements that the invisible elements belong to.

IV. Parallels 

  1. ( Euclid's Axiom):Let a be any line and A a point not on it. Then there is at least one line b in the plane, determined by a and A, that passes through A and does not intersect a. 

V. Continuity

  1. Axiom of Archimedes. If AB and CD are any segments of two lines , then there exists a number n such that n segments CD constructed contiguously from A, along the ray from through B, will pass beyond the point B, as long as Dω is at least larger than twice the AB.
  2. Axiom of line completeness. An extension of a set of visible points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible. (This axiom states that there is always a fixed and constant number of visible points on any  line.) 

VI. Resolution (or Density)

  1. Axiom of sufficient high  resolution or density Let a line a passing from the center of the space O and  the units of measurements  OA on it,  and let ω(a), Ω1(a), Ω2(a) demote the finite cardinal number of visible , invisible material points and invisible ethereal points respectively that belong to a. And let  ω(n) be the size of the model of the natural numbers constructed on the line a  through congruence and the above axioms. Then it holds that  
            2^ω(n) <= ω(a)
            2^ω(a) <=Ω1(a)  
            2^Ω1(a) <=Ω2(a)  



Let is denote by ω(E) the finite cardinal number of all visible points, and by Ω1(E) the finite cardinal number of all invisible material points and by Ω2(E) the finite cardinal number of all invisible ethereal  points . If a line a passes through the center of the space O, and let us denote by ω(n) is   the size of the model of the natural numbers constructed on the line a and the unit of measurements OA,  through congruence and the above axioms.



If two set-theoretical models M1, M2 of the above Ethereal Euclidean geometry have the same , ω(n), ω(E),Ω1(E), Ω2(E) , then they are isomorphic. 

In other words , the above axiomatic system is categorical up to the units of measurements and visible and invisible material and ethereal density




References

1) D. Hilbert “Grundlangen der Geometrie” Taubner Studienbucher  1977


2) V. Boltianskii “Hilbert’s 3rd problem” J. Wesley & Sons 1978
3) E. E. Moise “Elementary geometry from an advanced standpoint” Addison –Wesley 1963
4) Euclid The 13 books of the  Elements  Dover 1956


And here we present a corresponding axiomatic system of the real numbers without the difficulties of the infinite , with 3-levels of precision. It is a new integrity between thinking, feeling and acting in the physical worlds. It is an educational and conceptual revolution. For more in the Blog       thedigitalmathematics.blogspot.gr

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