SQT Review Text 005 18 Okt 09 Revised March 2011

8/3/2019 SQT Review Text 005 18 Okt 09 Revised March 2011

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Mathematical Physics

A studyto establish set theory asnatures logical description

Authors

Arno P.L.M. Gorgels

Shevkinaz Bulut

Introduction

Mathematical descriptions such as Lorentz-transformations or the mathematical formulation of quantum mechanicsare often taken as "mathematical physics". These descriptions, however, are neither more nor less than isolated

mathematical descriptions of specific experimental observations. In this paper, a mathematical theory of nature is

only called "Mathematical Physics" when its mathematical axioms are identifiable in nature. The authors believe -

based upon an idea of the mathematician Amir D. Aczel - that Set Theory complies with this requirement.

This studyproposes to introduceSet Theory, when based upon the set of natural numbers, as the most appropriatemathematical tool to describe nature. The authors believe that the axioms of Zermelo-Fraenkel[1] (1907-1930),

forming the standard form of axiomaticSet Theory (including Cantor's theory of infinities[2](1877)), allow for themost complete description of the universe; natural laws and nature's constants should be derivable from it. They

elaborate on an idea of Amir D. Aczel12. Set Theory allows real, virtual and imaginary (concrete and abstract)elements of nature to be arranged individuallyas well as in groups. It assumes the existence of any elements in amathematical environment that is called "the set of those elements" possibly with (actual) infinite numbers, andapplies Cantors mathematics of infinities. Unlike the fully symmetrical Lie-Algebras (around 1870, theory ofcontinued symmetry, [3]) and the KacMoody-Algebras, [4] and [5], Set Theory (when applied to physics) isasymmetrical with regards to the structure of its basic field (basic set). Full symmetry occurs only from its first

power set upwardsby natural selection ensuring enduring stability of the resulting closed system.Asymmetry of the

basic field is essential to the theory. Unlike KacMoody-Algebras, Set Theory does not require at t=0 the presence ofspace (or vacuum). On the contrary: Set Theory of natural numbers allows us to describe the early cosmic phase of

space inflation (assumed t=0-4 sec.), in a way which eventuallyresults in a stable residual basic field and the birth ofmyriads of photons (later to becomeenergy respectively matter). These photons are supposed to be caused due toinstability at the field's borders at actual infinity. This basic field is mathematically described by the set of rationalnumbers.It corresponds to the gravitational field (Isaac Newton(1642-1727), Wheeler (1911-2008)). It builds uponundetectable field elements beingvirtual directionless one-dimensional-volume-quanta of the format

Xn = X.sin(t + n) +j.X.cos(t + n) with n=1,2,3 actual (1)

X = non-directional length in m; = frequency; = angular phase shift


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Itargues that these virtual elements, numbered n=1,2,3 are eventually neatly lined up like a sphere surrounding afixed starting point according to an increasing n finally forminga structured vacuum in which all natural eventshappen. The structure resembles a classical spin network (as proposed by Roger Penrose) but has the advantage to

seem to enable non-commuting processes as measured by quantum mechanics. The word "virtual is here to beunderstood as not directly measurable. The basis of this study is therefore not directly falsifiable according to Karl

Popper's condition for falsifiability dealing with a decisive criterion for scientific theories (1902-1994).

In line with the established expressionsuperstrings, the authors call volume-quanta:superquanta. Superquantafactually correspond to three-dimensional strings. Their surface can mathematically be made visible in the Lorentz

equations of Albert EinsteinsSpecial Relativity as follows:

x+y+z+j.c.t= F (2)

for which we can write:

x+y+z+j.c.t= F.(sin(t + n)-jcos(t + n)) (3)

whereassin(t + n)-jcos(t + n) = sin(t + n)+cos(t + n) = 1.

F in m is the basic invariable introduced by Max Born7(1920). (It should be noted that the expression of the rightmember in (3) can (without F) be written as: (sin(t + n)+jcos(t + n))*(sin(t + n)-jcos(t + n)) or ej(t +

n)*e-j(t + n),i.e. the product of two undamped vibrations. Each volume-point in vacuum appears to possess thisfeature.) Through formula (3) Special Relativity (SRT),volume-quanta (Quantum Mechanics) and strings (one-dimensional vibrations in multidimensional space) can thus be seen to be interrelated at vacuum level.

The field asymmetry mentioned aboveshould explain the phenomenon of dark matter (the extra gravitationalforce that is being observed in addition to the gravitational force caused by visible matter, which pulls stars to the

centre of the galaxy they belong to). At material level,however, complete symmetry prevails. Thiscomplies with

Set Theory and is also in line with current exclusively symmetrical theories.

This studygives a number of examples of observations that can only be explained through the basic asymmetrymentioned here-above.Thoseobservations include, at the macroscopic level, the concept of dark matter,as wellas the concept of a galaxy being a closed space-time unit in a steady state universe with intergalactic spaceequal to 0. At the microscopic level, they include the allocation of positrons within the atom nucleus.Related to thisis the redefinition of protons, neutrons and quarks as being measured results of subatomic structures that, according

to Set Theory,can becompositions of positrons surrounded by newly proposed particles, called eons andpeons, thatcarry -1/3 e and +2/3 e electrical charge, as well as electrons orbiting in subatomic paths in respectively 4*c and 2*c

space-environments, for which c is the speed of light.

The article thusprovides argumentsto accept Set Theory as the most fundamental mathematical description ofnature.The related physical theory is proposed as Superquantum Theory (SQT). For a complete picture, please referto the diagram Generalized structure of mathematical physics.

1) Methods of investigation.

In 1959, Louise Volders demonstrated that spiral galaxy M33 does not spin as expected according to Keplerian

dynamics, a result which was extended to many other spiral galaxies during the seventies. This extra mass isproposed by astronomers to be dark matter within the galactic halo.

The phenomenon ofDark Matterfirst observed by Fritz Zwicky[6] in 1933,in addition tothe almost classical

problem of relating the theories of gravity (the General Relativity Theory[7] of Albert Einstein, 1912) and quantummechanics (Heisenberg, 1929), seems to urge a new approach in theoretical physics. The idea of a possible newapproach was given by Amir D. Aczel12. He suggested to investigate Set Theory based upon natural numbers together

with Cantor's Continuum as a theory that could fully describe nature. To do this, the field of gravity is, as a first


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step, to be understood as the basic set (a field of numbers with an originating point (midpoint of the galaxy) and, as

such, asymmetrical) which unfoldsintoseveral power sets that form the natural forces. What, then, can space andmatter be made of?

2) Matter composed of atoms

Only two centuries ago scientists believed the atom to be the smallest unit of matter. A short time later it was found

that the atom is composed of a nucleus surrounded by orbiting electrons. The present experiments with collidinghadrons (LHC) are an attempt to find further subatomic particles. Quantum physics assumes subatomic particles tobe elementary. The Standard Model describes this world of elementary particles.

Atoms are made up of a dense nucleus containing protons and neutrons. The nucleus is surrounded by one or more

electrons. Electrons are lightweight, negatively charged particles. The nucleus is made up of positively charged

particles called protons and neutral neutrons. Protons and neutrons are believed to be made up of even smaller particles called quarks. In this paper, a slightly different composition is proposed, taking two new elementary

particles called eons and peons into account.


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3) The standard model and gravity

The standard model (SM) has mainly been developed in the years 1961-1973. It has extensively been investigated

and tested (Glashow 1961, [8]; Weinberg/Salam 1967, [9]), and thus appearsto perfectly describe the buildingblocks of the atomic world and the interactions of at least three of the four known natural forces. However, there are

a number of open questions that it can not answer.There is, first of all,the question of a quantum related to gravity:so far gravity is excluded from the SM the reason being that gravitational field elements called gravitons have

never been observed. A hypothetical particle, a graviton called the Higgs Boson, has been proposed by Peter Higgs

(1964, [10]) to cause that gravitational interaction that gives mass to particles. Secondly, we mentioned above theproblem of Dark Matter: this substance which is necessary to explain fast star movements at the edge of galaxies can

neither be explained using the particles of the Standard Model nor by the General Relativity Theory.

The present understanding of nature describesfour natural forces: gravitation, electromagnetism, the weakinteraction and the strong interaction. Each of these forces is mediated by a fundamental particle (=quantum) known

as a force carrying particle.Three of the four forces are unified throughthe Standard Model of particle physics. SMdescribes the universe in terms of matter and force. If the Higgs Boson experimentally could be found SM would

gain considerably in terms of reliability and acceptance. However, SQT does not predict a Higgs Boson.

SQT argues that the genuine theoretical basis of nature is given by the mathematics of sets. The universal and simplestructure of set theory that, at the same time, allows for the most complicated constructions,is understood as beingable to formalize almost all mathematical concepts, [11]. Set theory is therefore generally taken to be the backbone

structure of mathematics[12]. SQT proposes that it may also constitute the mathematical description of nature. Like


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the string theory, SQT assumes virtual particles for which no experiment can be designed toallow direct observation.Moreoverthose virtual particles are associated solutions of partial differential equations, i.e. the solutions contain areal and an imaginary portion of the form given above in (1). They are grouped as in the infinite set of numbers that

leads to Cantor's set theory of infinities, also called Cantor's Continuum. In this paper the authors propose touse themathematical toolsof Set Theory to find solutions to some of the current problems of particle physics.

4)Power set of the gravitational field


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The set theory was conceived in 1874 by the mathematician Georg Cantor[13]. He especially investigated sets withan infinite number of elements. Cantor's perception was that several number infinities do exist; that was calledCantor's Continuum. He defines actual infinity and uses the power set axiom to build a large numbers of infinities.

Aczel (2000) suggests applying Set Theory to theoretical physics. We are convinced that Aczel is correct. Thisarticle proposes ways to test or falsify his hypothesis.

SQT suggests connecting the four known natural forces by arranging them within the power sets of the basic set ofreal numbers, already identified as the gravitational field. The following graphic shows how.

5) The correspondence between power set and natural forces

Assuming gravity as the basic field of the set of natural fields, it can be considered as a basic set with just one

element: gravity. The first power set of this basic set contains, according to set theory, 2elements that easily shouldbe identifiable as the electric and magnetic field. These three measurable fields (gravity, electric field and magnetic

field) can be arranged as elements of afurtherbasic set, being the set of measurable fields.Besides the function of itselements as causing the natural forces ofgravity (1), electricity (2) and magnetism (3), they allow for the existenceof a further power set, that exists of 2 elements, schematically made visible as follows:


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6) Power Set and Quantum Chromodynamics

Gluons are commonly described as combinations of three "colour charges". SQT suggests that the three gluon

colours seen in experiments (Weinberg, 1967) should therefore be identified as the equivalent of G, E, and M.


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7) Theatom nucleus

Cantor's continuum allows the calculation of the internal structure of elementary particles. It covers, in principle, thethree-dimensional space x, y, z of Galileo, the four-dimensional space time x, y, z, t of Albert Einstein, and the

extended six- and more dimensional spaces of Burkhard Heim and his successors.

Cantor understood that the continuum is unlimited, however it should be realized that the corresponding physicalcontinuum must be limited because its components and elements may physically be neither infinitely small nor

infinitely large even though this would mathematically be thinkable.

The physical continuum, limited to 10 levels as shown in the next diagram, allows, in addition to electrons andpositrons, the occurrence of electrically charged particles with +2/3e and -1/3e. These newly-postulated particles

now allow us to define the internal structure of protons and neutrons. The new elementary particles are called eon (-

1/3e) and peon (+2/3e).

Thus, the nucleus is defined as the following constellation: positrons veiled by peons and eons (which are bound byweak nuclear forces) with a surrounding electron. The interaction of positron, eon, peon, and electron are measured as proton, neutron, up-Quark and down-Quark as follows. The elements positron, peon and eon may be experimental

results of the coming hadron colliding experiments in CERN.

1- Proton = + Peon + 2*Eon

2- Neutron = p + e

3- Up Quark = + Peon + e

4- Down Quark = + Eon + e


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Thus, the nucleus is a combination of positrons and electrons which interact with peons and eons. Protons,neutrons as well as up-Quarks and down-Quarksmerely are experimental resultsof particle physics. In the course offurther study W-bosons and Quarks need to be reviewed, since in this article positrons and electrons appear to be an

integral part of atom nuclei. As can be seen from the picture, quarks and W-bosons are composed of elementaryparticles within the several space-time-structures. These space-time structures are shown in the following generalized

overall picture.


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8) Generalized structure of mathematical physics


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The mathematical/physical architecture of nature thus is a projection of Cantor's continuum that combines themathematical real axis and the imaginary axis. All matter as well as events can only be combinations of volume

quanta and are therefore sure to carry real and imaginary elements.

All events within Level L(x, y, z, t)aretherefore governed by the formula:

EL = EL.cos(Lt + nL)+j.EL.sin(Lt + nL)with L = 0, 1, 2, 3 9 and L = 2L.

Remark: according to Euler[14]this can be written for L=1 as:

E.e j (t + n ) =E.( cos(t + n) + j sin(t + n))

This equation is, among others, essential to understand the several velocities in vacuum of electromechanical waves

shown in the previous diagram. In this diagram the frequency and the speed of electomechanical waves c within L0

are mathematically and physically interrelated due to their their presence in the powerfactor of the mathematical

formula: i.e. e2c equals ec. It should be independently noted that an implicit precondition that is yet to beverified is that c and (and other natural constants related to L) will be found to be proportional to Cantors' first

infinite cardinal number called Aleph0 (shortly ). The authors believe that this can be proven without great

difficulty.

The portion of nature thus taken from the total of Cantor's continuum is seen to be limited to only 10 levels of theentire infinite continuum. The ten levels should be understood to be equal to genuine spaces, each featuring a f(x, y,

z, t). Each of these spaces, together forming the vacuum, carries its own distinguished (maximal) speed of

electromechanical waves c, 2*c, 4*c etc. in line with set theory. The relativity space with 2*c may carry the

background radiation. Einsteins General Relativity applies to gravitation between masses; an addition of field

gravitation should be considered as per Set Theory. An observer could only, by measuring interactions between theseveral spaces L (i.e. the phenomenon of tunnelling) or by identifying particles within or allocated to particular

spaces, become aware of the existence of L(x,y,z,t). The authors believe that the experiments in CERN will give

evidence of the standard existence of positrons within protons and, as such, will give a strong experimental sign of

the correctness of the proposed level 3 particles (eon and peon) and, in the following analysis, proof of thecorrectness of the proposed introduction of set theory as the most adequate mathematical description of nature. The

authors believe that the corresponding suggestion made by Amir D. Aczel will be proven to be substantiated.

This paper contains an approach to establish proof of the theorem of set theory being the most comprehensive

mathematical description of nature. Further theoretical study linked with experimental investigations should provide

the necessary evidence to consolidate this approach.

Correspondence address:

Arno GorgelsHans-Thoma-Strasse 3 14467 Potsdam GermanyTel. 0049 (0)331 9678361Mobile: 0049 (0)15772788149

Property and Copyright of ideas, texts, illustrations and elaborations:

Arno Gorgels /Potsdam Shevkinaz Bulut/ Cologne

Reviewed by Joshua Berkowicz


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9) Literature

[1] Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy, 1973 (1958). Foundations of Set Theory. NorthHolland. Fraenkel's final word on ZF and ZFC.

[2] Cohen, P. J. (1966). Set Theory and the Continuum Hypothesis. W. A. Benjamin.

[3] Igor Frenkel, James Lepowsky, Arne Meurman: Vertex Operator Algebras and the Monster, AcademicPress, New York (1989) ISBN 0-12-267065-5

[4] V. Kac Infinite dimensional Lie algebras ISBN 0521466938[5] R.V. Moody, A new class of Lie algebras J. of Algebra , 10 (1968) pp. 211230

[6] Richard Panek, The Father of Dark Matter. Discover. pp.81-87. January 2009

[7] Die Relativittstheorie Einsteins und ihre physikalischen Grundlagen (Springer, 1920) - Based on Bornslectures at the University of Frankfurt am Main. Available in English under the titleEinsteins Theory of

Relativity.

[8] Sheldon L. Glashow (1961). "Partial-symmetries of weak interactions". Nuclear Physics 22: 579588.doi:10.1016/0029-5582(61)90469-2

[9] Steven Weinberg (1967). "A Model of Leptons". Physical Review Letters 19: 12641266.doi:10.1103/PhysRevLett.19.1264.

Abdus Salam (1968). Nils Svartholm. ed. Eighth Nobel Symposium. Elementary Particle Physics:

Relativistic Groups and Analyticity. Stockholm: Almquvist and Wiksell[10] Peter W. Higgs (1964). "Broken Symmetries and the Masses of Gauge Bosons". Physical Review Letters

13: 508509. doi:10.1103/PhysRevLett.13.508[11] Johnson, Philip, 1972. A History of Set Theory. Prindle, Weber & Schmidt ISBN 0871501546

[12] Amir D. Aczel The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity, 2000.ISBN 1-56858-105-X

[13] Cantor, Georg (1955, 1915). Contributions to the Founding of the Theory of Transfinite Numbers. NewYork: Dover. ISBN 978-0486600451

[14] Dunham, William (1999) Euler: The Master of Us All, Washington: Mathematical Association ofAmerica. ISBN 0883853280

10)

[1] Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy, 1973 (1958). Foundations of SetTheory. North Holland. Fraenkel's final word on ZF and ZFC.

[2] Cohen, P. J. (1966). Set Theory and the Continuum Hypothesis. W. A. Benjamin.

[3] Igor Frenkel, James Lepowsky, Arne Meurman: Vertex Operator Algebras and the Monster,Academic Press, New York (1989) ISBN 0-12-267065-5

[4] V. Kac Infinite dimensional Lie algebras ISBN 0521466938

[5] R.V. Moody, A new class of Lie algebras J. of Algebra , 10 (1968) pp. 211230

[6] Richard Panek, The Father of Dark Matter. Discover. pp.81-87. January 2009


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[7] Die Relativittstheorie Einsteins und ihre physikalischen Grundlagen (Springer, 1920) - Based onBorns lectures at the University of Frankfurt am Main. Available in English under the titleEinsteins Theory of

Relativity.

[8] Sheldon L. Glashow (1961). "Partial-symmetries of weak interactions". Nuclear Physics 22: 579588. doi:10.1016/0029-5582(61)90469-2

[9] Steven Weinberg (1967). "A Model of Leptons". Physical Review Letters 19: 12641266.doi:10.1103/PhysRevLett.19.1264.

Abdus Salam (1968). Nils Svartholm. ed. Eighth Nobel Symposium. Elementary Particle Physics:

Relativistic Groups and Analyticity. Stockholm: Almquvist and Wiksell

[10] Peter W. Higgs (1964). "Broken Symmetries and the Masses of Gauge Bosons". PhysicalReview Letters 13: 508509. doi:10.1103/PhysRevLett.13.508

[11] Johnson, Philip, 1972. A History of Set Theory. Prindle, Weber & Schmidt ISBN 0871501546

[12] Amir D. Aczel The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search forInfinity, 2000. ISBN 1-56858-105-X

[13] Cantor, Georg (1955, 1915). Contributions to the Founding of the Theory of TransfiniteNumbers. New York: Dover. ISBN 978-0486600451

[14] Dunham, William (1999) Euler: The Master of Us All, Washington: Mathematical Associationof America. ISBN 0883853280

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SQT Review Text 005 18 Okt 09 Revised March 2011