It gives me great comfort to visualize this universe as the surface of an ever expanding four-dimensional sphere originating from a distant, but finite, past and growing indefinitely for ever. In this idealized model it easy to calculate the age of the universe by observing the velocity of the receding stars and also to make several other interesting conclusions. For more details, continue reading the presentation.
Quantum Geometry: A reunion of math and physicsRafa Spoladore
Caltech's professor Anton Kapustin "describes the relationship between mathematics and physics, mathematicians and physicists, and so on. He focuses on the noncommutative character of algebras of observables in quantum mechanics." via http://motls.blogspot.com.br/2014/11/anton-kapustin-quantum-geometry-reunion.html
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Quantum Geometry: A reunion of math and physicsRafa Spoladore
Caltech's professor Anton Kapustin "describes the relationship between mathematics and physics, mathematicians and physicists, and so on. He focuses on the noncommutative character of algebras of observables in quantum mechanics." via http://motls.blogspot.com.br/2014/11/anton-kapustin-quantum-geometry-reunion.html
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Nuclear Decay - A Mathematical PerspectiveErik Faust
Radioactivity as a phenomenon is often misunderstood: if one says ‘Radioactive’, most people will think about disastrous electrical plants, dangerous bombs and other forms of life-threatening details. In my native Germany, members of the Green party have been campaigning for a decade to put an end to nuclear energy. Only few think of the useful aspects of this unique actuality, although radiotherapy is most promising of tools in the fight against cancer, and radioactive dating allows us to identify the age of any historical item. But even fewer people see radioactivity as the natural process that it actually is: A spontaneous mechanism, in which one nucleus decays into another. As an aspiring Physicist and Engineer, Radioactivity is one my favourite topics in the realm of science. I am fascinated at how we are able to predict exactly how many Nuclei will decay in a certain amount of time, but not say for certain which Nuclei exactly will do so.
Formal Unification of Gravity and Particle Physics in Lagrangian Euclidean Sp...Hontas Farmer
Formal Unification of Gravity and Particle Physics in Lagrangian Euclidean Space with Experimental Predictions - http://meetings.aps.org/Meeting/APR19/Session/Z13.5
In tis slide, an introduction to string theory has been given. Apart from that, a simple proof of 26 dimensions of bosonic string theory is given (following Zwiebach's approach).
I explained this presentation in two parts (on my YouTube channel). Here are the links
_______________________________________________
Part 1
https://www.youtube.com/watch?v=QQA4JQ6Y-eo&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp
_______________________________________________
Part 2
https://www.youtube.com/watch?v=vhLCtLn79jE&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp&index=2
_______________________________________________
Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Informa...Vasil Penchev
Hilbert space underlying quantum mechanics and pseudo-Riemannian space underlying general relativity share a common base of quantum information. Hilbert space can be interpreted as the free variable of quantum information, and any point in it, being equivalent to a wave function (and thus, to a state of a quantum system), as a value of that variable of quantum information. In turn, pseudo-Riemannian space can be interpreted as the interaction of two or more quantities of quantum information and thus, as two or more entangled quantum systems. Consequently, one can distinguish local physical interactions describable by a single Hilbert space (or by any factorizable tensor product of such ones) and non-local physical interactions describable only by means by that Hilbert space, which cannot be factorized as any tensor product of the Hilbert spaces, by means of which one can describe the interacting quantum subsystems separately. Any interaction, which can be exhaustedly described in a single Hilbert space, such as the weak, strong, and electromagnetic one, is local in terms of quantum information. Any interaction, which cannot be described thus, is nonlocal in terms of quantum information. Any interaction, which is exhaustedly describable by pseudo-Riemannian space, such as gravity, is nonlocal in this sense. Consequently all known physical interaction can be described by a single geometrical base interpreting it in terms of quantum information.
Nuclear Decay - A Mathematical PerspectiveErik Faust
Radioactivity as a phenomenon is often misunderstood: if one says ‘Radioactive’, most people will think about disastrous electrical plants, dangerous bombs and other forms of life-threatening details. In my native Germany, members of the Green party have been campaigning for a decade to put an end to nuclear energy. Only few think of the useful aspects of this unique actuality, although radiotherapy is most promising of tools in the fight against cancer, and radioactive dating allows us to identify the age of any historical item. But even fewer people see radioactivity as the natural process that it actually is: A spontaneous mechanism, in which one nucleus decays into another. As an aspiring Physicist and Engineer, Radioactivity is one my favourite topics in the realm of science. I am fascinated at how we are able to predict exactly how many Nuclei will decay in a certain amount of time, but not say for certain which Nuclei exactly will do so.
Formal Unification of Gravity and Particle Physics in Lagrangian Euclidean Sp...Hontas Farmer
Formal Unification of Gravity and Particle Physics in Lagrangian Euclidean Space with Experimental Predictions - http://meetings.aps.org/Meeting/APR19/Session/Z13.5
In tis slide, an introduction to string theory has been given. Apart from that, a simple proof of 26 dimensions of bosonic string theory is given (following Zwiebach's approach).
I explained this presentation in two parts (on my YouTube channel). Here are the links
_______________________________________________
Part 1
https://www.youtube.com/watch?v=QQA4JQ6Y-eo&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp
_______________________________________________
Part 2
https://www.youtube.com/watch?v=vhLCtLn79jE&list=PLDpqC3uXLZGl0cDod6g30PcjeJ4DAZWhp&index=2
_______________________________________________
Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Informa...Vasil Penchev
Hilbert space underlying quantum mechanics and pseudo-Riemannian space underlying general relativity share a common base of quantum information. Hilbert space can be interpreted as the free variable of quantum information, and any point in it, being equivalent to a wave function (and thus, to a state of a quantum system), as a value of that variable of quantum information. In turn, pseudo-Riemannian space can be interpreted as the interaction of two or more quantities of quantum information and thus, as two or more entangled quantum systems. Consequently, one can distinguish local physical interactions describable by a single Hilbert space (or by any factorizable tensor product of such ones) and non-local physical interactions describable only by means by that Hilbert space, which cannot be factorized as any tensor product of the Hilbert spaces, by means of which one can describe the interacting quantum subsystems separately. Any interaction, which can be exhaustedly described in a single Hilbert space, such as the weak, strong, and electromagnetic one, is local in terms of quantum information. Any interaction, which cannot be described thus, is nonlocal in terms of quantum information. Any interaction, which is exhaustedly describable by pseudo-Riemannian space, such as gravity, is nonlocal in this sense. Consequently all known physical interaction can be described by a single geometrical base interpreting it in terms of quantum information.
Mathematical Universe: Our Search for the Ultimate Nature of RealityManjunath.R -
We become more aware of how little we know about how the cosmos functions as we learn more about it. The Universe is full of secrets, from the doorstep of our own Solar System to the far-off shores of the intergalactic ocean. With the help of a large number of telescopes and satellites, we have increased our understanding of the universe. We have been investigating the history of the cosmos, from the Big Bang through comets' peculiarities and our curiosity about the chemistry of stars. One thing unites many of the most prevalent theories: they start from a mathematical framework that aims to explain more than our existing leading theories can. We Humans, inquisitive creatures shaped by Darwin's theory of natural selection, are used to asking questions. The question is not 'do we know everything from the very nature of physical laws to the underlying discomfort of the ultimate question of our place in the Universe?' or it is 'do we know enough?' But how does the creative principle reside in mathematics? There's something very mathematical about our gigantic Cosmos, and that the more carefully we look, the more equations are built into nature: From basic arithmetic to the calculation of rocket trajectories, math provides a good understanding of the equations that govern the world around us. Our universe isn't just described by math, but that universe is a "grand book" written in the language of mathematics. We find it very appropriate that mathematics has played a striking role in our expanding understanding of the universe − its origin, composition and destiny.
Dynamical Systems Methods in Early-Universe CosmologiesIkjyot Singh Kohli
Talk I gave at The Southern Ontario Numerical Analysis Day (SONAD): http://www.math.yorku.ca/sonad2014/ on General Relativity, Dynamical Systems, and Early-Universe Cosmologies.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
NO ONE CAN FLY UNTILL LEARNED TO WALK.......
from monkeys to men ,,fro crawling to fly ..it took us centuries to take a small step..how we can evolve... and give our future what our ancestors given us ...try to learn the way THEY THINK... how beautifully they adore the nature and think the things .. differently.....
Power and Log sequences: Shannon's Log n and Ramanujan's Log Log n ver1904Kannan Nambiar
Shannon recognized log n as an appropriate measure of choice when there are n objects to choose from, and developed a whole theory of communication that we use today. Ramanujan in his collected papers states that "almost all numbers n are composed of about log log n prime factors". This presentation explores log log ... log n further.
Mathematical and Spiritual Universes with Millennia old MantrasKannan Nambiar
Since the dawn of civilization, we have attempted to restrict our attention to the palpable world, but we have failed to do so miserably. Anybody with some intellect always ends up with questions about his existence, usually with no solution in sight. This presentation gives some answers provided by our ancestors.
My understanding is that it is our linguistic ability, that distinguishes us from the rest of the living creatures. Scribbling on paper, we have found out that earth goes round the sun in an elliptical orbit and also there has to be electromagnetic waves. Paul Erdos conceives of a book, he calls it The Book, where the Almighty has recorded all the proofs (reasoning) we are capable of in lexical order so that mathematicians can refer to it for the validity of their arguments. I call The Book our intellectual space. The presentation here is about the duality between the intellectual space and the physical space.
In ancient cultures the favorite question of sishya to his guru was "Who Am I?" and in the end he learnt everything about Reality. If you want a similar question in mathematics, ask "Is Riemann Hypothesis true?", and you will learn almost everything about mathematics. This presentation gives an elementary introduction to zeta functions using natural functions.
In computer science, Turing machine (TM) defines computation.Since the NuMachine (NM) is equivalent to the Turing Machine, we can explain the concepts of programming in terms of NM. The advantage is that the programming will be based on the Peano axioms which forms the foundations of mathematics. A special matrix multiplication of NuAlgebra (NA) replaces the unintuitive movements of the TM.
Sentient Arithmetic and Godel's Incompleteness TheoremsKannan Nambiar
For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.
Infinitesimal is defined as an infinite recursive subset of positive integers. The definition visualizes the unit interval as a set of infinitesimals and the unreachable infinite universe as a set of black-wholes.
Explicit Formula for Riemann Prime Counting FunctionKannan Nambiar
Corresponding to every zeta function there is a delta series. We make use of this fact to derive an explicit formula for Riemann Prime Counting Function.
Unconventional View of Godel's Theorems to Accommodate HistoryKannan Nambiar
What happened with respect to Fermat's last theorem and four-color conjecture forces us to change our view on the significance of Godel's incompleteness theorems.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
1. THE MATHEMATICAL UNIVERSE IN A NUTSHELL
K. K. NAMBIAR
ABSTRACT. The mathematical universe discussed here gives models of possible
structures our physical universe can have.
Keywords—Cosmology, White Hole, Black Whole.
Date: July 12, 2002.
2. 1. Introduction 3
2. Mathematical Logic 5
3. Generalized Anthropic Principle and The Book 10
4. Expanding Universe 13
5. Intuitive Set Theory 15
6. Universal Number System 21
7. Conclusion 23
25
3. 1. INTRODUCTION
When we talk about the universe, we usually mean the physical universe around
us, but then we must recognize that we are capable of visualizing universes which
are quite different from the one we live in. The possible universes that we can
reasonably imagine is what we collectively call the mathematical universe. There
is yet another universe which we may call the spiritual universe, a universe hard
to avoid and to define. We will accept a weak definiton of the spiritual universe as
the collection of unverifiable persistent emotional beliefs within us that are difficult
to analyze. Our description makes it clear that spiritual universe is beyond and the
physical universe is within the mathematical universe. As a matter of record, here
are the three universes we are interested in:
4. • Physical universe
• Mathematical universe
• Spiritual universe
The purpose of this paper is to discuss the mathematical universe in some detail,
so that we may have a deeper understanding of the physical universe and a greater
appreciation for the spiritual universe.
5. 2. MATHEMATICAL LOGIC
We want to talk about what the mathematicians call an axiomatic derivation and
what the computer scientists call a string manipulation. A familiar, but unempha-
sized fact is that it is the string manipulation of the four Maxwell’s equations that
allowed us to predict the radiation of electromagnetic waves from a dipole antenna.
Similarly, it is the derivations in Einstein’s theory of relativity that convinced us
about the bending of light rays due to gravity. It is as though nature dances faith-
fully to the tune that we write on paper, as long as the composition strictly conforms
to certain strict mathematical rules.
It was the Greeks who first realized the power of the axiomatic method, when
they started teaching elementary geometry to their school children. Over the years
6. logicians have perfected the method and today it is clear to us that it is not necessary
to draw geometrical figures to prove theorems in geometry. Without belaboring the
point, we will make a long story short, and state some of the facts of mathematical
logic that we have learned to accept.
Mathematical logic reaches its pinnacle when it deals with Zermelo-Fraenkel set
theory (ZF theory). Here is a set of axioms which define ZF theory.
• Axiom of Extensionality: ∀x(x ∈ a ⇔ x ∈ b) ⇒ (a = b).
Two sets with the same members are the same.
• Axiom of Pairing: ∃x∀y(y ∈ x ⇔ y = a ∨ y = b).
For any a and b, there is a set {a, b}.
7. • Axiom of Union: ∃x∀y(y ∈ x ⇔ ∃z ∈ x(y ∈ z)).
For any set of sets, there is a set which has exactly the elements of the sets
of the given set.
• Axiom of Powerset: ∀x∃y(y ∈ x ⇔ ∀z ∈ y(z ∈ a)).
For any set, there is a set which has exactly the subsets of the given set as
its elements.
• Axiom of Infinity: ∃x(∅ ∈ x ∧ ∀y ∈ x(y ∪ {y} ∈ x)).
There is a set which has exactly the natural numbers as its elements.
• Axiom of Separation: ∃x∀y(y ∈ x ⇔ y ∈ a ∧ A(y)).
For any set a and a formula A(y), there is a set containing all elements of a
satisfying A(y).
8. • Axiom of Replacement: ∃x∀y ∈ a(∃zA(y, z) ⇒ ∃z ∈ xA(y, z)).
A new set is created when every element in a given set is replaced by a new
element.
• Axiom of Regularity: ∀x(x ∅ ⇒ ∃y(y ∈ x ∧ x ∩ y = ∅)).
Every nonempty set a contains an element b such that a ∩ b = ∅.
These axioms assume importance when they are applied to infinite sets. For finite
sets, they are more or less obvious. The real significance of these axioms is that
they are the only strings, other than those of mathematical logic itself, that can be
used in the course of a proof in set theory. Here is an example of a derivation using
the axiom of regularity, which saved set theory from disaster.
Theorem 2.1. a a.
9. Proof: a ∈ a leads to a contradiction as shown below.
a ∈ a ⇒ a ∈ [{a} ∩ a] ... (1).
b ∈ {a} ⇒ b = a ... (2).
Using axiom of regularity and (2),
{a} ∩ a = ∅, which contradicts, a ∈ [{a} ∩ a] in (1).
Even though we are in no position to prove it, over a period of time we have built
up enough confidence in set theory to believe that there are no contradictions in it.
10. 3. GENERALIZED ANTHROPIC PRINCIPLE AND THE BOOK
It is generally accepted that all of mathematics can be described in terms of the
concepts of set theory, which in turn means that we can, in principle, axiomatize
any branch of science, if we so wish. This allows us to conceive of an axiomatic
theory which has all of known science in it, and all the phenomena we observe in
the universe having corresponding derivations. Since every derivation in a theory
can be considered as a well-formed formula, we can claim that the set of derivations
in our all-encompassing theory can be listed in the lexicographic order, with formu-
las of increasing length. A book which lists all the proofs of this all-encompassing
mathematics is called The Book. The concept of The Book is an invention of the
11. mathematician, Paul Erd¨os, perhaps the most prolific mathematician of the twen-
tieth century. Note that the book contains an infinite number of proofs, and also
that the lexical order is with respect to the proofs and not with respect to theorems.
It was this book that David Hilbert, the originator of formalism, once wanted to
rewrite with theorems in the lexical order, which of course, turned out to be an
unworkable idea.
Note that a computer can be set up to start writing the book. We cannot expect the
computer to stop, since there are an infinite number of proofs in our theory. Thus,
a computer generated book will always have to be unfinished, the big difference
between a computer generated book and The Book is that it is a finished book.
When discussing cosmology, a notion that is often invoked is called the anthropic
principle. The principle states that we see the universe the way it is, because if it
12. were any different, we would not be here to see it [1]. We generalize this concept
as follows.
Generalized Anthropic Principle: Every phenomenon in the uni-
verse has a corresponding derivation in The Book.
Note that the generalized anthropic principle does not claim that there is a phenom-
enon corresponding to every derivation. Such derivations are part of mathematics,
but not of physics, in other words, we consider the physical universe as part of the
mathematical universe.
13. 4. EXPANDING UNIVERSE
As a preliminary to the understanding of the expanding universe, we will first
talk about a perfectly spherical balloon whose radius is increasing with velocity U,
starting with 0 radius and an elapsed time T. If we write UT as R, we have the
volume of the balloon as (4/3)πR3
, surface area as 4πR2
and the length of a great
circle as 2πR. In our expanding balloon it is easy to see that two points which are a
distance Rθ apart from each other will be moving away from each other with veloc-
ity Uθ. Since (R/U) = T, it follows that a measurement of the relative movement
of two spots on the balloon will allow us to calculate the age of the balloon.
The facts about the expanding universe is more or less like that of the balloon,
except that instead of a spherical surface, we have to deal with the hyper surface
14. of a 4-dimensional sphere of radius R. If we use the same notations as before, we
have the hyper volume of the hyper sphere as (π2
/2)R4
, the hyper surface as 2π2
R3
,
and the length of a great circle as 2πR. We can calculate the age of the universe by
measuring the velocity of a receding galaxy near to us. If we assume the velocity
of expansion of the hyper sphere as U, we have UT = R and the volume of the
universe as 2π2
R3
.
We would have been more realistic, if we were to start off our analysis with a
warped balloon, but then our intention here is only to discuss what is mathemati-
cally possible and not what the actual reality is.
15. 5. INTUITIVE SET THEORY
ZF theory which forms the foundations of mathematics gets simplified further to
Intuitive Set Theory (IST), if we add two more axioms to it as given below [2]. If
k is an ordinal, we will write ℵα
k
for the cardinality of the set of all subsets of ℵα
with the same cardinality as k.
Axiom of Combinatorial Sets:
ℵα+1 =
ℵα
ℵα
.
We will accept the fact that every number in the interval (0, 1] can be represented
uniquely by an infinite nonterminating binary sequence. For example, the infinite
16. binary sequence
.10111111 · · ·
can be recognized as the representation for the number 3/4 and similarly for other
numbers. This in turn implies that an infinite recursive subset of positive integers
can be used to represent numbers in the interval (0, 1]. It is known that the cardi-
nality of the set R of such recursive subsets is ℵ0. Thus, every r ∈ R represents a
real number in the interval (0, 1].
We will write
ℵα
ℵα r
,
17. to represent the cardinality of the set of all those subsets of ℵα of cardinality ℵα
which contain r, and also write
ℵα
ℵα r
r ∈ R =
ℵα
ℵα R
.
We will call ℵα
ℵα r
, a bonded sack, and define a bonded sack as a collection which
can appear only on the left side of the binary relation ∈ and not on the right side.
What this means is that a bonded sack has to be considered as an integral unit from
which not even the axiom of choice can pick out an element. For this reason, we
may call the elements of a bonded sack figments.
18. Axiom of Infinitesimals:
(0, 1] =
ℵα
ℵα R
.
The axiom of infinitesimals makes it easy to visualize the unit interval (0, 1].
We derive the generalized continuum hypothesis from the axiom of combinatorial
sets as below:
2ℵα
=
ℵα
0
+
ℵα
1
+
ℵα
2
+ · · ·
ℵα
ℵ0
+ · · ·
ℵα
ℵα
.
19. Note that ℵα
1
= ℵα. Since, there are ℵα terms in this addition and ℵα
k
is a mono-
tonically nondecreasing function of k, we can conclude that
2ℵα
=
ℵα
ℵα
.
Using axiom of combinatorial sets, we get
2ℵα
= ℵα+1.
The concept of a bonded sack is significant in that it puts a limit beyond which
the interval (0, 1] cannot be pried any further. The axiom of infinitesimals allows
us to visualize the unit interval (0, 1] as a set of bonded sacks, with cardinality ℵ0.
20. Thus, ℵα
ℵα r
represents an infinitesimal or white hole or white strip corresponding to
the number r in the interval (0, 1].
21. 6. UNIVERSAL NUMBER SYSTEM
A real number in the binary number system is usually defined as a two way binary
sequence around a binary point, written as
xxx.xxxxx · · ·
in which the left sequence is finite and the right sequence is nonterminating. Our
discussion earlier, makes it clear that the concept of a real number and a white strip
are equivalent. The two way infinite sequence we get when we flip the real number
around the binary point, written as
· · · xxxxx.xxx
22. we will call a supernatural number or a black stretch. The set of white strips we
will call the real line and the set of black stretches the black whole. Since there
is a one-to-one correspondence between the white strips and the black stretches, it
follows that there is a duality between the real line and the black whole.
The name black stretch is supposed to suggest that it can be visualized as a set
of points distributed over an infinite line, but it should be recognized as a bonded
sack, which the axiom of choice cannot access. Our description of the black whole
clearly indicates that it can be used to visualize what is beyond the finite physical
space. If we use the concept of point at infinity of complex analysis, the black
whole can be visualized as the usual black hole.
23. 7. CONCLUSION
We will conclude with a few remarks about mathematical logic, which give us
some indication why we cannot afford to ignore the spiritual universe. G¨odel tells
us that there is no logical way to establish that there are no contradictions in ZF
theory, which forms the foundations of mathematics. We are confident about our
mathematics only because it has worked well for us for the last two thousand years.
Since, any set of axioms is a set of beliefs, it follows that any theory is only a set of
beliefs. Since, any individual is the sum total of h(is)er beliefs (axioms) and rational
thoughts (derivations), no individual, including scientists, can claim to be infallible
on any subject matter. If an honest scientist is called to appear in the ultimate court
of nature, (s)he can use The Book for taking the oath, and the most (s)he can say
24. is: I solemnly swear that if I am sane, I will tell the truth, nothing but the truth, but
never the whole truth.
25. 1. S. W. Hawking, The Universe in a Nutshell, Bantam Books, New York, NY, 2001.
2. K. K. Nambiar, Visualization of Intuitive Set Theory, Computers and Mathematics with Applica-
tions 41 (2001), no. 5-6, 619–626.
FORMERLY, JAWAHARLAL NEHRU UNIVERSITY, NEW DELHI, 110067, INDIA