Transcript Zero ^0 infinity Sept 21

Zero & infinity
Mohsen Kermanshahi
Nothing
Nothing is more important than nothing
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Nothing is more puzzling than nothing
Nothing is more interesting than nothing
What lies at the heart of mathematics? You guessed it, nothing
Ian Stewart an Emeritus Professor of Mathematics at the University of Warwick,
England
Definitions
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Zero
nothingness
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Infinity
Beyond biggest number
Formalizing Natural Numbers
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Natural Numbers are positive whole numbers.
1, 2, 3, 4, …………..
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A set of axioms for the natural numbers presented by the 19th century Italian
mathematician Giuseppe Peano
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Peano axioms are the basis for the version of number theory known as Peano
Arithmetic
Peano Axioms
The principles of the Peano arithmetic are as follows,
1. Zero is a natural number.
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2. If a is a number, the successor of a is a number as well.
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3. Zero is not the successor of any number.
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4. Two numbers of which the successors are equal are
themselves equal.
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5. (induction axiom.) If a set s of numbers contains zero and
also the successor of every number in s, then every number is
in s.
Completeness and Consistency
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In mathematical logic a formal system is called complete
with respect to a particular property if every formula
having the property can be derived using that system,
otherwise the system is said to be incomplete.
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a consistent theory is one that does not contain a
contradiction.
Gödel's first incompleteness theorem
Gödel's incompleteness theorems are among the most important finding
in modern logic.
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The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried
out, there are statements of the language of F which can neither be
proved nor disproved in F.
Zero as Natural Number
What are few examples of incompleteness in the Peano arithmetic?
Can zero be considered a natural number?
Among Mathematicians, Cantor, Peano, and Burbaki thought that
zero is a natural number,
While Euler, Kronecker and Sloane thought otherwise.
Let us see why there are disagreements. Does inconsistencies appear
by inserting zero in the natural number system?
Here are few exceptions and therefore inconsistencies within Peano arithmetic
involving zero,
1/ adding two numbers should create a bigger number, whereas by adding
zero to zero or any other number, the said number will remain the same.
2/ the system is undecidable if both s + a and s - a are equal. Since s + 0 and s
- 0 are equal then piano arithmetic appears undecidable in respect to zero
3/ Logical trap; zero divide by any number is zero. This implies that, every
natural number is equal to another which appears contradictory.
4/ any number to the power of zero equals one, whereas zero to power of zero
equals 0. This also exhibit inconsistency in the system.
Inconsistencies regarding zero,
within Peano Arithmetic
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5/ similarly with n factorial such as 5! We do not follow it up to
zero because it defies the purpose. Mathematicians take 0! = 1.
Another exception.
Furthermore,
a/ if natural numbers represent physical entities, why one has to
correlate nothingness by a number?
b/unlike any other number, zero cannot take positive or negative
sign.
c/unlike any other number, zero is not analytical.
How about infinity?
The concept of infinity has been a major problem in mathematics.
From Aristotle to Galileo, Cantor, Gödel and the others had struggled with it.
Actually, Cantor and Gödel have had nervous breakdowns over it and ended up in
mental hospitals.
The debate is still going on and jury is out.
Some mathematicians believe that infinity is not a number. yet, the Zermelo–
Fraenkel set theory plus the axiom of choice (ZFC), the most common foundation of
mathematics contains the axiom of infinity3.
Axiom of infinity; There exists a set containing the empty set and the successor of
each of its elements.
Zermelo-Fraenkel set theory
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Zermelo–Fraenkel set theory, is one of several axiomatic
systems that were proposed in the early twentieth century to
formulate a theory of sets . Today ZFC is the standard form of
axiomatic set theory and as such is the most common
foundation of mathematics.
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By Gödel's first theorem, even the extremely powerful standard
axiom system of Zermelo-Fraenkel set theory (denoted as ZF,
or, with the axiom of choice, ZFC) which is more than sufficient
for the derivation of all ordinary mathematics is incomplete.
Few examples of encountering exceptions and inconsistencies if we
include infinity in the natural number system
1/ Different sets having the same cardinality, suggested by Cantor is
being observed by mathematicians although with reservations (1).
Intuitively, a limited set such as natural numbers should have less
members than a more extended set like real numbers.
2/ If infinity #1 + infinity # 2 is indistinguishable from infinity # 1 minus
infinity # 2 then the finding cannot be explained by natural number
system.
3/ Again a+a is supposed to be bigger than a. This rule does not apply to
infinity either. Adding any number to infinity doesn’t change it. This is
another exception within natural number system regarding infinity.
Similarly, infinity times infinity is infinity, another exception.
4/ infinity is not successor of any actual number, therefore it defies
the second Peano axiom
5/Different levels of infinities mentioned by Cantor is counterintuitive as
well.
Aleph zero being smaller than aleph one and so on and so forth …. up to
infinity of infinities raise many questions
Furthermore
a/while any other number is analyzable (for example, one tenth of
number 1 is meaningful), just like zero, infinity is not analytical (one
tenth of infinity does not relay any meaning).
b/There are unlimited points between zero and one in the number
line. As a matter of fact, there are unlimited numbers between every
two consecutive integers. This creates a big puzzle in mathematics.
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Rational Numbers, 1/2,1/4,1/5…
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Decimal numbers, 0.111111…,0.21111111…
What is Axiom of choice
Bertrand Russel has an analogy of AC.
“To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice,
but for shoes the Axiom is not needed. The idea is that the two socks in a pair are identical
in appearance, and so we must make an arbitrary choice if we wish to choose one of them.
For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe."
Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely
many pairs of socks, then AC is not needed -- we can choose one member of each pair
and we can repeat an operation finitely many times.
Axiom of choice has been a matter of debate for centuries.
Does infinity exist?
Stephen Simpson, a mathematician and logician at Pennsylvania State University
asks “What truly infinite objects exist in the real world?” If space-time universe is
quantized and made of discrete elements, how can we assign infinity to anything
at all?
Taking a view originally espoused by Aristotle, Simpson argues that actual
infinity doesn’t really exist and so it should not so readily be assumed to exist in
the mathematical universe.
The idea is that everything in space-time is quantized. Even atoms in the universe
is finite and measured at about 10^80
So there is no room for anything infinite in space-time universe
Does Zero Exist
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Zero represents nothingness
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Logically if there is a thing, there should be a nothing as well
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Thing without nothing makes no sense
What is mathematics
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A platonic domain, obscure. Physics can just use a part of it. Roger
Penrose, Charles Parsons
Or
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mathematics is not a separate domain. Rather mathematics is just a
simpler language to be used in describing physical domains. Therefore
any unreal part has to be discarded, such as -3 apples. Kronecker, Van
Orman
Or
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Applied mathematics; using unreal figures are acceptable as long as we
end up to real figures, such as
5 apples – 3 apples which equals 2 real apples
Is there a place for zero and infinity
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The fact is zero and infinity are needed and being used in
mathematical calculations and building mathematical theories.
they simply can’t be ignored.
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However, zero and infinity are affecting natural numbers and
mathematics in bizarre and unusual ways.
Therefore, one would assume that they should be accounted for.
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Is there a solution?
Adding Axioms
By Gödel's first theorem, there are arithmetical truths that are not provable
even in ZFC.
There is thus a sense in which such truths are not provable using today's
“ordinary” mathematical methods and axioms, nor can they be proved in a
way that mathematicians would today regard as unproblematic and
conclusive.
Proving them would require a formal system that incorporates methods
going beyond ZFC.
Objectivity
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Natalie Wolchover writes “Mathematics has a reputation for objectivity.
But without real-world infinite objects upon which to base abstractions,
mathematical truth becomes, to some extent, a matter of opinion.”(2)
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As Kurt Gödel the twenty century Logician, mathematician and
philosopher postulates, Mathematic which works with numbers
(discrete domain) cannot be complete. It needs un-calculable domain
outside it to make the whole scheme complete. Similarly Turing
halting problem suggests the same.
Adding extra axioms
New formal System,
Gödel's incompleteness theorem implies,
For any statement A unprovable in a particular formal
system F, there are, trivially, other formal systems in which
S is provable (but we need to take A as an extra axiom)
Continuum Hypothesis
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The continuum hypotheses (CH) is one of the most central open
problems in mathematics. Number one in Hilbert's list
The continuum hypothesis was advanced by Georg Cantor in 1878, his
first paper . But he couldn’t prove this “continuum hypothesis” using the
axioms of set theory. Nor could anyone else.
It is said that struggling with continuum hypothesis caused Cantor’s and
Gödel's nervous break downs.
Continuum Hypothesis
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how many points are there on a line?
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How many numbers are between 1 and 2
1.0001…, 1.1111…
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Cantor stipulated that there is no infinity in between countable
sets and the continuum.
Continuum hypothesis is unsolvable within current mathematical
systems
Controversy
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Gödel himself proved that the continuum hypothesis is consistent
with ZFC,
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and Paul Cohen, an American mathematician, proved the
opposite, that the negation of the hypothesis is also consistent
with ZFC.
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Their combined results demonstrated un-decidability. Therefore,
the continuum hypothesis is actually independent of the existing
axioms. Something beyond ZFC is needed to prove or refute it.
(2)
Adding Axioms
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A solution has to be found for the enigma.
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According to Gödel extra axioms has to be added to the system to
create an encompassing system.
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But what axiom has to be assigned to the system in order to return
consistency to common mathematical systems in dealing with
zero and infinity
Continuity versus discreteness
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What is discreteness? Numbered, Countable, quantized,
analyzable
1,2,3,4,5,…
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What is a continuum? Ultimately smooth, indivisible, not
calculable, not analytical
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Cavieleri’s dot, line, plane
Continuity and discreteness
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Zero is not analytical. 1/3 of zero doesn’t have any
meaning
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Similarly, infinity is not calculable. 1/3 of infinity is
meaningless.
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One may conclude that zero an infinity are continuums by
nature
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Natural numbers are countable , therefore discrete
Discreteness versus continuity
One can argue that, if zero and infinity are not countable,
then combining them with countable natural numbers, can
prove troublesome.
Let us explore as Gödel suggested what happens if we
extract un-calculable domains out of Peano arithmetic and
add them as separate axioms.
Is there a way out of the gridlock?
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While natural numbers do represent discrete domains, zero and infinity not
being analytical can represent a continuum.
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Here is an speculation. If we extract zero and infinity out of the natural number
system and insert them to added layer then we may have a system where
some of the inconsistencies are removed while a more encompassing theory is
achieved.
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In such a scheme, the discrete domain (countable) can overlay over a
continuum layer containing zero and infinity.
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The two layers are different form each other (discrete versus continuum). Yet
they can interact with each other with a different set of rules.
Supporting Points
Extracting zero and infinity from the natural number system and inserting them in an
underlying continuum layer, has so many advantages.
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For one, it offers a solution for the logical trap mentioned above where dividing
different numbers by zero makes them to appear equal.
X x 0 = 0,
Y x 0 = 0,
Then X = Y
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In such a model, one can conclude that in a Cartesian system where
X and Y coordinates represent discrete values, the image of any point projected
to point zero equals zero.
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In point zero images of all points in the field overlap each other that
demonstrate equality.
Calculus
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When calculus originally introduced by Newton and
Leibnitz, it was noticed that, the discipline frequently
encounters zeros and infinities. This was troublesome. The
19th century German mathematician Karl Weierstrass
introduced limits to calculus in order to bypass the
problem.
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Although, artificially placing limits in the field helps to
deal effectively with discrete elements, it ignores a good
portion of the domain (the continuum portion).
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Zero is scattered in the field. In calculus, one can choose
any point of the field as point zero, the practice is called
blowing up the origin
Infinities and Renormalization
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Certain phenomena such as self-energy of the electron as well
as vacuum fluctuations of the electromagnetic field seems to
require infinite energy.
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To avoid infinities different technics of renormalization has
been used to circumvent the divergence.
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One way is to cut off (renormalize) the integrals in the
calculations at a certain value Λ of the momentum which is
large but finite.
Infinity in the domain
Super-Space
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Many mechanisms—for example, electromagnetic fields—cannot be
explained in the context of a four-dimensional universe alone.
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To explain these mysteries, mainstream physicists chose to theorize
another space-like manifold in addition to ordinary space-time. This
manifold is called super-space. In basic terms, the idea of super-space
presumes that the points in space-time are actually cross-sections of
bundles which are extended into this proposed super-space.
This is to compensate for the
inconsistencies.
Hilbert Space
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In mathematics the separable up to infinite dimensional Hilbert inner space is an example
for the above model. In the formal model the Inner product spaces generalize Euclidean
spaces (in which the inner product is the dot product,) to vector spaces with up to infinite
dimensions.
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Hilbert space is indispensable tool in the theories of partial differential equations,
quantum mechanics, Fourier analysis and ergodic theory, which forms the mathematical
underpinning of thermodynamics.
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One may assume that the dot inner product opens the arena to the underlying continuum
layer.
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https://youtu.be/zv_reNX3puQ
Zero Point Energy
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Vacuum energy is the zero-point energy of all the fields in space, which in the
Standard Model includes the electromagnetic field, other gauge fields,
fermionic fields, and the Higgs field.
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It is the energy of the vacuum, which in quantum field theory is defined not as
empty space but as the ground state of the fields.
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In cosmology, the vacuum energy is one possible explanation for the
cosmological constant.[3] A related term is zero-point field, which is the lowest
energy state of a particular field.
point zero energy is not detected inside space-time universe
Few more current puzzles
In this scheme, some of the most puzzling question in theoretical physics may
find explanations,
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the cosmological constant problem will obtain an explanation. One can
hypothesis a source for the mysterious dark energy sipping through form
underlying layer
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Non-locality and quantum entanglement, quantum tunnelling, spin of
subatomic particles, virtual particles, nature of fields and alike can obtain
explanation as well.
References
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1/ Hermann Weyl a German mathematician, Theoretical physicist and
philosopher wrote:
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“… classical logic was abstracted from the mathematics of finite sets and
their subsets …. Forgetful of this limited origin, one afterwards mistook that
logic for something above and prior to all mathematics, and finally applied it,
without justification, to the mathematics of infinite sets. This is the Fall and
original sin of [Cantor's] set theory …."
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http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory#cite_not
e-11
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2/ http://www.scientificamerican.com/article/infinity-logic-law/
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3/ http://www.math.vanderbilt.edu/~schectex/ccc/choice.html