Zero & infinity Mohsen Kermanshahi Nothing Nothing is more important than nothing 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 Zero nothingness

Infinity Beyond biggest number Formalizing Natural Numbers Natural Numbers are positive whole numbers. 1, 2, 3, 4, .. A set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano

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. 2. If a is a number, the successor of a is a number as well.

3. Zero is not the successor of any number. 4. Two numbers of which the successors are equal are themselves equal. 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

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. a consistent theory is one that does not contain a contradiction. Gdel's first incompleteness theorem Gdel's incompleteness theorems are among the most important finding in modern logic.

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. Inconsistencies regarding zero, within Peano Arithmetic

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, Gdel and the others had struggled with it. Actually, Cantor and Gdel 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 ZermeloFraenkel set theory plus the axiom of choice (ZFC), the most common foundation of mathematics contains the axiom of infinity 3. Axiom of infinity; There exists a set containing the empty set and the successor of each of its elements. Zermelo-Fraenkel set theory ZermeloFraenkel 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.

By Gdel'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 doesnt change it. This is another exception within natural number system regarding infinity. 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. Rational Numbers, 1/2,1/4,1/5 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 doesnt 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 Zero represents nothingness Logically if there is a thing, there should be a nothing as well Thing without nothing makes no sense

What is mathematics A platonic domain, obscure. Physics can just use a part of it. Roger Penrose, Charles Parsons Or 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

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 The fact is zero and infinity are needed and being used in mathematical calculations and building mathematical theories. they simply cant be ignored.

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. Is there a solution? Adding Axioms By Gdel'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 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) As Kurt Gdel 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, Gdel'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 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 couldnt prove this continuum hypothesis using the axioms of set theory. Nor could anyone else. It is said that struggling with continuum hypothesis caused Cantors and Gdel's nervous break downs. Continuum Hypothesis how many points are there on a line? How many numbers are between 1 and 2

1.0001, 1.1111 Cantor stipulated that there is no infinity in between countable sets and the continuum. Continuum hypothesis is unsolvable within current mathematical systems Controversy Gdel himself proved that the continuum hypothesis is consistent with ZFC,

and Paul Cohen, an American mathematician, proved the opposite, that the negation of the hypothesis is also consistent with ZFC. 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 A solution has to be found for the enigma.

According to Gdel extra axioms has to be added to the system to create an encompassing system. 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 What is discreteness? Numbered, Countable, quantized,

analyzable 1,2,3,4,5, What is a continuum? Ultimately smooth, indivisible, not calculable, not analytical Cavieleris dot, line, plane Continuity and discreteness Zero is not analytical. 1/3 of zero doesnt have any

meaning Similarly, infinity is not calculable. 1/3 of infinity is meaningless. One may conclude that zero an infinity are continuums by nature 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 Gdel 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? While natural numbers do represent discrete domains, zero and infinity not being analytical can represent a continuum.

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. In such a scheme, the discrete domain (countable) can overlay over a continuum layer containing zero and infinity. 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. 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 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. In point zero images of all points in the field overlap each other that demonstrate equality. Calculus 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. 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). 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 Certain phenomena such as self-energy of the electron as well as vacuum fluctuations of the electromagnetic field seems to require infinite energy. To avoid infinities different technics of renormalization has been used to circumvent the divergence. One way is to cut off (renormalize) the integrals in the calculations at a certain value of the momentum which

is large but finite. Infinity in the domain Super-Space Many mechanismsfor example, electromagnetic fields cannot be explained in the context of a four-dimensional universe alone. 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 spacetime are actually cross-sections of bundles which are extended

into this proposed super-space. This is to compensate for the inconsistencies. Hilbert Space 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. 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. One may assume that the dot inner product opens the arena to the underlying continuum layer. https://youtu.be/zv_reNX3puQ Zero Point Energy

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. 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. 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, the cosmological constant problem will obtain an explanation. One can hypothesis a source for the mysterious dark energy sipping through form underlying layer Non-locality and quantum entanglement, quantum tunnelling, spin of subatomic particles, virtual particles, nature of fields and alike can obtain explanation as well.

References 1/ Hermann Weyl a German mathematician, Theoretical physicist and philosopher wrote: 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 ."

http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory#cit e_note-11 2/ http://www.scientificamerican.com/article/infinity-logic-law/ 3/ http://www.math.vanderbilt.edu/~schectex/ccc/choice.html