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A History of Infinity

Wendi Clouse

Math G Final Project

Due April 29, 2002

File written by Adobe Photoshop® 4.0

A Photo of Georg Cantor

Courtesy ofhttp://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Cantor.html

Infinityis a fathomless gulf, into which all things vanish.

MarcusAurelius (121-180) Roman Emperor and Philosopher

Infinity is where things happenthat don’t.

An anonymous schoolboy

When wesay anything is infinite, we signify only that we are not able to conceive theends and bounds of the thing named.

Thomas Hobbes (1588-1679) EnglishPhilosopher

Theinfinite! No other question hasever moved so profoundly the spirit of man; no other idea has so fruitfullystimulated his intellect; yet no other concept stands in greater need ofclarification than that of the infinite . . .

DavidHilbert (1862-1943)

Artists, philosophers, mathematicians and common manhave contemplated the idea of the infinite from the beginning of writtenhistory, and perhaps before. Famous intellectuals throughout the course of time have made many quotesabout the idea of infinity, however to read them all, one would still wonderwhat the actual definition of infinity should be, because the ideas presentedare as varied as the concepts that these famous people put forth in their life’sworks. It seems as though infinityhas a different meaning for each application; Rudy Rucker inhis book Infinity and Beyond discusses “different types of infinity” heintroduces “potential and actual, mathematical and physical, theological andmundane” infinities. Rucker’swritings show just how versatile the concept of infinity can be when applies todifferent disciplines. In theworld of infinity the mystical and the logical go hand in hand.

The first association with the word infinity is oftenthe idea of a number, a great number, a number so great that no type ofnotation can define it. Some wouldargue that the idea of infinity is not a number; but instead the idea of timecontinuing for an eternity, time that is not measurable. For the pious, infinity might representthe divine, the all knowing and all seeing God. The most conceivable idea of infinity resides in the fieldof mathematics, as it is here that we are able to assign a definition andnotation that can somehow make a concept so broad and complicatedfathomable. It is in mathematicsthat the infinite has practical applications to real world functions. In a Nova film presentation aboutunsolved mathematical problems, it was said, “that without the infinite,today’s mathematics would not exist”.

The Greeks first acknowledged the concept of infinityin approximately sixth century B.C. The Greek word for infinite is Apeiron and it translates into numerousmeanings. The literal translationmeans unbounded, but Apeiron was also used to describeinfinite. “Apeiron was a negative, even pejorative word. The original chaos out of which theworld was formed was Apeiron. An arbitrary crooked line was Apeiron. A dirty crumpled handkerchief was Apeiron. Thus, Apeiron need not only mean indefinitely large, but can also meantotally disordered, infinitely complex, subject to no finite determination… InAristotle’s words “… being infinite is a privation, not a perfection but theabsence of limit…”[footnote 1]. TheGreeks were the first to take mathematical ideas from practical applications toan intellectual and philosophical science. However, by the example of the translation of Apeiron, their acknowledgement was not the same as acceptance; theyrequired extensive mathematical proofs to support any mathematical formulas:because their contribution to algebra was very small in comparison to theircontribution to geometry, they did not have the correct notation to expressinfinity in mathematical language. The Greeks commonly practiced avoidance in respect to infinity; it wasdismissed with arguments of “ad absurdum”[footnote2]. A famous Greek philosopher named Zeno makes arguments of ad absurdumclear. He proposed that a runnermust make an infinite number of steps to cross a finish line, therefore makingmotion impossible. In order toreach his destination he must cover half the distance, then again half thedistance, then half again; the process would be repeated an infinite amount oftimes without the runner ever reaching his destination, while it could bephysically proven that motion was possible and the destination would bereached. The mathematical language of the time showed a discrepancy in theory,therefore the idea of infinity was trivialized. Zeno’s paradox would be left unsolved for twenty centuries.

Aristotlechose to view Apeiron as the potentially infiniteinstead of the actual infinite. Inmaking this distinction he was able to avoid addressing the obvious questionsof infinity, such as the continuation of time, and that space is infinitelydivisible. Although the Greeks avoided infinity and did not have the notationto use it in their mathematics, they were able to devise the ratio of thecircumference of a circle to its diameter (or as we call it today pi). Amazingly, pi is the first numberdevised from mathematical process that has a dependant relationship withinfinity (although at the time the mathematical method for finding this ratiowas geometric, not algebraic). Theessential of idea of infinity and its application to mathematics was not tochange again until the renaissance.

Eli Maor tells us “like most other sciences, Europeanmathematics came to a virtual standstill during the long, dark Middleages. It was not until thesixteenth century that the notion of infinity-long since forgotten as ascientific issue and having become instead the subject of theologicalspeculations-underwent its revival” François Viète (1540-1603) inductedinfinity into the official realm of mathematics with a formula showing that pican be calculated from the number 2 with elementary arithmetic operationsinstead of geometry. Mostimportant are the three dots at the end, which tell us to continue theoperation forever. This formula isthe first that expresses a function with the use of infinity.

See the formula below:

2 √2 √2+√2 √2+√2+√2

– = — ∙ ————∙ ———— …

π 2 2 2

Viète’s formula is also the first in a series offormulas that show a relationship between π and infinity. Pi was one of the first issues thatmathematicians pursued in the beginning of the renaissance. In 1650, mathematician John Wallisdiscovered another formula that involved both infinity and π:

π 2∙2∙4∙4∙6∙6…

─ ═ ——————

2 1∙3∙3∙5∙5∙7…

Wallis is the mathematician thatproposed the symbol ∞ to use for infinity, it is conjecture that Wallisborrowed the symbol from the Roman numeral one hundred million, which is alemniscates with a frame around it. See figure one. Then in 1674 Gregory and Leibniz workingindependently found an infinite series.

π/4=1/1-1/3+1/5-1/7+-… All three formulas were the result of finding anapproximation of the value of pi. The exact value of pi will never be defined because it would requireinfinite amount of digits, due to this characteristic, pi is defined as atranscendental number [footnote 3]. Without the help of infinitymathematicians would still be searching to define the expansion of thisinteresting number. Figure above shows the Romannumeral representing one hundred million taken from an inscription dating fromthe year 36 A.D. Number Wordsand Number Symbols-A Cultural History of Numbers, the MIT press, 1977

Figure to the leftgives visual representation of the method of exhaustion and method ofindivisibles. To Infinity andBeyond.

The next big development with infinity took placeduring the second half of the seventeenth century. Sir Isaac Newton and Gottfried Wilhelm Leibniz workingindependently invented differential and integral calculus. This new math was based loosely on theold ideas of the method of exhaustion and method of indivisibles, which hadpreviously been used to approximate the areas and volumes of plane figures andsolids [footnote 4]. Infinity was a very large part of this new math and appearedin the form of the infinitesimal (verysmall). Not only could we now findarea and volume of plane figures, but also we could now have a way to use theseareas and volumes for practical applications such as mechanics and optics. Physicist, astronomers and engineershad a new tool to solve previously unanswered questions.

Calculusalso includes the ideas of convergence and limit. It was with these concepts that it became possible tofinally solve the ancient Greek paradox put forth be Zeno centuries ago. Thesolution is put forth by Eli Maor “by first covering one-half the distancebetween the runner’s starting and ending points, then half the remainingdistance, and so on, the runner will cover the total distance of the sum, thisinfinite series, has the property that no matter how many terms we add up, wewill never reach one, let alone exceed one; and yet we can make the sum get asclose to one as we please, simply by adding a large number of terms. We say the series converges to one orthat it has the number one as its limit. Assuming that the runner maintains a steady speed, the time intervalsthat it takes him to cover these distances will also cover the same series;therefore he will cover the entire amount of distance in a finite span of time”The explanation of the runner’s paradox offers a simple insight into thedefinitions of limit and convergence and how they operate with infinitely smallspaces on the number line. Theconcept of convergence and limit is expressed mathematically as: an→L as n →∞ thus a sequence a1,a2, a3…,an…converges to the limit L. To obtain an infinite series from asequence, we derive an ever-increasing sum described by Maor as “the series hasthe (infinite) sum S” or a1+a2+a3…=S is the proper notation. To determine whether or not a series has a limit,understanding of calculus is necessary, but it is possible to tell when aseries does not converge to the limit. A series of whole numbers such as 2+4+6… does not converge because thesum will grow beyond all bounds, however terms can get smaller and still notconverge. The harmonic series is agood example of the phenomenon. This series is derived by adding reciprocals of the natural numbers (1/1+1/2+1/3+1/4…); the terms become smaller, butdon’t converge, after enough time they will reach an infinite value. There are other types of series thatmerit exploration; unfortunately due to the broad range of informationavailable with regards to infinity they will not be included in this research.

In 1847 a mathematician by the name of Georg Cantorpublished the first of many papers that would change the concept ofinfinity. Cantor was the first toaccept the idea of actual infinity instead of the potentially infinite. Instead of looking at infinite as the largestor smallest group of numbers, he chose to see infinity as a completeentity. He determined thatdifferent levels of infinity exist mathematically and defined them in settheory as countable and uncountable. Cantor’s conclusions revolved around two main issues the first was thatof set theory and the second was one to one correspondence. Prior to Cantor’s publishing’s, theonly way to denote the idea of infinity was the lemniscates or lazy eight, butCantor would provide several new symbols of notation for the concept ofinfinity.

One can without qualification say that the transfinite numbers stand or fall with the infinite irrationals; their inmost essence is the same, for these are definitely laid out instances or modifications of the actual infinite.
– Georg Cantor

Quoteabove provided by http://www.mathacademy.com/pr/minitext/infinity/

In the study of natural numbers, integers, rational numbers and wholenumbers Cantor determined they all were the same size infinity and used the notationof aleph-null (א0) to define them. The method used to determine this “size” is described as a one to onecorrespondence between two sets of numbers that have the same cardinality. This method is easy to see if we use afinite set for an example. When weuse an infinite set, the one to one correspondence behaves in the same manner.

Example of one to one correspondence with a finite set

(1,2,3,4,5,) This set has a cardinality of five.

↕ ↕ ↕ ↕ ↕

(3,3,3,2,8) This sethas a cardinality of five

Both sets one and twohave the same cardinality as they contain 5 members each thus (set) = (set).

Example of one to one correspondence with an infiniteset found in Mathematical Ideas

(1,2,3,4…,n, …) contains an infinite number ofcounting numbers, thus cardinality א0

↕ ↕ ↕ ↕ ↕↕ ↕…

(0,1,2,3…,n-1,…)contains an infinite number of whole numbers, thus cardinality א0

Both sets have thecardinality of א0, because there is a corresponding number represented byn in set one, and n-1 in set two on the number line. No matter what point is chosen in set one (n), (n-1) willrepresent the correct correspondence in set.

Cantor also shows one toone correspondence of rational numbers. This method is a little more complex than the original one to one lineup.

See figure belowshowing rational number one to one correspondence courtesy of http://www.mathacademy.com/pr/minitext/infinity/

After determining the size ofnatural numbers, integers, rational numbers and whole numbers Cantor made astartling discovery involving irrational numbers and real numbers. Cantor could not put these sets intoone to one correspondence with a countable set whose size was aleph-null. Afterdetermining that these sets had different cardinality, he put forth theContinuum Hypothesis and produced transfinite set theory.

Irrational numbers are a propersubset of real numbers, thus intuition tells us that if rational numbers aresize aleph-null, irrational numbers must be size aleph-null. However, if you were to add the set of rationalnumbers and the set of irrational numbers the sum is all real numbers. Because rational numbers and irrationalnumbers make up all real numbers, it would be determined that the size of realnumbers and irrationals would be a larger size of infinity constructed of thesets from proper subsets of aleph- null. Cantor named this larger infinity c for continuum.

The following examplesare taken from Infinite Ink Mathematics website and help explain inmathematical terms the idea of the Continuum Hypothesis:

R= Real

N= Natural

Z= Integers

Q= Rational

1) א0<C=2aleph-null

2) The following is a simplified model that will helpexplain the continuum hypothesis.

aleph₀ < card(R) = c = card((0,1)) =card(P(N)) = 2^aleph₀

3) Cantor showed that real numbers couldn’t beput in one to one correspondence with natural numbers. Real numbers are a superset of naturalstheir size is larger.

aleph₀ < card(R) = c

Inessence Cantor had discovered sets that have size greater than c. This was accomplished by showing theset of all subsets of a defined set will have more members than the originalset; then, the process can be repeated …Maor calls this phenomenon an “infinitehierarchy of sets, in which each new set (of subsets) has a greater power thanthe one from which it was derived”. Cantor used the notation of 2א0, 22 א0 and so on. Cantor also defined the uniquearithmetic used in regards to infinity sets [footnote 6].

Cantor’swork on set theory was unfortunately met with peer criticism, especially fromhis former teacher Leopold Kronecker. Bitter public correspondence from Kronecker undermined Cantors standingin the mathematical community. This new concept Cantor proposed, was a juxtaposition of everything thatthe mathematical community had previously established. Although he was a brilliant man he wasplagued with frequent bouts of depression that hindered his work. He was institutionalized many times inhis lifetime, and in 1918 he was committed for the last time, it was there thathe died unaware of the importance of the effect his life’s work would have onthe history of mathematics.

DavidHilbert at the turn of the century provided the Second International Congressof Mathematicians a list of 23 unsolved problems that were in his opinion ofthe utmost importance. One of thefirst questions was one that Cantor himself had wrestled with. The question isposed best on page 64 of Beyond Infinity “Cantor created a hierarchy ofinfinities represented by the transfinite cardinal aleph-null, two aleph-null,two to the second power of aleph-null… But he also showed that the real numbers have a transfinite cardinal c,which is greater than aleph-null, and in fact he was able to prove that 2aleph-null is = to c, i.e. that the set of all the subsets of the natural numbershas exactly as many elements as the set of all real numbers. The question which presented itself toCantor was, can one find a set with a power between aleph-null and c?” The answer to Cantors question wouldhave to wait for over 60 years before it would be answered. It was 1963 when the question wasfinally addressed, as it turns out the hypothesis could be either true, orfalse [footnote7]. The answer hinged on the axioms of set theory.

In1902 the Russell paradox was introduced. The question was, is A an element of A? If A is an element of A, and A is not a member of A, A leadsto a contradiction, Set construction itself becomes a paradox. The key to Russell’s paradox lieswithin the unrestricted comprehension axiom. This axiom states P(x) as a freevariable, determines a set whose members satisfy P(x). Most attempts at solvingRussell’s paradox make attempts to restrict this axiom.

In1908 Ernst Zermelo was “the first to attempt an “axiomatisation” of set theory”(St. Andrews). The axiom of choiceis the basis of Zermelo’s proof that every set can be well ordered. The definition for “well ordered” is asfollows: A set S is well orderedif it has a relationship<defined on which it satisfied 3 properties.

1. for any element A, B in S A=B, A<b or B<A

2. for every A,B,C in S with A<B and B<C then A,C.

3. Every non-empty subset of S has at least an element. The set of a natural number with theusual ordering is well ordered, but the set of integers is not since the subsetof negative integers has no least elements.

Zermeloproved that every set could be well ordered and the axiom of Choice is thebasis for his proof.

In1940 K. Godel showed the axiom of choice couldn’t be disproved using the otheraxioms of set theory. Then in 1963a mathematician named Paul Cohen used the method of forcing to prove the axiomof choice was independent of set theory and of the general continuumhypothesis. Cohen’s research tellsus that the continuum hypothesis can be regarded as an additional axiom thatcan be accepted or rejected by choice.

Infinity’sjourney doesn’t end with Cohen; it has found many applications in geometry,cosmology and even in art. It is asubject almost as expansive as its definition. Great artists such as M.C. Escher and Bach have used infinityfor a muse; Bach with his canon diversi and Escher with a multitude ofworks. Even ancient peoples ofmany civilizations used infinitely repeating patterns in their artisticexpression. In geometry infinityshows up in graphing functions, the inversion of a circle, geographic maps,fractals and of course more paradox. In cosmology infinity explores the ancient world, the expanding universeand the modern atomist. At thisvery moment in time the Hubble Telescope continues its journey deep into spaceto discover phenomenon humanspreviously would not have believed. Is this the type of research that the concept of infinity has in storefor us in the future?

Spiral Galaxy photo to the right courtesy ofhttp://heritage.stsci.edu/2002/03/table.html

Footnote 1- Quote taken from Infinity of the MindPage 2 and 3

Footnote 2- “The infinite was taboo” said Tobias Dantzig in Number-the Language of Science “ithad to be kept out at all costs; or, failing this, camouflaged by arguments ad absurdumand the like”.

Footnote 3- a number is called algebraic if it is asolution of an algebraic equation, i.e., a polynomial equation whosecoefficients are integers. Thusthe number 5, -2/3,Ö2, and 2+Ö3are all algebraic because they are the solutions of the equations x-5=0,3x+2=0, x squared-2=0 and x squared-4x+1=0, respectively. Transcendental number- a number istranscendental if it is not algebraic; that is, if it is not a solution of anyalgebraic expression” pg11 To Infinity and Beyond.

Footnote 4-Method of exhaustion was the first way tomeasure the area of a segment of a parabola. This method took small segments of the “shape” and segmentedit into figures that could be measured, and then the smaller units werecombined to approximate the total volume or area of the shape in question.

Footnote 5- Solution to Zeno’s paradox, “by firstcovering one-half the distance between the runner’s starting and ending points,then half the remaining distance, and so on, the runner will cover the total distanceof the sum, this infinite series, has the property that no matter how manyterms we add up, we will never reach 1 let alone exceed one; and yet we canmake the sum get as close to one as we please, simply by adding a large numberof terms. We say the seriesconverges to one or that it has the number one as its limit. Assuming that the runner maintains asteady speed, the time intervals that it takes him to cover these distanceswill also cover the same series; therefore he will cover the entire amount ofdistance in a finite span of time”

Footnote 6- an example of the arithmetic in question is“א0+ א0= א0 (if we combine two denumerablesets, the united set will still be denumerable)… and

א0·א0=א0 (the union of a denumerably infinite number of a denumerable sets isstill denumerable)” ToInfinity and Beyond.

Footnote 7- Continuum hypothesis isindependent of the axioms of set theory. This hypothesis cannot be proven, and it also cannot be refuted. It can be accepted as an additionalaxiom to set theory or rejected.The Continuum Hypothesis was for long regarded the most famous unsolved problemin mathematics. In 1963, the works of Godel and Cohen proved the independenceof the Continuum Hypothesis within the framework of an axiomatic set theory.