Mary O'Malley

Math G - Dr. Ian Walton

November 25, 2002

Math G Final

Exploring Infinity

∞

I. Introduction

            Infinity.  Just the word commands respect.  Using it implies that you knowsomething awe-inspiring; something not easily comprehended but instantlyrecognizable.  It has power.  It was one of the first "big"words I remember using as a kid; a word we used with authority. "I cancount to infinity!" was the ultimate trump card on the playground.   I use it now when I tell my sonhow much I love him.  "I loveyou more than infinity" means it's not possible to love you any morebecause that is everything there is. You don't need to understand the math behind it, the concept isclear.  Or is it? 

            Acloser look at infinity reveals that there is much, much more.  Infinity is no longer just a word thatrolls easily off the tongue.  It isa word of "infinite"possibilities.  It is math, yet it goes beyond math.  If mathematics can prove that space andtime are infinite, then maybe mathematics can answer all those unansweredquestions the nuns couldn't.  All Ineed is a little proof.  

            Sothis paper will be my first exploration of infinity.  I will take a look at the history of infinity; I will searchfor a definition of infinity; and for proof that it exists mathematically; andfinally, I will take a look at some of the applications of infinity.

II.  Early History of Infinity

            Manhas contemplated infinity in one way or another since the beginning of time,but the Greeks were the first to acknowledge the existence of infinity as anissue in mathematics.  The firstreal debate on infinity started among the Greeks sometime between the fourthand fifth century (depending on your reference book).  There were two schools of thought in mathematics during thisperiod. The Atomists believed that you could continue to divide matter intosmaller and smaller pieces infinitely many times.  Pythagoras on the other hand, believed that everything wasexplained by numbers and that the universe was made up of finite naturalnumbers.

            TheGreek philosopher Zeno of Elea (495-435 B.C.) presented a series of paradoxes(arguments) intended to show that Pythagoras was correct.  However, Zeno's paradoxes seemed toprove both sides of the argument. While Zeno's paradoxes did not solve thequestion of infinity, they did lead to the surprising conclusion that aninfinite number of steps could have a finite sum, which is known as"convergence"[1].  They also formed the basis for furtherdebates on infinity which would continue for centuries.

            TheGreek scientist Archimedes (287-212 B.C.) developed several concepts whichindirectly addressed the idea of infinity.  He was the first to devise a mathematical method ofmeasuring the ratio of the circumference of a circle to its diameter. Archimedes found that byincreasing the number of sides of a polygon, the area of the polygon becamecloser and closer to that of the circle. By continually increasing the numberof sides on the polygons, he was able to "fill the circle"or reduce thearea of the circle not covered by the polygon. Using this method he was able tomeasure the radius of a circle.  He established that the area of the circle was exactly proportional tothe square of its radius, and defined the constant of proportionality –what we now know as 'π'. His method of approximating π came veryclose to the measurement still used today.

            Archimedesalso developed a method for finding the area of a segment of the parabola.  He used a series of smaller and smallertriangles to fill up the area of the parabola.  By making the triangles smaller and smaller he was able tomake them fit the figure. He could then add them up and measure them.  This is known as his "method ofexhaustion."

            Somebelieve that Archimedescarefully avoided any direct reference to the actual term infinity because itwas not generally accepted by the Greeks who based their mathematics on theteachings of Aristotle.  Aristotlewas increasingly suspicious of infinity.  He called it "imperfect, unfinished, unlimited andtherefore unthinkable”, and finally rejected it on the basis of the beliefthat "nothing should be accepted into the body of mathematical knowledgethat could not be logically proven from previously established facts”[2].

            Althoughthe Greeks discovered a great deal about infinity, after Archimedes theexploration of infinity seemed to come to an end. 

            Itwas not until the sixteenth century that the idea of infinity again becamepopular among mathematicians, and once again it was π that sparked theinterest.  In 1593, Frenchmathematician Francois Viete (1540-1603) proved for the first time that aninfinite process could be explicitly expressed as a mathematical formula.  Viete's formula (shown below) provedthat π could be calculated solely from the number 2 by a succession ofadditions, multiplications, divisions, and square root extractions.

            Byexpressing an infinite process mathematically, Viete allowed infinity to becomea legitimate part of mathematics and other mathematicians began to explore theconcept of infinity. 

            Sometimebetween 1650 and 1655, John Wallis (1616-1703) approximated the area of a quarter circle using infinitelysmaller rectangles which is represented by the formula below:  

            Itwas John Wallis who first used the symbol ∞to represent infinity.  He chose itbecause it represented the fact that one could traverse the curve infinitelyoften.

            In1671 James Gregory (1638-1675) found another formula involving π which wasan infinite series:

p/₄ = 1 - ¹/₃ + ¹/₅- ¹/₇ + ....

            Thisformula was also discovered by Gottfried Wilhelm Leibniz (1646-1716)(approximately four years later and completely independent of Gregory's work)and the formula is sometimes referred to as the Gregory-Leibniz series.

            Alsoemerging during the Renaissance Period was the "method ofindivisibles"which was an improvement on Archimedes earlier "methodof exhaustion." Using indivisibles, mathematicians such as Galileo Galilei(1564-1642), Johannes Kepler (1571-1630) and Bonaventura Cavalieri (1598-1647)discovered new ways of measuring various figures and solids of geometry, aswell as a variety of uses in mechanics and optics, many of which were publishedby Cavalieri in his book "Geometria indivisibilibuscontinuorum"published in 1635.

            Inthe second half of the seventeenth century Sir Isaac Newton (1642-1727) andGottfried Wilhelm Leibniz (independent of each other) introduced thecontroversial concept of "infinitesimals"which was the key element intheir newly discovered differential and integral calculus.  The term infinitesimal describes atheory of diminishing quantities or the idea of things growing infinitelysmaller and smaller. This new method ofcalculus was controversial and was initially rejected by mathematicians but wasquickly adopted by physicists, astronomers and engineers because of its abilityto solve previously unsolvable problems in mathematics, physics andastronomy.  It was the developmentof infinitesimals and calculus which led to a branch of mathematics known asanalysis which deals with continuity and change.  It was becoming increasingly difficult to ignore thepossibility of infinity in mathematics. These developments ultimately resulted in the deeper explorations of theconcept of infinity by Georg Cantor and others.

III.          Georg Cantor and Set Theory

            Priorto Georg Cantor (1945-1918) infinity was regarded as a number which was eitherlarger or smaller than all other numbers, but always within some type of undefinedlimit.  Georg Cantor's work,beginning in 1874, changed the entire foundation on which the concepts ofinfinity were based.  Cantoraccepted the infinite as a part of mathematics by insisting that a set, and aninfinite set in particular, must be regarded as a whole.   He also showed that there aredifferent sizes or classes of infinity which can be treated much like othernumbers are.  Cantor's theorieswere in direct opposition to the mathematical beliefs of the time, but Cantorpersevered, eventually proving his theories on infinity through his use of settheory.

IV.  How Did He Prove It?

            FirstCantor defined "set"as "any collection of well-distinguished andwell-defined objects considered as a single whole." He showed that setscan be large or small; finiteor infinite. He used the term cardinalityto describe the "size"of a set. 

            Ina finite set, no matter how many objects or numbers there are, given enoughtime, you can count them all. Comparing finite sets was easy because all he had to do was count themembers of each set and then compare them. 

            Aninfinite set however contains a never ending number of members which cannot becounted. Examples of infinite sets are sets that contain all of the countingnumbers, whole numbers, integers or rational numbers.  Cantor needed a way to compare the size or cardinality ofinfinite sets.  To accomplish thishe used the idea of a one-to-one correspondence.

V. One-to-One Correspondence

            Thefollowing is a simple illustration of a one-to-one correspondence in a finiteset.

Take your hands and match each ofyour fingers on your right hand with the corresponding fingers on your lefthand.  Each hand represents a setcontaining five members, and each member of one set corresponds to a member ofthe other set. 

            Nowlet's expand the idea to infinite sets. 

            Thefirst set contains natural numbers and the second set contains even numberswhich are different.  But if weassign the number n to the set of natural numbers and 2n to the set of evennumbers, we can take any number in the first set and multiply it by 2 to get acorresponding even number in the second set. This shows that the each set hasthe same cardinality (contains the same infinite amount of numbers), and eachset has a one-to-one correspondence with the other. 

            Rememberthat Cantor said we must consider the set as a whole?  Cantor further said that "a collection is infinite if someof its parts are as big as the whole." Looking at the example above, wesee that the "parts"of the whole (or the subset of even numbers) areequal in size to the complete set (or natural numbers) as shown by theirone-to-one correspondence.  We can remove some ofthe elements of an infinite set and it will still be infinite.  

            Thissame concept can be applied between the natural numbers and the set of squares, or the set ofmultiples of five, or the set of prime numbers, or the set of numbers greaterthan 37 as shown in the examples below. [3]

            Cantorshowed for the first time that it was possible to compare and distinguishinfinities.  He used the term"aleph null”, denoted as Ào, to describe the cardinality of the set of natural or countingnumbers.  Cantor called  Ào a "transfinite number" whichmeans "beyond the finite.”

            NextCantor needed to prove that rational numbers would show a one-to-onecorrespondence just as natural numbers had.  Rational numbers were more difficult because there wereinfinitely more of them.  Lookbetween any two numbers on a number line and you will see that there are infinitely more rationalnumbers.  But in 1874, Cantor madethe historic discovery that there are just as many natural numbers as there arerational numbers.  Cantor said thatin order to see numbers as sets, we must abandon the tendency to arrange themor see them according to their magnitude. Cantor was able to move away from looking at numbers in theirtraditional order such as 1, 2, 3…. And see them in alternative orderssuch as 1, -1, 2, -2….  Thisnew perspective enabled Cantor to arrange the rational numbers in an infinitearray as shown below. 

            Bytraversing down and across and then back up again and then repeating theprocedure, the example above contains all the rational numbers (withduplications).  Using this exampleas his first set, Cantor could show a one-to-one correspondence with naturalnumbers in his second set. [4]   Even Cantor was amazed by this discoveryand is quoted as saying "I see it, but I can't believe it!"

VI.  Real Numbers and the Continuum

            Sofar, Cantor has shown a one-to-one correspondence in all of his sets ofinfinity.  However, real numbers(which can be shown as decimals) presented unique problems.

In order to visualize real numbers, think of anumber line.  How many numbers arethere between 1 and 2?  Each pointon the line represents a number and there are infinitely many points between 1and 2.   Real numbers are alsocalled continuous because there is no break between each number or point.   It is impossible to know whereone number ends and another begins and it is impossible to count real numberswithout leaving some of the real numbers out.  Cantorbelieved that this was a bigger type of infinity and he proved it with a newtype of proof called "Cantor's diagonalization argument."

            Using"proof by contradiction"Cantor was able to prove that real numbershave a higher order of infinity than natural numbers and he called thecardinality of this set "c"which stood for continuum.

            Cantorcontinued to expand his search for infinity and discovered yet anothertransfinite number which was found to be even larger that c.  Hecalled this set d.  d is the cardinality of all sets of points in space and includesall possible sets of points in the plane and all possible sets of points inthree-dimensional space.

            Cantornow had three transfinite numbers: Àoand c and d, with Ào being a subset of c, and both Ào and c being subsets of d.  Ào _is defined as being countablyinfinite and c and d are defined as being uncountably infinite.  Cantor now had the beginning of a newset, a set containing infinite members. Cantor called these cardinal numbers.

            ButCantor was convinced that there were infinitely many levels of infinity.  Using what is now called Cantor's Theorem, Cantor statedthat 

"IfX is any set, then there exists at leastone set, the power set of X, which is cardinally larger than X.”

            Cantorbelieved that the set of all subsets always produced a bigger set than theoriginal, a set of higher cardinality. He could see that his cardinal numbers were endless, since for any sethe could produce there was a subset that had greater cardinality than the originalset.  This process would continuewith no end and it is shown as follows:

א₀, א₁, א₂,א₃, א₄,…….

The following tableillustrates how the sets multiply:

VI.  From Here to Infinity

            Cantorhad shown not only infinity, but infinite infinities, each one leading to yet abigger one. But where did it end? If his theories were true there could be no largest set in infinity.  There must always be somethingbigger.  

            Cantordeeply believed that infinity was not just mathematics, infinity wasGod-given.  Cantor believed (as didothers before him in one form or another) that infinity consisted of variouslevels.  On the lowest transfinitelevel were the integers and the rational and algebraic numbers.  The transcendental numbers and thecontinuous real number line belonged to a higher level of infinity and beyondthat was the absolute infinity ….God himself.  Hespent his life, and ultimately his mental health, trying to prove the existenceof infinity one step at a time. 

            Ofcourse Cantor was not the first to try and prove the existence of God.  Theologians before Cantor had tried toexpress God as infinite using mathematics as a stepping stone.  For example Nicholas of Cusa, anecclesiastic and mathematician who studied circles and polygons likened theknowledge of God to a circle.  Hevisualized human knowledge as a polygon within the circle, with the polygongaining more and more sides as knowledge increased.  But he concluded that "no matter how much suchknowledge grows, it could never reach God's knowledge." It is interestingto me that if the polygon did become a perfect circle, then it would cease toexist as a polygon.   This issimilar to the concept of the vanishing point which is thought to be the pointat infinity.   

            Inanother comparison of God to infinity, St. Gregory said that "No matterhow far our mind may have progressed in the contemplation of God, it does notattain to what he is, but to what is beneath him."

            AsI researched infinity I realized several things:  First, that infinity is not just mathematics, it isalso philosophy and theology.  Itis beautifully clear and yet completely confusing at the same time.   Secondly, that much like countingreal numbers, I could not possibly process all of the available information oninfinity in a mere four weeks.

             I also realized that mathematics is not just about algebra and geometry.  To truly understand mathematicalconcepts is to understand the principles of the universe.   Admittedly it would take me aninfinitely long time to reach that level of understanding, but it is somethingto work on. And even though my research on infinity did not answer thequestions the nuns avoided, it did give me ideas for further thought andexploration.  More importantly, itopened up my mind to a whole host of possibilities I had never consideredbefore.

            Andfinally, from now on, when I tell my son how much I love him, instead ofsaying  "I love you more thaninfinity"I will say "I love you absolutely!”

References:

Aczel, Amir D., "The Mystery of the Aleph:  Mathematics, the Kabbalah, and theSearch for Infinity,"2000, Four Walls, Eight Windows, New York, ISBN156858105X.

Guillen, Michael, "Bridges to Infinity:  The Human Side ofMathematics,"1983, ISBN 0874772338.

Swetz, Frank J., "From Five Fingers to Infinity –A Journey through the History of Mathematics"Open Court Publishing, 1994,ISBN 0812691938.

Dauben, Joseph Warren, "Georg Cantor: His Mathematicsand Philosophy of the Infinite"Princeton University Press, 1979, ISBN0691024472.

Maor, Eli, "To Infinity and Beyond: A Cultural Historyof the Infinite"Birkhauser, Boston, 1986, ISBN 0817633251.

Gamow, George, "One Two Three…Infinity: Facts andSpeculations of Science,"New York, Dover Publications, 1988, ISBN0486256642.

Rucker, Rudy, "Infinity and the Mind: The Science andPhilosophy of the Infinite,"Boston, Birkhäuser, 1982, ISBN3764330341.

Grier, David Alan, "A Brief Look at Infinity,"TheChristian Science Monitor, September 24, 2002.

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http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html"Welcome to the Hotel Infinity"(First accessed 11/15/02).

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http://www.shu.edu/projects/reals/infinity/index.html"Countable Infinity"and related web pages that following including"Uncountable Infinity"and "The Principle ofInduction"(First accessed 11/15/02).

http://www.mathacademy.com/pr/minitext/infinity/"You can't get there from here! (First accessed 11/15/02).

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