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 know something awe-inspiring; something not easily comprehended but instantly recognizable.  It has power.  It was one of the first "big" words I remember using as a kid; a word we used with authority. "I can count to infinity!" was the ultimate trump card on the playground.   I use it now when I tell my son how much I love him.  "I love you more than infinity" means it's not possible to love you any more because that is everything there is.  You don't need to understand the math behind it, the concept is clear.  Or is it? 

 

            A closer look at infinity reveals that there is much, much more.  Infinity is no longer just a word that rolls easily off the tongue.  It is a word of "infinite"possibilities.  It is math, yet it goes beyond math.  If mathematics can prove that space and time are infinite, then maybe mathematics can answer all those unanswered questions the nuns couldn't.  All I need is a little proof.  

 

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

 

II.  Early History of Infinity

 

            Man has 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 an issue in mathematics.  The first real debate on infinity started among the Greeks sometime between the fourth and fifth century (depending on your reference book).  There were two schools of thought in mathematics during this period. The Atomists believed that you could continue to divide matter into smaller and smaller pieces infinitely many times.  Pythagoras on the other hand, believed that everything was explained by numbers and that the universe was made up of finite natural numbers.

 

            The Greek 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 to prove both sides of the argument. While Zeno's paradoxes did not solve the question of infinity, they did lead to the surprising conclusion that an infinite number of steps could have a finite sum, which is known as "convergence"[1].  They also formed the basis for further debates on infinity which would continue for centuries.

 

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

 

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

 

            Some believe that Archimedes carefully avoided any direct reference to the actual term infinity because it was not generally accepted by the Greeks who based their mathematics on the teachings of Aristotle.  Aristotle was increasingly suspicious of infinity.  He called it "imperfect, unfinished, unlimited and therefore unthinkable”, and finally rejected it on the basis of the belief that "nothing should be accepted into the body of mathematical knowledge that could not be logically proven from previously established facts”[2] .

 

            Although the Greeks discovered a great deal about infinity, after Archimedes the exploration of infinity seemed to come to an end. 

 

            It was not until the sixteenth century that the idea of infinity again became popular among mathematicians, and once again it was π that sparked the interest.  In 1593, French mathematician Francois Viete (1540-1603) proved for the first time that an infinite process could be explicitly expressed as a mathematical formula.  Viete's formula (shown below) proved that π could be calculated solely from the number 2 by a succession of additions, multiplications, divisions, and square root extractions.

            By expressing an infinite process mathematically, Viete allowed infinity to become a legitimate part of mathematics and other mathematicians began to explore the concept of infinity. 

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

            It was John Wallis who first used the symbol ∞ to represent infinity.  He chose it because it represented the fact that one could traverse the curve infinitely often.

 

            In 1671 James Gregory (1638-1675) found another formula involving π which was an infinite series:

p/₄ = 1 - ¹/₃ + ¹/₅ - ¹/₇ + ....

 

            This formula 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.

 

            Also emerging during the Renaissance Period was the "method of indivisibles"which was an improvement on Archimedes earlier "method of 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, as well as a variety of uses in mechanics and optics, many of which were published by Cavalieri in his book "Geometria indivisibilibus continuorum"published in 1635.

 

            In the second half of the seventeenth century Sir Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (independent of each other) introduced the controversial concept of "infinitesimals"which was the key element in their newly discovered differential and integral calculus.  The term infinitesimal describes a theory of diminishing quantities or the idea of things growing infinitely smaller and smaller. This new method of calculus was controversial and was initially rejected by mathematicians but was quickly adopted by physicists, astronomers and engineers because of its ability to solve previously unsolvable problems in mathematics, physics and astronomy.  It was the development of infinitesimals and calculus which led to a branch of mathematics known as analysis which deals with continuity and change.  It was becoming increasingly difficult to ignore the possibility of infinity in mathematics.  These developments ultimately resulted in the deeper explorations of the concept of infinity by Georg Cantor and others.

 

 

 

III.           Georg Cantor and Set Theory

 

            Prior to Georg Cantor (1945-1918) infinity was regarded as a number which was either larger or smaller than all other numbers, but always within some type of undefined limit.  Georg Cantor's work, beginning in 1874, changed the entire foundation on which the concepts of infinity were based.  Cantor accepted the infinite as a part of mathematics by insisting that a set, and an infinite set in particular, must be regarded as a whole.   He also showed that there are different sizes or classes of infinity which can be treated much like other numbers are.  Cantor's theories were in direct opposition to the mathematical beliefs of the time, but Cantor persevered, eventually proving his theories on infinity through his use of set theory.

 

IV.  How Did He Prove It?

            First Cantor defined "set"as "any collection of well-distinguished and well-defined objects considered as a single whole." He showed that sets can be large or small; finite or infinite. He used the term cardinality to describe the "size"of a set. 

            In a finite set, no matter how many objects or numbers there are, given enough time, you can count them all.  Comparing finite sets was easy because all he had to do was count the members of each set and then compare them. 

            An infinite set however contains a never ending number of members which cannot be counted. Examples of infinite sets are sets that contain all of the counting numbers, whole numbers, integers or rational numbers.  Cantor needed a way to compare the size or cardinality of infinite sets.  To accomplish this he used the idea of a one-to-one correspondence.

V.  One-to-One Correspondence

            The following is a simple illustration of a one-to-one correspondence in a finite set.

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

 

            Now let's expand the idea to infinite sets. 

 

 

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

 

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

 

            This same concept can be applied  between the natural numbers and the set of squares, or the set of multiples of five, or the set of prime numbers, or the set of numbers greater than 37 as shown in the examples below. [3]

 

 

 

 

 

 

 

 

 

 

 

 

            Cantor showed for the first time that it was possible to compare and distinguish infinities.  He used the term "aleph null”, denoted as Ào, to describe the cardinality of the set of natural or counting numbers.  Cantor called  Ào a "transfinite number" which means "beyond the finite.”

 

            Next Cantor needed to prove that rational numbers would show a one-to-one correspondence just as natural numbers had.  Rational numbers were more difficult because there were infinitely more of them.  Look between any two numbers on a number line and you will see that there are infinitely more rational numbers.  But in 1874, Cantor made the historic discovery that there are just as many natural numbers as there are rational numbers.  Cantor said that in order to see numbers as sets, we must abandon the tendency to arrange them or see them according to their magnitude.  Cantor was able to move away from looking at numbers in their traditional order such as 1, 2, 3…. And see them in alternative orders such as 1, -1, 2, -2….  This new perspective enabled Cantor to arrange the rational numbers in an infinite array as shown below. 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

 

VI.  Real Numbers and the Continuum

 

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

In order to visualize real numbers, think of a number line.  How many numbers are there between 1 and 2?  Each point on the line represents a number and there are infinitely many points between 1 and 2.   Real numbers are also called continuous because there is no break between each number or point.   It is impossible to know where one number ends and another begins and it is impossible to count real numbers without leaving some of the real numbers out.  Cantor believed that this was a bigger type of infinity and he proved it with a new type of proof called "Cantor's diagonalization argument."

  

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

            Cantor continued to expand his search for infinity and discovered yet another transfinite number which was found to be even larger that c.  He called this set d.  d is the cardinality of all sets of points in space and includes all possible sets of points in the plane and all possible sets of points in three-dimensional space.

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

 

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

 

"If X is any set, then there exists at least one set, the power set of X, which is cardinally larger than X.”

 

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

 

א₀, א₁, א₂, א₃, א₄,…….

The following table illustrates how the sets multiply:

Set	Number of Subsets
2	22
3	23
4	24
5	25
.	.
.	.
א0	א1
א1	א2

 

VI.  From Here to Infinity

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

            Cantor deeply believed that infinity was not just mathematics, infinity was God-given.  Cantor believed (as did others before him in one form or another) that infinity consisted of various levels.  On the lowest transfinite level were the integers and the rational and algebraic numbers.  The transcendental numbers and the continuous real number line belonged to a higher level of infinity and beyond that was the absolute infinity  ….God himself.  He spent his life, and ultimately his mental health, trying to prove the existence of infinity one step at a time. 

            Of course Cantor was not the first to try and prove the existence of God.  Theologians before Cantor had tried to express God as infinite using mathematics as a stepping stone.  For example Nicholas of Cusa, an ecclesiastic and mathematician who studied circles and polygons likened the knowledge of God to a circle.  He visualized human knowledge as a polygon within the circle, with the polygon gaining more and more sides as knowledge increased.  But he concluded that "no matter how much such knowledge grows, it could never reach God's knowledge." It is interesting to me that if the polygon did become a perfect circle, then it would cease to exist as a polygon.   This is similar to the concept of the vanishing point which is thought to be the point at infinity.   

            In another comparison of God to infinity, St. Gregory said that "No matter how far our mind may have progressed in the contemplation of God, it does not attain to what he is, but to what is beneath him."

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

 

             I also realized that mathematics is not just about algebra and geometry.  To truly understand mathematical concepts is to understand the principles of the universe.   Admittedly it would take me an infinitely long time to reach that level of understanding, but it is something to work on. And even though my research on infinity did not answer the questions the nuns avoided, it did give me ideas for further thought and exploration.  More importantly, it opened up my mind to a whole host of possibilities I had never considered before.

 

            And finally, from now on, when I tell my son how much I love him, instead of saying  "I love you more than infinity"I will say "I love you absolutely!”

References:

Aczel, Amir D., "The Mystery of the Aleph:  Mathematics, the Kabbalah, and the Search for Infinity,"2000, Four Walls, Eight Windows, New York, ISBN 156858105X.

 

Guillen, Michael, "Bridges to Infinity:  The Human Side of Mathematics,"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 Mathematics and Philosophy of the Infinite"Princeton University Press, 1979, ISBN 0691024472.

 

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

 

Gamow, George, "One Two Three…Infinity: Facts and Speculations of Science,"New York, Dover Publications, 1988, ISBN 0486256642.

 

Rucker, Rudy, "Infinity and the Mind: The Science and Philosophy of the Infinite,"Boston, Birkhäuser, 1982, ISBN 3764330341.

 

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

 

 

Websites:

 

http://www-history.mcs.st-and.ac.uk/history/HistTopics/Infinity.html "Infinity"(First accessed 11/15/02).

 

http://mathforum.org/dr.math/faq/faq.large.numbers.html "The Math Forum: Ask Dr. Math: FAQ – Large Numbers and Infinity"(First accessed 11/15/02).

 

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

 

http://scidiv.bcc.ctc.edu/Math/infinity.html "Counting to Infinity"(First accessed 11/15/02).

 

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

 

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

 

http://www.math.lsa.umich.edu/~mathsch/courses/Infinity/index.shtml "Infinity"(First accessed 11/15/02).

 

http://members.shaw.ca/quadibloc/math/inf01.htm "Infinite Ordinals"and http://members.shaw.ca/quadibloc/math/infint.htm "Infinity"(Both first accessed on 11/15/02).

 

http://www.c-parr.freeserve.co.uk/hcp/infinity.htm "Infinity"by Hector C. Parr (First accessed 11/15/02).

 

http://www.xrefer.com/entry/552421  "The Oxford University Press – Infinity"(First accessed 11/15/02).

 

http://mathworld.wolfram.com/Infinity.html "Infinity"and all related webpages (First accessed 11/15/02).

 

http://www.sosmath.com/calculus/limcon/limcon04/limcon04.html "Limits and Infinity"(First accessed 11/15/02).