This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it.

A History of Infinity

Wendi Clouse

Math G Final Project

Due April 29, 2002

File written by Adobe PhotoshopÆ 4.0

A Photo of Georg Cantor

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

Infinity is a fathomless gulf, into which all things vanish.

Marcus Aurelius (121-180) Roman Emperor and Philosopher

Infinity is where things happen that donít.

An anonymous schoolboy

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

Thomas Hobbes (1588-1679) English Philosopher

The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite . . .

David Hilbert (1862-1943)

Artists, philosophers, mathematicians and common man have contemplated the idea of the infinite from the beginning of written history, and perhaps before. Famous intellectuals throughout the course of time have made many quotes about the idea of infinity, however to read them all, one would still wonder what the actual definition of infinity should be, because the ideas presented are as varied as the concepts that these famous people put forth in their lifeís works. It seems as though infinity has a different meaning for each application; Rudy Rucker in his book Infinity and Beyond discusses ìdifferent types of infinityî he introduces ìpotential and actual, mathematical and physical, theological and mundaneî infinities. Ruckerís writings show just how versatile the concept of infinity can be when applies to different disciplines. In the world of infinity the mystical and the logical go hand in hand.

The first association with the word infinity is often the idea of a number, a great number, a number so great that no type of notation can define it. Some would argue that the idea of infinity is not a number; but instead the idea of time continuing for an eternity, time that is not measurable. For the pious, infinity might represent the divine, the all knowing and all seeing God. The most conceivable idea of infinity resides in the field of mathematics, as it is here that we are able to assign a definition and notation that can somehow make a concept so broad and complicated fathomable. It is in mathematics that the infinite has practical applications to real world functions. In a Nova film presentation about unsolved mathematical problems, it was said, ìthat without the infinite, todayís mathematics would not existî.

The Greeks first acknowledged the concept of infinity in approximately sixth century B.C. The Greek word for infinite is Apeiron and it translates into numerous meanings. The literal translation means unbounded, but Apeiron was also used to describe infinite. ìApeiron was a negative, even pejorative word. The original chaos out of which the world 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 mean totally disordered, infinitely complex, subject to no finite determinationÖ In Aristotleís words ìÖ being infinite is a privation, not a perfection but the absence of limitÖî[footnote 1]. The Greeks were the first to take mathematical ideas from practical applications to an intellectual and philosophical science. However, by the example of the translation of Apeiron, their acknowledgement was not the same as acceptance; they required extensive mathematical proofs to support any mathematical formulas: because their contribution to algebra was very small in comparison to their contribution to geometry, they did not have the correct notation to express infinity in mathematical language. The Greeks commonly practiced avoidance in respect to infinity; it was dismissed with arguments of ìad absurdumî[footnote 2]. A famous Greek philosopher named Zeno makes arguments of ad absurdum clear. He proposed that a runner must make an infinite number of steps to cross a finish line, therefore making motion impossible. In order to reach his destination he must cover half the distance, then again half the distance, then half again; the process would be repeated an infinite amount of times without the runner ever reaching his destination, while it could be physically proven that motion was possible and the destination would be reached. The mathematical language of the time showed a discrepancy in theory, therefore the idea of infinity was trivialized. Zenoís paradox would be left unsolved for twenty centuries.

Aristotle chose to view Apeiron as the potentially infinite instead of the actual infinite. In making this distinction he was able to avoid addressing the obvious questions of infinity, such as the continuation of time, and that space is infinitely divisible. Although the Greeks avoided infinity and did not have the notation to use it in their mathematics, they were able to devise the ratio of the circumference of a circle to its diameter (or as we call it today pi). Amazingly, pi is the first number devised from mathematical process that has a dependant relationship with infinity (although at the time the mathematical method for finding this ratio was geometric, not algebraic). The essential of idea of infinity and its application to mathematics was not to change again until the renaissance.

Eli Maor tells us ìlike most other sciences, European mathematics came to a virtual standstill during the long, dark Middle ages. It was not until the sixteenth century that the notion of infinity-long since forgotten as a scientific issue and having become instead the subject of theological speculations-underwent its revivalî FranÁois ViËte (1540-1603) inducted infinity into the official realm of mathematics with a formula showing that pi can be calculated from the number 2 with elementary arithmetic operations instead of geometry. Most important are the three dots at the end, which tell us to continue the operation forever. This formula is the first that expresses a function with the use of infinity.

See the formula below:

2 √2 √2+√2 √2+√2+√2

ñ = ó ∙ óóóó ∙ óóóó Ö

π 2 2 2

ViËteís formula is also the first in a series of formulas that show a relationship between π and infinity. Pi was one of the first issues that mathematicians pursued in the beginning of the renaissance. In 1650, mathematician John Wallis discovered another formula that involved both infinity and π:

π 2∙2∙4∙4∙6∙6Ö

─ ═ óóóóóó

2 1∙3∙3∙5∙5∙7Ö

File written by Adobe PhotoshopÆ 5.0 Wallis is the mathematician that proposed the symbol ∞ to use for infinity, it is conjecture that Wallis borrowed the symbol from the Roman numeral one hundred million, which is a lemniscates with a frame around it. See figure one. Then in 1674 Gregory and Leibniz working independently found an infinite series.

π/4=1/1-1/3+1/5-1/7+-Ö All three formulas were the result of finding an approximation of the value of pi. The exact value of pi will never be defined because it would require infinite amount of digits, due to this characteristic, pi is defined as a transcendental number [footnote 3]. Without the help of infinity mathematicians would still be searching to define the expansion of this interesting number. Figure above shows the Roman numeral representing one hundred million taken from an inscription dating from the year 36 A.D. Number Words and Number Symbols-A Cultural History of Numbers, the MIT press, 1977

Figure to the left gives visual representation of the method of exhaustion and method of indivisibles. To Infinity and Beyond.

The next big development with infinity took place during the second half of the seventeenth century. Sir Isaac Newton and Gottfried Wilhelm Leibniz working independently invented differential and integral calculus. This new math was based loosely on the old ideas of the method of exhaustion and method of indivisibles, which had previously been used to approximate the areas and volumes of plane figures and solids [footnote 4]. Infinity was a very large part of this new math and appeared in the form of the infinitesimal (very small). Not only could we now find area and volume of plane figures, but also we could now have a way to use these areas and volumes for practical applications such as mechanics and optics. Physicist, astronomers and engineers had a new tool to solve previously unanswered questions.

Calculus also includes the ideas of convergence and limit. It was with these concepts that it became possible to finally solve the ancient Greek paradox put forth be Zeno centuries ago. The solution is put forth by Eli Maor ìby first covering one-half the distance between the runnerís starting and ending points, then half the remaining distance, and so on, the runner will cover the total distance of the sum, this infinite series, has the property that no matter how many terms we add up, we will never reach one, let alone exceed one; and yet we can make the sum get as close to one as we please, simply by adding a large number of terms. We say the series converges to one or that it has the number one as its limit. Assuming that the runner maintains a steady speed, the time intervals that 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 timeî The explanation of the runnerís paradox offers a simple insight into the definitions of limit and convergence and how they operate with infinitely small spaces on the number line. The concept of convergence and limit is expressed mathematically as: an→L as n → ∞ thus a sequence a1, a2, a3Ö,anÖ converges to the limit L. To obtain an infinite series from a sequence, we derive an ever-increasing sum described by Maor as ìthe series has the (infinite) sum Sî or a1+a2+a3Ö=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 a series does not converge to the limit. A series of whole numbers such as 2+4+6Ö does not converge because the sum will grow beyond all bounds, however terms can get smaller and still not converge. The harmonic series is a good example of the phenomenon. This series is derived by adding reciprocals of the natural numbers (1/1+1/2+1/3+1/4Ö); the terms become smaller, but donít converge, after enough time they will reach an infinite value. There are other types of series that merit exploration; unfortunately due to the broad range of information available with regards to infinity they will not be included in this research.

In 1847 a mathematician by the name of Georg Cantor published the first of many papers that would change the concept of infinity. Cantor was the first to accept the idea of actual infinity instead of the potentially infinite. Instead of looking at infinite as the largest or smallest group of numbers, he chose to see infinity as a complete entity. He determined that different levels of infinity exist mathematically and defined them in set theory as countable and uncountable. Cantorís conclusions revolved around two main issues the first was that of set theory and the second was one to one correspondence. Prior to Cantorís publishingís, the only way to denote the idea of infinity was the lemniscates or lazy eight, but Cantor would provide several new symbols of notation for the concept of infinity.

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.
ñ Georg Cantor

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

In the study of natural numbers, integers, rational numbers and whole numbers Cantor determined they all were the same size infinity and used the notation of aleph-null (א0) to define them. The method used to determine this ìsizeî is described as a one to one correspondence between two sets of numbers that have the same cardinality. This method is easy to see if we use a finite set for an example. When we use 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 set has a cardinality of five

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

Example of one to one correspondence with an infinite set found in Mathematical Ideas

(1,2,3,4Ö,n, Ö) contains an infinite number of counting numbers, thus cardinality א0

↕ ↕ ↕ ↕ ↕ ↕ ↕Ö

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

Both sets have the cardinality of א0, because there is a corresponding number represented by n 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) will represent the correct correspondence in set.

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

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

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

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

The following examples are taken from Infinite Ink Mathematics website and help explain in mathematical 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 help explain the continuum hypothesis.

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

3) Cantor showed that real numbers couldnít be put in one to one correspondence with natural numbers. Real numbers are a superset of naturals their size is larger.

aleph₀ < card(R) = c

In essence Cantor had discovered sets that have size greater than c. This was accomplished by showing the set of all subsets of a defined set will have more members than the original set; then, the process can be repeated ÖMaor calls this phenomenon an ìinfinite hierarchy of sets, in which each new set (of subsets) has a greater power than the one from which it was derivedî. Cantor used the notation of 2א0 , 22 א0 and so on. Cantor also defined the unique arithmetic used in regards to infinity sets [footnote 6].

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

David Hilbert at the turn of the century provided the Second International Congress of Mathematicians a list of 23 unsolved problems that were in his opinion of the utmost importance. One of the first questions was one that Cantor himself had wrestled with. The question is posed best on page 64 of Beyond Infinity ìCantor created a hierarchy of infinities represented by the transfinite cardinal aleph-null, two aleph-null, two to the second power of aleph-nullÖ 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 2 aleph-null is = to c, i.e. that the set of all the subsets of the natural numbers has exactly as many elements as the set of all real numbers. The question which presented itself to Cantor was, can one find a set with a power between aleph-null and c?î The answer to Cantors question would have to wait for over 60 years before it would be answered. It was 1963 when the question was finally addressed, as it turns out the hypothesis could be either true, or false [footnote7]. The answer hinged on the axioms of set theory.

In 1902 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 leads to a contradiction, Set construction itself becomes a paradox. The key to Russellís paradox lies within the unrestricted comprehension axiom. This axiom states P(x) as a free variable, determines a set whose members satisfy P(x). Most attempts at solving Russellís paradox make attempts to restrict this axiom.

In 1908 Ernst Zermelo was ìthe first to attempt an ìaxiomatisationî of set theoryî (St. Andrews). The axiom of choice is the basis of Zermeloís proof that every set can be well ordered. The definition for ìwell orderedî is as follows: A set S is well ordered if 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 the usual ordering is well ordered, but the set of integers is not since the subset of negative integers has no least elements.

Zermelo proved that every set could be well ordered and the axiom of Choice is the basis for his proof.

In 1940 K. Godel showed the axiom of choice couldnít be disproved using the other axioms of set theory. Then in 1963 a mathematician named Paul Cohen used the method of forcing to prove the axiom of choice was independent of set theory and of the general continuum hypothesis. Cohenís research tells us that the continuum hypothesis can be regarded as an additional axiom that can be accepted or rejected by choice.

Infinityís journey doesnít end with Cohen; it has found many applications in geometry, cosmology and even in art. It is a subject almost as expansive as its definition. Great artists such as M.C. Escher and Bach have used infinity for a muse; Bach with his canon diversi and Escher with a multitude of works. Even ancient peoples of many civilizations used infinitely repeating patterns in their artistic expression. In geometry infinity shows 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 universe and the modern atomist. At this very moment in time the Hubble Telescope continues its journey deep into space to discover phenomenon humans previously would not have believed. Is this the type of research that the concept of infinity has in store for us in the future?

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

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

Footnote 2- ìThe infinite was tabooî said Tobias Dantzig in Number- the Language of Science ìit had to be kept out at all costs; or, failing this, camouflaged by arguments ad absurdum and the likeî.

Footnote 3- a number is called algebraic if it is a solution of an algebraic equation, i.e., a polynomial equation whose coefficients are integers. Thus the number 5, -2/3,÷2, and 2+÷3 are 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 is transcendental if it is not algebraic; that is, if it is not a solution of any algebraic expressionî pg11 To Infinity and Beyond.

Footnote 4-Method of exhaustion was the first way to measure the area of a segment of a parabola. This method took small segments of the ìshapeî and segmented it into figures that could be measured, and then the smaller units were combined to approximate the total volume or area of the shape in question.

Footnote 5- Solution to Zenoís paradox, ìby first covering one-half the distance between the runnerís starting and ending points, then half the remaining distance, and so on, the runner will cover the total distance of the sum, this infinite series, has the property that no matter how many terms we add up, we will never reach 1 let alone exceed one; and yet we can make the sum get as close to one as we please, simply by adding a large number of terms. We say the series converges to one or that it has the number one as its limit. Assuming that the runner maintains a steady speed, the time intervals that 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 timeî

Footnote 6- an example of the arithmetic in question is ìא0+ א0= א0 (if we combine two denumerable sets, the united set will still be denumerable)Ö and

א0·א0= א0 (the union of a denumerably infinite number of a denumerable sets is still denumerable)î To Infinity and Beyond.

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