the history and evolution of the concept of infinity
TRANSCRIPT
The History and Evolution of the Concept of Infinity
John H. Batchelor
Honors 499 – Senior Project
Dr. Bruce Mericle, Project Advisor
July 26, 2002
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Table of Contents.
Page Chapter / Section
3 Introduction.
4 1. A Brief History of Infinity.
7 2. The Contributions of Euclid and Euler.
16 3. Gauss’s Contributions.
18 4. The Contributions of Georg Cantor.
29 5. Cardinal Numbers and One-to-One Correspondence.
36 6. The Uncountable Set of Real Numbers.
37 7. Limits and Convergence.
39 8. The Geometric and Harmonic Series.
40 9. Fermat’s Theorems.
42 10. Geographic Maps and Inversion in a Circle.
43 11. Einstein’s General Theory of Relativity.
44 12. Special Categories of Numbers.
48 13. Non-Euclidean Geometry.
51 14. Religion and Infinity.
52 15. Astronomy and Infinity.
55 16. Infinity and Art.
57 17. Paradoxes and Antinomies.
59 18. Unsolved Problems Regarding Infinity.
62 Afterword.
63 Appendices.
63 I. Timeline of Major Events.
68 II. Proof that2 is an Irrational Number.
70 Works Cited / Bibliography Page.
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Introduction.
People interpret the concept of infinity in a variety of ways. People often see
infinity as a “number” that is greater than all other numbers. Many philosophers and
theologians believe that infinity represents God or eternity. In some primitive tribes,
infinity began at three, since the members referred to anything larger than three as
“many” and did not considered it countable (Maor 2). The early Greeks were the first
people to study infinity in a formal setting. They began to acknowledge the concept of
infinity during the sixth century B.C. (Maor 3). The Greeks feared infinity, but later
mathematicians such as Euler and Cantor studied it enthusiastically (Dunham, Master
47). Georg Cantor had a profound impact on the study of the infinite. He made the
remarkable discovery of the one-to-one correspondence between the square and the
interval (Dauben 55). In 1900, David Hilbert emphasized the importance of Cantor’s
Continuum Hypothesis. The Continuum Hypothesis states that there is no transfinite
cardinal number strictly between 0א and c (Dunham, Genius 282). Other mathematicians
such as Gauss, Newton, and Fermat have also made significant contributions to the study
of infinity (Laubenbacher 13). Numerous paradoxes related to infinity exist, and they
continue to puzzle mathematicians. For example, Russell’s Paradox involves sets that do
not contain themselves as elements (Maor 255). Mathematicians continue to study many
unsolved problems. Goldbach’s Conjecture, the infinitude of twin primes, and the search
for an odd perfect number all represent unsolved problems related to infinity (Maor 23).
The concept of infinity has had a major effect on religion, art, and astronomy.
Astronomers have long wondered whether the universe is finite or infinite. Based on
Einstein’s general theory of relativity, it appears that the universe is finite but unbounded
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(Laubenbacher 16). The study of the infinitesimal has led to questions of the existence of
an “ultimate particle” from which all matter is created (Maor 225). Artists such as M.C.
Escher have creatively depicted infinity through their paintings and sculptures
(Laubenbacher 53). Infinity is a fascinating concept that has evolved and matured since
the early Greeks first studied it.
1. A Brief History of Infinity.
The early Greeks were the first people to study the concept of infinity in a formal
setting. They were the first people to use a mathematical process to determine the value
of π. The previous attempts to determine this value involved actual measurements of a
circle’s diameter and circumference (Maor 5). Despite the good intentions and
enthusiasm of the Greek mathematicians, other mathematicians accused them of being
too ambiguous in their study of infinity (Dauben 107). The Greeks believed that
everything in nature could be represented using ratios of integers, and they assumed that
√2 was a rational number. One can express rational numbers as ratios of integers, but one
cannot express irrational numbers in this way. After they discovered that √2 is an
irrational number, they temporarily refused to accept that √2 is a number at all (Maor 46).
According to Tobias Dantzig, “The attempt to apply rational arithmetic to a problem in
geometry resulted in the first crisis in the history of mathematics. The two relatively
simple problems – the determination of the diagonal of a square and that of the
circumference of a circle – revealed the existence of new mathematical beings for which
no place could be found within the rational domain” (qtd. in Maor 44). Mathematicians
have since discovered many other important irrational numbers. The golden ratio, which
is denoted φ, is an irrational number (Maor 51). Archimedes developed a method for
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determining the circumference and area of a circle. To determine a circle’s
circumference, he circumscribed a circle by regular polygons with progressively more
sides. Polygons with greater numbers of sides provide a better approximation of a circle’s
circumference. The Greeks used this method to determine a value for π. To approximate
π, they simply divided the value for the polygon’s perimeter by the circle’s diameter. The
value for π represents the limit of these values as the number of sides of the regular
polygon approaches infinity (Dunham, Genius 29, 91). Archimedes made the following
statement about large but finite numbers in The Sand-Reckoner: “Many people believe,
King Gelon, that the grains of sand are without number. Others think that although their
number is not without limit, no number can ever be named which will be greater than the
number of grains of sand. But I shall try to prove to you that among the numbers which I
have named there are those which exceed the number of grains in a heap of sand the size
not only of the earth, but even of the universe” (qtd. in Maor 16). Aristotle also studied
the concept of infinity, particularly with regard to its paradoxes. He believed in the
potential infinite, but he did not believe in the idea of the actual infinite (Laubenbacher
54). Bertrand Russell was originally skeptical of the concept of the actual infinite, but he
later recognized the similarities between Cantor’s theory of the infinite and his own
theory (Miller 2).
Bernhard Bolzano, a theologian, also studied the concept of infinity. He took a
particular interest in the paradoxes related to infinity. Bolzano emphasized the property
of a one-to-one correspondence between a set and a proper subset of the set. He is famous
for his distinction between potential and actual infinities (Laubenbacher 55). A one-to-
one correspondence can exist between an infinite set and a subset of the set.
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Mathematicians refer to this as Galileo’s Paradox (Maor 57). An important connection
exists between mathematics and philosophy with regard to the study of the infinite. Georg
Cantor took a particular interest in developing a philosophical approach to analyzing
infinite sets. Many philosophers and theologians were opposed to Cantor’s theory of the
actual infinite. Cantor defended his theory against such arguments as Aristotle’s concept
of “annihilation of number.” Cantor pointed out that the arguments against his theory
relied on the assumption that infinite sets will behave according to the properties of finite
sets. He showed that this assumption is not always true, and he proved that the
“annihilation of number” concept was not true for infinite sets (Dauben 122).
Georg Cantor made many important contributions to the study of set theory. He
proved the important Uniqueness Theorem with regard to finite exceptional sets (Dauben
35). Ernst Zermelo, a German mathematician, successfully axiomatized Cantor’s set
theory (Laubenbacher 67). He developed a system of seven axioms for set theory.
Zermelo is particularly famous for his proof of the Well-Ordering Theorem (Dauben
253). Adolf Fraenkel contributed to set theory, and his contributions included the
Substitution Axiom. Fraenkel revised Zermelo’s set theory, and mathematicians now
refer to the resulting theory as Zermelo-Fraenkel set theory (Cohen 76). The Axiom of
Choice is a particularly fascinating part of the Zermelo-Fraenkel set theory (Cohen 85).
In 1963, Paul Cohen showed that the Axiom of Choice is independent of the other set
theory axioms (Laubenbacher 67). Many mathematicians have contributed to the study of
infinity, including Euclid, Euler, Fermat, Cantor, and Gauss. Non-Euclidean geometry is
an important concept related to infinity. There are many paradoxes and unsolved
problems involving the concept of infinity. Some theologians have objected to the study
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of the actual infinite, since they feel that it contradicts their understanding of God (Maor
190). Astronomers are also concerned with infinity, since they have long wondered
whether the universe is finite. Artists such as M.C. Escher and Max Bill have depicted
infinity in their drawings and sculptures (Laubenbacher 52).
2. The Contributions of Euclid and Euler.
Between 440 B.C. and 300 B.C., several mathematicians and philosophers
contributed to the study of infinity. When Plato was young, he studied in Athens under
the direction of Socrates. Plato’s work represents historians’ primary source of
information with regard to Socrates. In 387 B.C., Plato founded the Academy in Athens.
This was a well-respected institution, and people regarded it as one of the finest
intellectual centers in Greece. Although historians generally do not regard Plato as a
mathematician, he believed that mathematics was the perfect training for the human
mind. He took a particular interest in geometry, and he believed that a solid
understanding of geometry was essential for his students’ success (Dunham, Genius 28).
Eudoxus was a great mathematician, and he was one of the Academy’s best students. He
attended Plato’s lectures at the Academy, and he was enthusiastic about the motion of the
moon and the planets. Sir Thomas Heath once stated, “[Eudoxus] was a man of science if
ever there was one” (qtd. in Dunham, Genius 28). Eudoxus developed the theory of
proportion, and his contributions to the method of exhaustion represent his most
remarkable work. He developed this method in order to prove some of Democritus’s
discoveries with regard to the volumes of pyramids and cones. Book XII of Euclid’s
Elements includes a section about the method of exhaustion (Laubenbacher 99).
Mathematicians can use the method of exhaustion to determine the volumes and areas of
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complex geometric figures. Archimedes applied this method when he determined the
formula for a circle’s area. The method of exhaustion is similar to the concept of limits in
calculus (Maor 12). In 332 B.C., Alexander the Great established the city of Alexandria
in Egypt. He formed the Alexandrian Library, which became the most prestigious
academic institution in the world (Dunham, Genius 29).
Euclid went to Alexandria in 300 B.C., and he intended to establish a school of
mathematics. He is most famous for the Elements, which is a collection of 465
propositions related to plane geometry, solid geometry, and number theory (Dunham,
Master 1). The Elements became the accepted text for geometry, and mathematicians
often refer to it as the “Bible of mathematics” (Dunham, Genius 30). There have been
approximately 2,000 different editions of the Elements. Isaac Newton studied Euclid’s
Elements, and Abraham Lincoln became very interested in the Elements. Carl Sandburg,
Lincoln’s biographer, stated that Lincoln “… bought the Elements of Euclid, a book
twenty-three centuries old … [It] went into his carpetbag as he went out on the circuit. At
night … he read Euclid by the light of a candle after others had dropped off to sleep”
(qtd. in Dunham, Genius 30). Bertrand Russell was also fascinated with the Elements. In
his autobiography, he stated, “At the age of eleven, I began Euclid, with my brother as
tutor. This was one of the great events of my life, as dazzling as first love” (qtd. in
Dunham, Genius 31). Euclid presented his mathematical theory in a very clear and
logical manner (Maor 121). He used a system of axioms as the foundation of his theory.
First, he stated five general axioms, five postulates, and 23 definitions. Then, he used
these foundations as the basis for proving his propositions. He developed the axiomatic
method in a careful manner in order to avoid circular reasoning (Laubenbacher 174).
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In Book I of the Elements, Euclid stated the following definitions concerning
plane geometry:
Definition. A point is that which has no part.
Definition. A line is breadthless length.
Definition. A straight line is a line that lies evenly with the points on itself (Dunham,
Genius 32).
In modern geometric theory, mathematicians regard point and line as undefined
concepts. Euclid stated several postulates and common notions. Some mathematicians
were skeptical of Euclid’s work. Bertrand Russell commented, “I had been told that
Euclid proved things, and was much disappointed that he started with axioms. At first, I
refused to accept them unless my brother could offer me some reason for doing so, but he
said, ‘If you don’t accept them, we cannot go on,’ and, as I wished to go on, I reluctantly
admitted them” (qtd. in Dunham, Genius 37). In 1902, Russell offered the following
additional criticism of Euclid’s work: “[Euclid’s] definitions do not always define, his
axioms are not always indemonstrable, his demonstrations require many axioms of which
he is quite unconscious. A valid proof retains its demonstrative force when no figure is
drawn, but very many of Euclid’s earlier proofs fail before this test … The value of his
work as a masterpiece of logic has been very grossly exaggerated” (qtd. in Dunham,
Genius 38). Despite these harsh criticisms, Euclid’s Elements is clearly the top
“bestseller” in the history of mathematical literature (Laubenbacher 173). Euclid was not
the first person to discover the Pythagorean Theorem, but he provided a unique proof of
the theorem. Mathematicians regard his proof of the Pythagorean Theorem as one of the
most important mathematical proofs ever completed (Hartshorne 8). The figure that
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Euclid used to demonstrate the proof resembles the structure of a windmill, and people
often refer to the proof as The Windmill for this reason (Dunham, Genius 51). Euclid also
proved the converse of the Pythagorean Theorem. He used the Pythagorean Theorem in
the process of proving its converse. The Pythagorean Theorem is one of the most well
known and widely used mathematical theorems (Maor 44).
Euclid stated the following postulate in Book I of the Elements.
Postulate 5. If a straight line falling on two straight lines make the interior angles on the
same side less than two right angles, the two straight lines, if produced indefinitely,
meet on that side on which are the angles less than the two right angles.
This is Euclid’s famous Parallel Postulate, and it is the most controversial statement
associated with Greek mathematics (Dunham, Genius 35). Other mathematicians did not
challenge the fact that the Parallel Postulate was true. However, they challenged the
classification of the Parallel Postulate as a postulate instead of as a proposition. Adrien-
Marie Legendre attempted to prove the Parallel Postulate, and Gauss tried to show that
the Parallel Postulate was really a theorem (Laubenbacher 24). According to the writer
Proclus, “This [Parallel Postulate] ought even to be struck out of the Postulates
altogether; for it is a theorem …” (qtd. in Dunham, Genius 53). However, no one
succeeded in proving the Parallel Postulate. Wolfgang Bolyai warned his son, Johann, not
to try to prove the Parallel Postulate. He stated, “You must not attempt this approach to
parallels. I know this way to its very end. I have traversed this bottomless night, which
extinguished all light and joy of my life …. I entreat you, leave the science of parallels
alone” (qtd. in Dunham, Genius 56). This statement reflects the profound frustration that
mathematicians experienced when they were unable to prove the Parallel Postulate.
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Legendre offered several proofs of the Parallel Postulate, but other mathematicians
showed that his proofs were not valid (Laubenbacher 26).
Georg Friedrich Bernhard Riemann strongly questioned the assumption that
geometric lines have infinite length. He stated, “… We must distinguish between
unboundedness and infinite extent …. The unboundedness of space possesses … a
greater empirical certainty than any external experience. But its infinite extent by no
means follows from this” (qtd. in Dunham, Genius 57). Gauss, Bolyai, Riemann, and
Nikolai Lobachevski were very influential with regard to the early study of non-
Euclidean geometry. Lobachevski was a Russian mathematician who published an early
interpretation of non-Euclidean geometry (Maor 125). In 1868, Eugenio Beltrami proved
that non-Euclidean geometry is just as logically valid as Euclidean geometry (Dunham,
Genius 57). In non-Euclidean geometry, the sum of the angles of a triangle is not the
same for all triangles.
In Book VII of the Elements, Euclid addressed the concept of number theory.
He defined prime numbers, composite numbers, and perfect numbers. A perfect number
is a whole number that equals the sum of its proper divisors (Moews 1). For example, 6 is
a perfect number, since 6 = 1 + 2 + 3 (Dunham, Master 2). Euclid stated the Fundamental
Theorem of Arithmetic, which says that a number can be factored as a product of prime
numbers in only one way. Euclid’s Proposition IX.20 states that the set of prime numbers
is an infinite set. Euclid proved this statement, which is called the infinitude of primes.
G.H. Hardy, a British mathematician, made the following comment regarding Euclid’s
proof of this theorem: “[The proof is] as fresh and significant as when it was discovered –
two thousand years have not written a wrinkle on [it]” (qtd. in Dunham, Genius 73). The
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following theorem represents Proposition IX.36 of the Elements, stated using modern
terminology.
Theorem. If 2k – 1 is prime and if N = 2
k–1(2
k – 1), then N is a perfect number.
This theorem provides a method that one can use to generate perfect numbers (Dunham,
Master 3). In his Proposition XII.10, Euclid proved that a cone’s volume is one third of
the volume of the cylinder with the same base and height. This relationship can be
expressed using the formula, Vcone = (1/3)πr2h. Archimedes commented, “… though these
properties were naturally inherent in the figures all along, yet they were in fact unknown
to all the many able geometers who lived before Eudoxus, and had not been observed by
anyone” (qtd. in Dunham, Genius 77). In the final proposition of the Elements, Euclid
showed that there are exactly five regular solids. The five regular solids are the
tetrahedron, cube, octahedron, dodecahedron, and icosahedron (Maor 104). This proof
relied on the following formula, which Euler discovered in 1752: Every simple
polyhedron must satisfy the equation, V – E + F = 2, where V = number of vertices, E =
number of edges, and F = number of faces (Maor 103).
Leonhard Euler made major contributions to the study of analytic number theory.
This branch of mathematics applies the concepts of calculus and analysis to the study of
the whole numbers. While analysis traditionally involves the study of continuous
phenomena such as convergence and divergence, number theory generally involves
discrete phenomena. Analytic number theory is particularly important because it
combines these two concepts (Dunham, Master 61). Euler also studied the field of
indeterminate analysis (Euler 299). Euclid originally proved that the set of prime
numbers is an infinite set. Every odd prime number has the form 4k+1 or 4k–1.
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Throughout the first 200 positive integers, there are more 4k–1 odd prime numbers than
4k+1 odd primes. However, this majority switches in favor of the 4k+1 odd primes after
the first 26,861 positive integers. J.E. Littlewood proved that this majority switches back
and forth between the 4k–1 odd prime numbers and the 4k+1 odd primes infinitely many
times as one progresses through the positive integers (Dunham, Master 63). Fermat
proposed that a 4k+1 prime number can be expressed as the sum of two perfect squares in
exactly one way, while a 4k–1 prime number cannot be expressed as the sum of two
perfect squares in any manner. Euler proved this proposition, and he was very interested
in the study of prime numbers. Fermat made the conjecture that every number of the form
22n + 1 is a prime number. In 1732, Euler disproved this conjecture by showing that
225 + 1 = 2
32 + 1 = 4,294,967,297 = (641)(6,700,417) (Laubenbacher 160). Euler studied
the concepts of indeterminate equations of the first degree and the second degree (Euler
465).
In 1774, Euler completed a paper called On a table of prime numbers up to a
million and beyond. He wished to determine an exact sum for the infinite series,
1/15 + 1/63 + 1/80 + 1/255 + 1/624 + . Euler clarified the nature of this series by stating
that the terms of the series represented reciprocals “whose denominators are one less than
all perfect squares which simultaneously are other powers” (qtd. in Dunham, Master 65).
For example, 15 is a denominator in the series, since 16 = 42 = 2
4. This means that 16 is
both a perfect square and a fourth power at the same time. He previously proved that
(π2)/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + 1/64 + 1/81 + . Euler applied this
result and proved that the sum of the series equals 7/4 – (π2)/6. Euler studied the
harmonic series, and he proved the following result.
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1 + 1/2 + 1/3 + 1/4 + 1/5 + = (2*3*5*7*11*13*)/(1*2*4*6*10*12*) (Dunham,
Master 67). This result established a significant connection between the prime numbers
and the harmonic series. On the equation’s right side, the numerator represents the
product of all prime numbers. The denominator represents the product of all numbers that
are one less than prime numbers. In 1737, Euler considered the sum of the reciprocals of
the prime numbers, which can be expressed as
S = 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + . Euler proved that the set S diverges
(Ingham 10).
Now suppose that one wished to find the sum of the reciprocals of all of the
positive integers whose only prime number factors are 2 and 3. This is the sum,
T = 1 + 1/2 + 1/3 + 1/4 + 1/6 + 1/8 + 1/9 + 1/12 + 1/16 + 1/18 + 1/24 + 1/27 + 1/32 + .
Euler proved that this sum can be expressed in the following way:
T = [1/(1 – 1/2)]*[1/(1 – 1/3)] = (2*3)/(1*2). Another way to express Euler’s result is
1k
1/k)( = p
1/p)]-[1/(1 , where the sum on the left side represents positive integers k,
and the product on the right side represents prime numbers p. Euler’s result involved a
combination of number theory and analysis. The fact that the set of prime numbers is an
infinite set follows as a corollary to Euler’s result (Dunham, Master 70).
In 1775, Euler considered the infinite series,
V = 1/3 – 1/5 + 1/7 + 1/11 – 1/13 – 1/17 + 1/19 + 1/23 – 1/29 + . This series consists of
the reciprocals of the odd prime numbers. In the series, a positive sign precedes each
prime number with the form 4k – 1. A negative sign precedes each prime number with
the form 4k + 1. Euler showed that the sum V has an approximate value of 0.3349816.
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After proving that the series converges, Euler claimed that the set of 4k + 1 prime
numbers is an infinite set (Dunham, Master 77). In 1837, Peter Gustav Lejeune-Dirichlet
proved that every arithmetic progression with the form,
a, a + b, a + 2b, a + 3b, , a + kb, , must include an infinite number of prime numbers,
as long as a and b are selected as relatively prime numbers. Two numbers are relatively
prime if there does not exist any number greater than 1 that is a factor of both of the
numbers (Moews 4).
Leonhard Euler made several important contributions to the study of number
theory. Number theory involves the study of the positive integers. In 1772, Euler wrote to
Daniel Bernoulli and stated that he had proved that 231
– 1 is a prime number. This
implies that 230
(231
– 1) = 2,305,843,008,139,952,128 is a perfect number (Laubenbacher
159). Euler attempted to find four different numbers such that when any two numbers are
added, the sum is a perfect square. He provided a correct solution when he showed that
the numbers 18,530; 38,114; 45,986; and 65,570 satisfy this condition. Four of the
volumes of Euler’s Opera Omnia cover the topic of number theory. Harold Edwards
stated that even if this were Euler’s only mathematical accomplishment, “his
contributions to number theory alone would suffice to establish a lasting reputation in the
annals of mathematics” (qtd. in Dunham, Master 7). In 1747, Euler stated, “Whether …
there are any odd perfect numbers is a most difficult question” (qtd. in Dunham, Master
13). The question of whether or not any odd perfect numbers exist represents an unsolved
problem in mathematics. While no odd perfect numbers have been found, no one has ever
proved that they cannot exist (Laubenbacher 160). In 1748, Leonhard Euler discovered
the formula, eπi
+ 1 = 0. This is called Euler’s formula, and it is significant because it
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provides a relation between the constants of arithmetic, geometry, analysis, and complex
numbers (Maor 53). During the same year, Euler discovered the following formula for
the irrational constant, e:
e = 1 + 1/1 + 1/(1*2) + 1/(1*2*3) + 1/(1*2*3*4) + 1/(1*2*3*4*5) + (Maor 52).
3. Gauss’s Contributions.
In 1777, Carl Friedrich Gauss was born in Brunswick. He demonstrated
outstanding mathematical ability from a very young age. His first major accomplishment
occurred in 1796, when he showed that a straightedge and compass could be used to
construct a 17-sided regular polygon (Hartshorne 249). He later showed that an N-sided
regular polygon could be constructed using a straightedge and compass, if N is prime and
has the form 22n + 1 (Maor 123). This means that one can use a straightedge and compass
to construct a 257-sided regular polygon or even a 65,537-sided regular polygon. In 1799,
Gauss received his doctoral degree from the University of Helmstadt. His dissertation
was entitled, “A New Proof of the Theorem That Every Integral Rational Algebraic
Function Can Be Decomposed into Real Factors of the First or Second Degree”
(Dunham, Genius 239). Gauss is famous for his proof of the Fundamental Theorem of
Algebra. Let P denote a polynomial whose coefficients are real. The theorem states that
one can factor P into a product of real quadratic and/or real linear factors. In other words,
every real polynomial with degree n can be decomposed into n linear factors, some of
which may be complex (Laubenbacher 216). Euler had unsuccessfully attempted to prove
this theorem in 1749. In 1801, Gauss published his Disquisitiones Arithmeticae. This was
a major work on the topic of number theory. Gauss made a conjecture with regard to the
distribution of the prime numbers among the whole numbers. Mathematicians now refer
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to this conjecture as the Prime Number Theorem (Ingham 25). Let σ(N) denote the
number of prime numbers below the integer N. The Prime Number Theorem states that
σ(N) approaches 1/ln(N) as N approaches infinity. In 1896, Jacques Hadamard and de la
Vallée Poussin proved the Prime Number Theorem (Maor 23). Gauss believed in the
potential infinite, but he did not believe in the actual infinite. In 1831, he told
Schumacher, “I must protest most vehemently against your use of the infinite as
something consummated, as this is never permitted in mathematics. The infinite is but a
façon de parler, meaning a limit to which certain ratios may approach as closely as
desired when others are permitted to increase indefinitely” (qtd. in Maor 55). Gauss
stated, “Mathematics is the queen of the sciences, and the theory of numbers is the queen
of mathematics” (qtd. in Dunham, Genius 240). Gauss was often concerned about how
his ideas would be received in the mathematical community, and he was afraid of public
scrutiny of his ideas.
Carl Friedrich Gauss encouraged the work of Sophie Germain, who was a
mathematician during the early part of the 1800s. At this time, society held the opinion
that women were not supposed to become mathematicians. Germain’s parents did not
permit her to study mathematics. Sophie Germain used the pseudonym, Antoine LeBlanc,
in order to avoid revealing her gender. She made significant efforts toward developing a
proof of Fermat’s Last Theorem (Laubenbacher 185). In 1807, Gauss learned Sophie
Germain’s true identity. Gauss continued to encourage her, and he recognized the gender
inequity that existed in mathematics. Gauss wrote,
The taste for the abstract sciences in general and, above all, for the mysteries of
numbers, is very rare: this is not surprising, since the charms of this sublime
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science in all their beauty reveal themselves only to those who have the courage to
fathom them. But when a woman, because of her sex, our customs, and prejudices,
encounters infinitely more obstacles than men in familiarizing herself with their
knotty problems, yet overcomes these fetters and penetrates that which is most
hidden, she doubtless has the most noble courage, extraordinary talent, and
superior genius. (qtd. in Dunham, Genius 242)
Gauss lived by the motto, “Pauca sed matura.” This means, “Few but ripe.” Gauss
published a relatively small number of papers. He refused to publish his results until they
were perfect enough to satisfy his standards (Dunham, Genius 243). He was hesitant to
publish his ideas because he feared public scrutiny. When he discovered non-Euclidean
geometry, he did not receive recognition because he refused to publish his results.
Because of this, Lobachevsky and Bolyai took credit for the discovery (Maor 125).
4. The Contributions of Georg Cantor.
In 1845, Georg Cantor was born in Russia (Dunham, Genius 252). Cantor studied
under Kronecker, Kummer, and Weierstrass (Golba 1). His primary interest was in
number theory. A mathematician named Edward Heine inspired Cantor to study
trigonometric series. Cantor proved the following requirements for integrating a function:
“If one sets f(x) = A0 + A1 + + An + Rn, for any given quantity there must exist an
integer m such that, for n ≥ m, the absolute value of Rn is less than for all values of x
which come into consideration” (qtd. in Dauben 32). Cantor proved the following
important theorem.
Uniqueness Theorem. If a function of a real variable f(x) is given by a trigonometric
series convergent for every value of x, then there is no other series of the same form
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which likewise converges for every value of x and represents the function f(x)
(Dauben 34). In 1871, Cantor began to consider exceptional sets. He attempted to
generalize his previous results, and this led him to the rigorous study of algebraic and
transcendental numbers (Laubenbacher 57). In 1874, Cantor showed that most real
numbers are transcendental numbers. Cantor objected to other mathematicians’ previous
efforts to define irrational numbers with respect to infinite series. He stated, “Here there
would be a logical error, since the definition of the sum ∑av would first be won by
equating it with the finished number b, necessarily defined beforehand. I believe that this
logical error, first avoided by Weierstrass, was earlier committed quite generally, and was
not noticed because it belongs to the rare cases in which real errors can cause no
significant harm in calculations” (qtd. in Dauben 37). Cantor defined fundamental
sequences and the order relations between them. Cantor observed that when comparing
two sequences a and b, either b = a, b < a, or b > a. This represents a trichotomy for the
fundamental sequences (Dauben 39).
Georg Cantor wished to develop a method through which one could precisely
identify complex groups of points distributed in specific ways throughout the geometric
continuum. When he introduced the concept of derived sets of the first species, Cantor
emphasized that a point should always be associated with a specific numerical value.
Cantor stated the following definition of limit points: “By a limit point of a point set P I
mean a point of the line for which in any neighborhood of same, infinitely many points of
P are found, whereby it can happen that the [limit] point itself also belongs to the set. By
a neighborhood of a point is understood an interval which contains the point in its
interior. Accordingly, it is easy to prove that a point set consisting of an infinite number
Batchelor 20
of points always has at least one limit point” (qtd. in Dauben 41). This definition is
related to the Bolzano-Weierstrass Theorem, which states that all infinite bounded point
sets must contain at least one limit point (Laubenbacher 71). Every set of points P must
have a set of limit points. Cantor denoted the set of limit points P′, and he referred to this
set as the “first derived point set of P.” Let P′ denote an infinite set of points. Then there
must exist a second derived point set, P′′. Cantor stated that this progression continues
until “one finds through v such transitions the concept of the vth
derived point set P(v)
of
P” (qtd. in Dauben 41). This progression is related to Cantor’s uniqueness theorem. After
v repetitions of the process, the derived set of points P(v)
contains a finite number of
points. Therefore, the set P(v+1)
cannot exist. Now consider the Riemann function,
F(x) = C0*(xx/2) – C2/4 – – Cn/(nn) – . Cantor stated that F(x) is continuous in the
neighborhood of any value of x. In addition, he claimed that F(x) is a linear function
throughout the interval (0,2π). When considering derived sets of the second species,
Cantor considered sets such as P(∞)
. He demonstrated the direct relationship between the
concepts of derived sets and limit points. As he developed the uniqueness theorem,
Cantor frequently referred to the previous work of Heine, Riemann, and Schwarz
(Dauben 45).
In 1883, Georg Cantor published a paper about set theory. He developed the paper
to address the mathematical and philosophical issues associated with transfinite set
theory. In 1874, Cantor had proved the existence of infinite sets that have different and
distinct magnitudes. He demonstrated how to count infinite sets, how to determine
powers, and how to define equivalence. Cantor stated the following definition of set
equivalence: “Two sets M and N are equivalent … if it is possible to put them, by some
Batchelor 21
law, in such a relation to one another that to every element of each one of them
corresponds one and only one element of the other” (qtd. in Dunham, Genius 253). In
1878, Cantor proved that one could restrict the analysis of continuity to the domain of the
real line (Maor 59). He showed that the same number of points exist on an infinite
straight line as on a finite segment of a line. Cantor considered the linear interval (0,1),
and he considered the square that is bounded by the interval (0,1) on the x-axis and by the
interval (0,1) on the y-axis. When he discovered that there is a one-to-one
correspondence between the square and the interval, he stated, “I see it but I do not
believe it!” (qtd. in Dunham, Genius 273). In 1879, Cantor published the first of a group
of six papers about infinite linear sets of points. He defined sets of the second species as
sets for which P(v)
is not an empty set for any finite value of v. Cantor provided the
following definition of everywhere-dense sets.
Definition. If P lies partially or entirely in the interval (α, β), then the remarkable case
can occur that any arbitrarily small interval (γ, δ) in (α, β) contains points of P. In
such a case we will say that P is everywhere-dense in the interval (α, β) (Dauben 78).
This definition states that a set P is everywhere-dense on an interval (α, β) whenever its
first derived set P′ contains the interval (α, β). Everywhere-dense sets are always sets of
the second species. In 1879, Paul du Bois-Reymond made the following statement
regarding the terminology for this concept.
A distribution of points is called pantachisch if, for any arbitrarily small interval,
points of the distribution in question occur. One is led to such distributions of
points (of which I have various examples) if one studies accumulation points of
infinite order, the existence of which I wrote to Cantor in Halle years ago. At
Batchelor 22
another opportunity I will discuss these distributions, the accumulation points of
finite and infinite order of increasingly diminishing intervals, and finally my choice
of the expression pantachisch compared with the expression of everywhere-dense
adopted later by Cantor. (qtd. in Dauben 93)
Cantor defined sets of the same power as sets for which a one-to-one correspondence
between the respective elements is possible (Maor 60). Cantor identified the concepts of
denumerable sets and non-denumerable sets. Denumerable sets have the same power as
the set of natural numbers, . Non-denumerable sets have the same power as the set of
real numbers, . Denumerable sets include the rational numbers and the algebraic
numbers; every set of the first species is denumerable. Cantor proved that and are
distinct sets (Dauben 79). Cantor rejected the concept of infinitesimal numbers. He
formally presented this rejection in the following theorem.
Theorem. Non-zero linear numbers ζ (in short, numbers which may be thought of as
bounded, continuous lengths of a straight line) which would be smaller than any
arbitrarily small finite number do not exist, that is, they contradict the concept of
linear numbers (Dauben 130).
In 1880, Georg Cantor published a short paper about transfinite symbols and
derived sets. He showed that sets of the first species could be completely described by
their derived sets, but second species sets could not be completely described by their
derived sets. Let P denote a general point set of the second species. Cantor showed that
P′, the first derived set of P, could be disjointly decomposed into two sets, Q and R:
P′ {Q, R}. In this decomposition, Q denotes the set of points that belong to the first
species sets of P′. R denotes the set of points that are contained in all derived sets of P′;
Batchelor 23
therefore, R is a second species set. He showed that R P(∞)
, and he referred to P(∞)
as the
derived set of P with order ∞ (Dauben 80). Cantor developed a method that one could use
to produce a hierarchy of infinite sets (Maor 64). He believed that the concept of
denumerability was based on the following two theorems.
Theorem I. Every infinite part of a denumerable set constitutes an infinitely
denumerable set (Dauben 84).
Theorem II. Given a finite or denumerably infinite set of sets (E), (E′), (E′′), , each of
which is denumerable, then the union of all elements of (E), (E′), (E′′), is likewise
a denumerable set (Dauben 84).
Next, Cantor provided the following definition of sets of negligible content.
Definition: A point set P is said to be of negligible content if its elements can be
enclosed in intervals (cv,dv) such that the sum of the lengths of these intervals,
(dv – cv) may be made arbitrarily small, in other words, lim (dv – cv) = 0 (Dauben
89).
Georg Cantor provided proofs of several additional theorems related to point sets of the
first and second species.
Cantor believed that an important connection existed between mathematics and
philosophy. Georg Cantor suffered several nervous breakdowns, and he made the
following statement to Mittag-Leffler after the first breakdown: “I thank you heartily for
your kind letter of May 15; I would have answered it sooner, but recently I have not felt
so fresh as I should, and consequently I don’t know when I shall return to the
continuation of my scientific work. At the moment I can do absolutely nothing with it,
and limit myself to the most necessary duty of my lectures; how much happier I would be
Batchelor 24
to be scientifically active, if only I had the necessary mental freshness!” (qtd. in Dauben
136). Leopold Kronecker was a professor in Berlin. He was a very harsh critic of
Cantor’s work, and Cantor was not able to convince Kronecker that his transfinite
numbers were legitimate (Laubenbacher 74). After this, Cantor became somewhat
disillusioned with his study of mathematics. In 1884, Cantor resumed his effort to prove
the Continuum Hypothesis, which states that there is no transfinite cardinal number that
falls strictly between 0א and c. The cardinal number c represents the cardinal number of
the continuum (Maor 60). The Continuum Hypothesis can be expressed as 2א
The .1א = 0
Continuum Hypothesis can be generalized to state that 2א
α = אα+1 (Laubenbacher 63).
Cantor announced that he had proved that the real line has the same power as the second
class of numbers. He stated, “Thus you see that everything comes down to defining a
single closed set of the second power. When I’ve put it all in order, I will send you the
details” (qtd. in Dauben 137). Gösta Mittag-Leffler was a Swedish mathematician who
had previously supported Cantor’s efforts (Laubenbacher 75). Cantor became frustrated
when he was unable to prove the Continuum Hypothesis. Later in 1884, Cantor told
Mittag-Leffler that the Continuum Hypothesis was a complete failure. He stated, “The
eventual elimination of so fatal an error, which one has held for so long, ought to be all
the greater an advance” (qtd. in Dauben 137). In 1900, David Hilbert spoke at the Second
International Congress of Mathematicians, which was held in Paris. During his lecture, he
described twenty-three unsolved mathematical problems that he considered important for
the twentieth century. The first problem on Hilbert’s list was Cantor’s Continuum
Hypothesis (Laubenbacher 64). In 1963, Kurt Gödel and Paul Cohen showed that the
Batchelor 25
Continuum Hypothesis is independent of the other axioms in the Zermelo-Fraenkel
system (Maor 258).
In 1885, Gösta Mittag-Leffler made the following statement to Cantor, cautioning
him against publishing his new work: “I am convinced that the publication of your new
work, before you have been able to explain new positive results, will greatly damage your
reputation among mathematicians. I know very well that basically this is all the same to
you. But if your theory is once discredited in this way, it will be a long time before it will
again command the attention of the mathematical world. It may well be that you and your
theory will never be given the justice you deserve in our lifetime” (qtd. in Dauben 138).
Cantor previously had a close professional relationship with Mittag-Leffler, and Cantor
had published his work in Mittag-Leffler’s journal, Acta Mathematica (Laubenbacher
75). After receiving Mittag-Leffler’s criticism, Cantor nearly decided to abandon
mathematics entirely. He decided that he would no longer publish in the journal, Acta
Mathematica. Whenever someone seriously criticized Cantor’s work, he took the
criticism very personally. One of the original criticisms was the concept of “annihilation
of number.” Aristotle and other intellectuals believed that based on Cantor’s theory of the
infinite, finite numbers would be “annihilated” by infinite numbers. They pointed out the
property that for finite positive numbers m and n, m + n > m and m + n > n. If infinite
numbers were allowed, the equation m + ∞ = ∞ would appear to violate this property. In
this way, the infinite number would “annihilate” the finite number. However, Cantor
disproved this argument by pointing out that one cannot assume that infinite numbers will
behave according to the properties of finite numbers. In addition, Cantor distinguished
between the infinite numbers ω and ω + 1. This distinction proved that adding a finite
Batchelor 26
number to an infinite number does not result in the “annihilation” of the finite number
(Dauben 122). Mittag-Leffler’s letter appears to be the primary source of Cantor’s
disillusionment with mathematics (Dauben 139). Cantor began to devote his energy to the
study of theology and philosophy (Golba 1). He found that he experienced a greater sense
of belonging and encouragement among Roman Catholic theologians than he did among
mathematicians (Golba 2).
Pope Leo XIII tried to reconcile scientific discoveries with religious scripture and
with the leaders of the Catholic Church. This led several Catholic intellectuals to pursue
the study of natural science. He encouraged the study of Thomistic philosophy. Pope Leo
XIII believed that science could function to further the goals of the Catholic Church. He
stated, “Above all [education] must be wholly in harmony with the Catholic faith in its
literature and system of training, and chiefly in philosophy, upon which the foundation of
other sciences in great measure depends” (qtd. in Dauben 142). Cantor’s work received
criticism from Mittag-Leffler and Leopold Kronecker, who did not believe in the concept
of the actual infinite (Maor 65). In addition, Aristotle stated, “The infinite has a potential
existence …. There will not be an actual infinite” (qtd. in Maor 55). Some members of
the Roman clergy strongly resisted Pope Leo XIII’s effort to encourage the study of
Thomistic philosophy. Constantin Gutberlet studied theology and philosophy at the
Collegio Romano. In 1888, Gutberlet founded the journal, Philosophisches Jahrbuch der
Görres-Gesellschaft. In 1886, he had published an article based on Cantor’s set theory.
He recognized that Cantor’s work represented a new phase for the study of infinity.
Gutberlet worried that mathematical infinity could represent a challenge to the absolute
infinity associated with God’s existence. Previous mathematicians and theologians
Batchelor 27
believed that absolute infinity represented God’s greatness and omnipotence (Maor 8).
However, Cantor believed that the transfinite numbers did not diminish the extent of
God’s greatness. He believed that the existence of these numbers enhanced it (Dauben
143). Gutberlet supported the concept of the actual infinite. He presented Cantor’s work
in order to support his own ideas against the opposition of some theologians. He stated
that he hoped that after he described Cantor’s work, readers could decide, “if they were
correct, when they supposed they could dispose of my theory of actually infinite
magnitude so easily. Above all we now want to explain the Cantorian theory and then to
defend our conception against criticism, which this journal published, with Cantor’s
corresponding interpretation of infinite magnitude” (qtd. in Dauben 143). Gutberlet
encouraged Cantor’s interest in the theological and philosophical aspects of his own
work. In addition, other philosophers in the Catholic Church showed an interest in
Cantor’s work (Golba 2).
Thomas Esser studied the implications of Cantor’s work for theology. Cardinal
Johannes Franzelin originally objected to Cantor’s views regarding infinity. Later, he
accepted Cantor’s theory. He reassured Cantor that his theory did not pose any
theological threats to religious beliefs (Dauben 146). Pope Leo XIII’s enthusiasm for
Cantor’s work helped Cantor to overcome his previous setbacks and become interested in
mathematics again. Cantor believed that God inspired him to do his work (Laubenbacher
76). In 1888, Cantor wrote to Jeiler and stated, “I entertain no doubts as to the truth of the
transfinites, which I have recognized with God’s help and which, in their diversity, I have
studied for more than twenty years; every year, and almost every day brings me further in
this science” (qtd. in Dauben 147). Cantor turned to the study of theology after he
Batchelor 28
became disillusioned with mathematics. However, his study of theology actually
encouraged him to continue with his study of infinite numbers (Laubenbacher 76).
Charles Hermite was a French mathematician who studied the nature of the number π
(Dunham, Genius 24). In 1894, Cantor told Hermite, “But now I thank God, the all-wise
and all-good, that He always denied me the fulfillment of this wish [for a position at a
university in Göttingen or Berlin], for He thereby constrained me, through a deeper
penetration into theology, to serve Him and His Holy Roman Catholic Church better than
I have been able with my exclusive preoccupation with mathematics” (qtd. in Dauben
147). Georg Cantor helped to demystify the concept of infinity by encouraging a
systematic study of the infinite. Cantor believed that his mathematical talent was given to
him so that he could serve the Catholic Church. In 1896, he told Esser, “From me,
Christian Philosophy will be offered for the first time the true theory of the infinite” (qtd.
in Dauben 147). Cantor emphasized the study of pure mathematics, and he encouraged
the study of mathematical theory with no restrictions based on the applications of
mathematics to other sciences. However, Cantor preferred to use the term free
mathematics instead of pure mathematics. He stated, “Because of this extraordinary
position which distinguishes mathematics from all other sciences, and which produces an
explanation for the relatively free and easy way of pursuing it, it especially deserves the
name of free mathematics, a designation which I, if I had the choice, would prefer to the
now customary pure mathematics” (qtd. in Dauben 132). Other mathematicians were
impressed with Cantor’s work. David Hilbert stated, “No one will expel us from the
paradise that Cantor has created” (qtd. in Dunham, Genius 281).
Batchelor 29
5. Cardinal Numbers and One-to-One Correspondence.
Georg Cantor recognized that mathematicians needed a method for comparing the
sizes of sets. The term equinumerosity is used to describe sets that have the same size
(Dunham, Genius 253). Cantor studied the concept of one-to-one correspondence. He
developed several examples to show that one does not need the ability to count objects in
order to make a determination of whether two sets are equinumerous. The cardinality or
power of a set is the number of elements that are contained in the set (Maor 253). Two
subsets belong to the same equivalence class if they have the same cardinal number
(Cohen 65). Cantor stated the following definition of equivalent sets.
Definition. Two sets M and N are equivalent … if it is possible to put them, by some
law, in such a relation to one another that to every element of each one of them
corresponds one and only one element of the other (Dunham, Genius 253).
Mathematicians say that two sets have the same cardinality or power if they satisfy this
definition and are equivalent (Maor 253). Cantor’s definition is very important because it
does not require the sets to be finite. One can use the definition to compare infinite sets.
Other mathematicians were not pleased with Cantor’s concept of the completed infinite.
They preferred to restrict the analysis of infinity to the study of the potential infinite.
Gauss stated, “… I protest above all against the use of an infinite quantity as a completed
one, which in mathematics is never allowed. The Infinite is only a manner of speaking
…” (qtd. in Dunham, Genius 254). Consider the two sets of numbers (the set of natural
numbers) and E, the set of even natural numbers. Cantor showed that based on his
definition, the sets and E have the same cardinality. This means that the two sets have
the same size. Cantor also showed that the sets and (the set of all integers) have the
Batchelor 30
same cardinality. Georg Cantor stated that any set that could be placed in a one-to-one
correspondence with was denumerable or countably infinite. He introduced the
transfinite cardinal number 0א to represent the number of items contained in a countably
infinite set (Maor 57). Cantor studied the concept of well-ordered sets. All nonempty
subsets of a well-ordered set must contain a least element. The concept of well-ordered
sets was the basis for Cantor’s theory of ordinal numbers. Cantor explained that the
ordinal numbers represent the different order types for well-ordered sets. Cantor showed
that one can add ordinal numbers, but this kind of addition is not commutative
(Laubenbacher 62).
Georg Cantor used the set of natural numbers, , as the basis for extending the
number system beyond the finite realm. Let M denote the cardinality of the set M.
Cantor next considered , the set of rational numbers. Between any two integers, there
exist an infinite number of rational numbers. Based on this information, it would appear
that there are more rational numbers than natural numbers. However, Cantor showed that
Q .This means that the set of rational numbers is countably infinite (Dauben 79) .0א =
Based on Cantor’s definition, the number of rational numbers equals the number of
natural numbers. However, the collection of all real numbers is a non-denumerable set. In
1874, Cantor showed that no interval of real numbers can be placed in a one-to-one
correspondence with the set of natural numbers. A continuum is an interval containing
real numbers. Mathematicians refer to the interval (0,1) as the unit interval. Cantor
proved that the interval of real numbers between 0 and 1 (the unit interval) is not
countably infinite (Dauben 51). This theorem can be proved using the method of proof by
contradiction. Cantor used a diagonalization process to show that many infinite sets have
Batchelor 31
cardinality 0א. The interval of real numbers between 0 and 1 is infinite, and Cantor proved
that it has a higher “degree” of infinity than the set of natural numbers. The entire set of
real numbers, , has the same cardinality as the interval (0,1) (Dunham, Genius 263).
Georg Cantor tried to formalize the concepts of less than and greater than with
regard to transfinite cardinal numbers. Cantor stated the following definition for the
ordering of transfinite cardinal numbers:
Definition. For two sets A and B, A B if there is a one-to-one correspondence from
all points of A to a subset of the points of B (Dunham, Genius 268).
Cantor then defined strict orderings, stating that A B if A B , but there is not a one-
to-one correspondence between the elements of A and B. Cantor proved that there does
not exist a one-to-one correspondence between and (0,1). He showed that N < (0,1) . By
substitution, this implies that 0א < c. After establishing that it is possible to compare the
cardinalities of two sets, Cantor made the following assertion: If A B and B A , then
A = B (Dunham, Genius 270). Ernst Schröder and Felix Bernstein independently proved
this result, so mathematicians now call it the Schröder-Bernstein Theorem. One can use
the Schröder-Bernstein Theorem in order to determine the cardinality of the set I (the set
of all irrational numbers). The fact that I is not a denumerable set implies that I .0א <
Therefore, the set of irrational numbers represents a subset of the real numbers. One can
express this relationship symbolically as I c. In addition, there exists a one-to-one
correspondence between all real numbers and some irrational numbers. One can express
this relationship as c I . Applying the Schröder-Bernstein theorem, the conclusion is that
I = c (Maor 63). Cantor wanted to determine whether there were any cardinal numbers
Batchelor 32
greater than c. He considered the interval (0,1) and the square bounded by the interval
(0,1) on the x-axis and by the interval (0,1) on the y-axis. In 1877, Cantor discovered that
a one-to-one correspondence exists between the square and the interval. After making
this discovery, Cantor exclaimed, “I see it, but I do not believe it!” (qtd. in Dauben 55).
The power set of a set consists of all subsets of the original set. The power set of
the set A is denoted P[A]. For example, let A = {1,2,3}. Then,
P[A] = {φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}. A set containing n elements will
always have 2n subsets. One can express this as P[A] = 2
n. Georg Cantor showed that a
power set always contains more elements than the set from which it was generated (Maor
253). This is called Cantor’s Theorem, and it states that for every set A, A < P[A].
Cantor’s Theorem applies to finite sets and infinite sets. In order to find a set that has
cardinality greater then c, Cantor considered the set P[(0,1)]. Applying Cantor’s
Theorem, one can obtain the result P[(0,1)] > (0,1) = c. This implies that P[(0,1)] > c.
Next, Cantor considered the power set of P[(0,1)]. Applying Cantor’s Theorem,
]P[P[(0,1)] > P[(0,1)] . By repeating this process indefinitely, one can obtain the following
increasing sequence of numbers:
c < P[(0,1)] > 0א < )]]1,0[(P[P < )]]]P[P[P[(0,1 < ,1)]]]]P[P[P[P[(0 < (Dunham,
Genius 277). Georg Cantor made the following statement about his views with regard to
the infinite: “This view, which I consider to be the sole correct one, is held by only a few.
While possibly I am the very first in history to take this position so explicitly, with all of
its logical consequences, I know for sure that I shall not be the last!” (qtd. in Dunham,
Genius 280). The following logical paradox is a direct result of Cantor’s Theorem. Let U
represent the universal set, or the set of all sets. Since U contains all possible sets, one
Batchelor 33
cannot possibly enlarge it. Next, consider the power set P[U]. Cantor’s Theorem implies
that P[U]> U . Therefore, P[U] is a larger set than U. This represents a logical
contradiction (Maor 255). When mathematicians formally axiomatized set theory, they
carefully chose the axioms in such a way that they prohibited this kind of logical paradox.
During the 1600s, Galileo Galilei discovered an important paradox related to one-to-one
correspondences. He showed that the number of perfect squares equals the number of
natural numbers. He reasoned that since not every natural number is a perfect square,
there must be more natural numbers than perfect squares. He also observed that the
perfect squares become progressively less abundant as the natural numbers increase.
Galileo stated, “The attributes larger, smaller, and equal have no place either in
comparing infinite quantities with each other or comparing infinite with finite quantities”
(qtd. in Laubenbacher 55).
In his Contributions to the Founding of the Theory of Transfinite Numbers,
Cantor addressed the concepts of less than and greater than with regard to cardinal
numbers. He established that for cardinal numbers a, b, and c, if a < b and b < c, then
a < c. Cantor also stated the trichotomy principle, which says that for any two cardinal
numbers m and n, either m = n, m < n, or m > n. Cantor addressed the concepts of adding
and multiplying cardinal numbers. Let A and B denote two sets that have no elements in
common. The notation AB represents the union of A and B. Let A and B have cardinal
numbers a and b, respectively. Cantor showed that a + b = BA , and the following
additional properties apply to the addition of cardinal numbers (Dauben 173). For
cardinal number addition, a + b = b + a. For cardinal numbers a, b, and c,
Batchelor 34
a+(b+c) = (a+b)+c. The following equation represents the definition of multiplication of
cardinal numbers: a*b = BA , where AB denotes the Cartesian product of A and B.
For the multiplication of cardinal numbers, a*b = b*a, and a*(b*c) = (a*b)*c. In addition,
a*(b+c) = a*b + a*c. These equations represent the associative, commutative, and
distributive properties for the addition and multiplication of cardinal numbers. In Section
4 of his Contributions to the Founding of the Theory of Transfinite Numbers, Cantor
addressed the concept of exponentiation with regard to cardinal numbers. The following
properties apply to the exponentiation of cardinal numbers. Let a, b, and c represent
cardinal numbers. Then ab*a
c = a
b+c, (a*b)
c = a
c*b
c, and (a
b)c = a
b*c (Dauben 174). Now
let X represent the totality of real numbers x for which 0 x 1. Let X denote the
cardinal number of X. Then, X = 2א
0. Georg Cantor defined the exponentiation of cardinal
numbers based on the set of functions from one set to another set (Laubenbacher 84). In
Section 6 of his Contributions to the Founding of the Theory of Transfinite Numbers,
Cantor addressed the concept of 0א, the smallest cardinal number that is transfinite.
Cantor referred to sets that have finite cardinal numbers as finite aggregates. He referred
to sets that have transfinite cardinal numbers as transfinite aggregates. Cantor stated the
equation, 0א = v , referring to the totality of cardinal numbers v, which are finite. He
showed that 0א = 1 + 0א, and he established that 0א is greater than μ, where μ denotes any
finite number. Cantor also showed that 0א = 0א*0א (Laubenbacher 86). In 1891, Cantor
wrote a paper entitled On an Elementary Question in the Theory of Sets. Let a denote any
cardinal number. In this paper, Cantor established that for all a, 2a > a. More specifically,
2א
Therefore, the set of real numbers has cardinality greater than the cardinality of .0א < 0
Batchelor 35
the set of natural numbers. In addition, the set of real numbers has cardinality greater than
the cardinality of the set of rational numbers (Maor 64).
Bernhard Bolzano, an Austrian mathematician, emphasized the importance of a
one-to-one correspondence between a set and a proper subset of the set (Hall 45). A one-
to-one correspondence between sets exists if one can pair the elements of two sets in a
manner such that every element of one set is matched with a unique element of the other
set. Cantor’s Continuum Hypothesis states that there does not exist a transfinite cardinal
number strictly between 0א and c (Dunham, Genius 281). David Hilbert referred to
Cantor’s Continuum Hypothesis as “a very plausible theorem, which, nevertheless, in
spite of the most strenuous efforts, no one has succeeded in proving” (qtd. in Dunham,
Genius 282). Later, Kurt Gödel used the axiomatized set theory to prove that Cantor’s
Continuum Hypothesis is logically consistent with all of the other axioms of set theory.
In 1963, Paul Cohen showed that one could not use the axiomatized set theory to prove
the Continuum Hypothesis. Therefore, Cantor’s Continuum Hypothesis is independent of
the other axioms of set theory (Dauben 269). Cantor was very confident about the
validity of his theory. In 1888, he stated, “My theory stands as firm as a rock; every
arrow directed against it will return quickly to its archer. How do I know this? Because I
have studied it from all sides for many years; because I have examined all objections
which have ever been made against the infinite numbers; and above all because I have
followed its roots, so to speak, to the first infallible cause of all created things” (qtd. in
Dunham, Genius 283).
Batchelor 36
6. The Uncountable Set of Real Numbers.
Georg Cantor studied the properties of rational and irrational numbers. The
decimal representation of a rational number either terminates or has a group of repeating
digits. The decimal representation of an irrational number does not terminate or have a
group of repeating digits. Between any two rational numbers, there exist an infinite
number of irrational numbers. Between any two irrational numbers, there exist an infinite
number of rational numbers. However, the set of rational numbers and the set of irrational
numbers are not interchangeable collections of numbers. Cantor recognized the need for a
method of comparing the sizes of sets. Mathematicians use the term equinumerosity to
describe sets that have the same size. Cantor defined set equivalence and emphasized the
importance of one-to-one correspondences. He studied the concept of infinity, and he
believed that the concept of the completed infinite was a legitimate idea (Dauben 172).
Cantor analyzed the set of natural numbers and the set of even natural numbers. Based on
his definition, the two sets have the same cardinality. In addition, the set of natural
numbers has the same cardinality as the set of all integers. Cantor’s definition stated that
any set that could be placed in a one-to-one correspondence with the set of natural
numbers was denumerable or countably infinite. He introduced the transfinite cardinal
number 0א to represent the number of items contained in a countably infinite set
(Dunham, Genius 255).
Georg Cantor proved that the set of rational numbers is denumerable. However,
the collection of all real numbers is a non-denumerable set (Laubenbacher 57). In 1874,
Cantor showed that no interval of real numbers can be placed in a one-to-one
correspondence with the set of natural numbers (Dunham, Genius 259). A continuum is
Batchelor 37
an interval containing real numbers, and the interval (0,1) is called the unit interval.
Cantor proved that the unit interval is not countably infinite. One can prove this theorem
using the method of proof by contradiction (Dunham, Genius 259). He showed that the
interval of real numbers between 0 and 1 has a higher “degree” of infinity than the set of
natural numbers, even though both sets are infinite sets. Cantor formalized the concept of
ordering of cardinal numbers. He succeeded in producing a hierarchy of infinite sets, and
he made several important discoveries regarding transfinite cardinal numbers. In 1910,
Bertrand Russell stated, “The solution of the difficulties which formerly surrounded the
mathematical infinite is probably the greatest achievement of which our age has to boast”
(qtd. in Maor 64).
7. Limits and Convergence.
The concept of the limit is one of the most important concepts of calculus (Hall
146). Newton tried to address the concept of continuous functions, and he developed his
idea of ultimate ratios. The ultimate ratios that Newton described were actually the limits
of mathematical ratios. Newton stated that a quantity’s ultimate ratio “… is to be
understood as the ratio of the quantities, not before they vanish, nor after, but that with
which they vanish” (qtd. in Dunham, Genius 248). Newton’s theory of ultimate ratios
was flawed because the ratio 0/0 is undefined. Leibniz also studied limits, and he
developed a theory of infinitely small quantities. An infinitely small quantity was a
number that was not equal to zero but had the property that one could not decrease it any
further. Leibniz’s statements about the infinitely small quantities were often imprecise.
For example, he stated,
It will be sufficient if, when we speak of … infinitely small quantities (i.e., the very
Batchelor 38
least of those within our knowledge), it is understood that we mean quantities that
are … indefinitely small …. If anyone wishes to understand these [the infinitely
small] as the ultimate things …, it can be done …, ay even though he think that
such things are utterly impossible; it will be sufficient simply to make use of them
as a tool that has advantages for the purpose of calculation, just as the algebraists
retain imaginary roots with great profit. (qtd. in Dunham, Genius 249)
Bishop George Berkeley was not pleased with the ambiguity of the mathematicians’
statements. He made the following famous comment: “And what are these fluxions? The
velocities of evanescent increments. And what are these same evanescent increments?
They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we
not call them the ghosts of departed quantities…?” (qtd. in Dunham, Genius 250).
Mathematicians recognized that they needed to make the concept of the limit
more logically precise. Augustin-Louis Cauchy defined limits and used his definition to
prove major theorems of calculus (Laubenbacher 141). Karl Weierstrass provided the
following definition, which is called the static definition of the limit.
Definition. L is the limit of the function f(x) as x approaches a if for any > 0, there
exists a > 0 so that, if 0 < x – a < , then f(x) – L < (Dunham, Genius 251).
The idea of convergence is an important concept of calculus. Consider the sum,
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + . When a finite number of terms of the series are added,
the sum will never equal 1. This series has the property that it converges to 1. In other
words, the number 1 is the limit of the series as the number of terms approaches infinity.
The concepts of convergence and limits are fundamental for the study of calculus.
Batchelor 39
8. The Geometric and Harmonic Series.
The geometric series is the most famous convergent series. A geometric
progression grows very quickly, and compound interest is an example of a geometric
progression. The geometric series has the form, α + α2 + α
3 + α
4 + α
5 + + α
k + , where
-1 < α < 1 (Dunham, Genius 194). This restriction on the values of α is necessary because
the values of the terms of the series must become progressively smaller as one progresses
through the series. Radioactive decay represents an infinite geometric progression. A
substance’s half-life is the amount of time necessary for a quantity of the substance to
decay to one-half of the initial quantity (Maor 31). One can predict the limit of some
geometric series using the following formula: a + aq + aq2 + aq
3 + = a/(1 – q), if and
only if -1 < q < 1 (Dunham, Genius 194). A convergent series has a distinct sum, but a
divergent series does not. Applications of infinite geometric series occur in geometry,
engineering, and physics. Euler devoted a great deal of attention to geometric series. He
discovered that 1 + 1/4 + 1/9 + 1/16 + 1/25 + = (π2)/6 (Dunham, Master 47). This is
considered one of the most remarkable theorems of mathematical analysis (Maor 35).
Niels Henrik Abel once said, “With the exception of the geometric series, there does not
exist in all of mathematics a single infinite series whose sum has been determined
rigorously” (qtd. in Maor 29).
The harmonic series consists of the sum of the reciprocals of the counting
numbers. One can express this series as 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + (Maor 26). The
first person to prove that the harmonic series diverges was the French scholar Nicolae
Oresme. The terms of the harmonic series become progressively smaller, but the series
does not converge. When one adds the first million terms of the series, the sum is
Batchelor 40
approximately equal to 14.357. When one adds the first trillion terms, the sum is
approximately equal to 28. While the value of the sum of the series increases very slowly,
the series does not converge to a finite sum. Johann Bernoulli studied the harmonic
series, and mathematicians referred to the series as a “pathological counterexample.” This
means that mathematicians considered the behavior of the series to be very bizarre and
counterintuitive (Dunham, Genius 194). The sum of the reciprocals of all prime numbers
also diverges. However, the sum of the reciprocals of all of the twin primes converges to
a finite value (Maor 28).
9. Fermat’s Theorems.
Pierre de Fermat was a great French mathematician. In 1631, he received a degree
in law, and he practiced law in France. During his spare time, he conducted research
regarding mathematical concepts. During the 1600s, he made significant contributions to
the study of calculus, and he contributed to the study of analytic geometry. However, he
is most famous for his work in number theory (Laubenbacher 159). He studied the perfect
numbers extensively, and he analyzed the previous work of Diophantus. He wrote the
following comments in his own copy of the Arithmetica: “But it is impossible to divide a
cube into two cubes, or a fourth power [quadratoquadratum] into two fourth powers, or
generally any power beyond the square into two like powers; of this I have found a
remarkable demonstration. This margin is too narrow to contain it” (qtd. in Dunham,
Genius 159). Fermat had the irritating habit of claiming to have proven theorems without
providing proofs of them. Fermat’s statement represents an unsolved problem in
mathematics. Fermat stated the following theorem, which Euler proved in 1736.
Batchelor 41
Fermat’s Little Theorem. Given a prime number p and an integer a that is not divisible
by p, then ap-1
has remainder 1 under division by p. Furthermore, there exists a least
positive integer n such that an has remainder 1 under division by p, n divides p – 1,
and akn
has remainder 1 under division by p for all positive integers k (Laubenbacher
160). Based on this theorem, if a number with the form 2n – 1 is prime, then the
number n must also be prime.
In 1729, Goldbach wrote a letter to Euler, in which he made the following
statement: “Is Fermat’s observation known to you, that all numbers 22n + 1 are primes? He
said he could not prove it; nor has anyone else to my knowledge” (qtd. in Dunham,
Genius 229). Fermat’s observation would have represented a formula that
mathematicians could use to generate prime numbers. Euler considered the number
225 + 1 = 2
32 + 1 = 4,294,967,297. He showed that 4,294,967,297 = (641)(6,700,417).
Therefore, this is not a prime number, and Fermat’s claim is not valid in all cases. Primes
with the form 22n + 1 are called Fermat primes (Laubenbacher 160). Fermat stated the
following theorem, but he did not provide a proof of it.
Fermat’s Last Theorem. No cube can be split into two cubes, nor any biquadrate into
two biquadrates, nor generally any power beyond the second into two of the same kind
(Laubenbacher 164). This theorem states that for n > 2, the equation xn + y
n = z
n does
not have any integer solutions. Fermat claimed that he had proved this theorem, but he
did not provide a proof to other mathematicians. Many mathematicians made
unsuccessful attempts to prove Fermat’s Last Theorem. In 1993, Andrew Wiles finally
proved the theorem. He solved a mathematical problem that had stumped mathematicians
for about 300 years (Laubenbacher 156).
Batchelor 42
10. Geographic Maps and Inversion in a Circle.
Inversion in the unit circle is a fascinating concept related to the study of infinity.
Definition. An inversion is a specific transformation from a two-dimensional point set to
another two-dimensional set of points (Maor 88).
Consider the circle where O denotes the center, and the radius is 1. Let P denote a point
in the circle’s interior, and let OP denote the distance from O to P. The point P will be
“mapped” to a point Q on the ray OP . The distance from O to Q, denoted OQ, will be
determined by the formula, OQ = 1/OP. All points inside the circle will be mapped to
points outside the circle. All points outside the circle will be mapped to points inside the
circle. This inversion is symmetric. If Q denotes the image of P, then P will be the image
of Q. In addition, inversion in a circle is conformal or angle-preserving (Hartshorne 338).
The points along the unit circle will be mapped onto themselves. A straight line that does
not pass through the origin will be mapped onto a circle that passes through the origin.
Lines closer to the center O will result in larger image circles. The image of a parabola is
a cardioid. The image of an ellipse is a curve that corresponds to a fourth-degree equation
(Maor 90).
One important application of inversion is the making of geographic maps that
represent the earth on a two-dimensional surface. Cartography is the science of making
maps. A projection method must be used in order to represent the earth’s surface on a
two-dimensional map. In order to use a projection method, a cartographer must transform
each point on the earth’s surface onto a unique point on the map. The stereographic
projection is often used to represent the earth’s surface on a flat map. The parallels of
latitude are represented by concentric circles surrounding the north or south pole. The
Batchelor 43
meridians of longitude are represented by straight lines passing through the north or south
pole. The stereographic projection is conformal or angle-preserving (Maor 98). One can
use a sphere to represent a finite model of a plane, using the concept of inversion. A path
that crosses every meridian of longitude at the same angle is called a loxodrome. On a
stereographic projection map, a loxodrome will be represented by a logarithmic spiral.
Gerhardus Mercator, a Flemish geographer, developed a world map in which loxodromes
are represented by straight lines. In 1569, he published his map, which represents the
earth using a rectangular grid. Inversion is an important concept related to the science of
cartography (Maor 99).
11. Einstein’s General Theory of Relativity.
Albert Einstein is famous for his general theory of relativity. Astronomers
consider this theory very important, since it provides an explanation of the nature of the
universe. Einstein’s theory is consistent with Riemann’s theory of non-Euclidean
geometry (Laubenbacher 16). According to Einstein’s theory of gravitation, space and
time represent a single, four-dimensional entity. Scientists refer to this entity as the
space-time continuum. Every point in the space-time continuum is called an event.
Einstein defined a straight line as the path traveled by a ray of light. While Newton
believed in an infinite universe, Einstein believed that the universe is finite but
unbounded. Sir Arthur Eddington made the following comment regarding Einstein’s
theory.
There was just one place where [Einstein’s] theory did not seem to work properly,
and that was – infinity. I think Einstein showed his greatness in the simple and
drastic way in which he disposed of difficulties at infinity. He abolished infinity.
Batchelor 44
He slightly altered his equations so as to make space at great distances bend round
until it closed up. So that, if in Einstein’s space you keep going right on in one
direction, you do not get to infinity; you find yourself back at your starting-point
again. Since there was no longer any infinity, there could be no difficulties at
infinity. (qtd. in Maor 221)
Although many scientists were originally skeptical of Einstein’s theory, it has
withstood many tests. Most scientists now accept it as the best modern theory of the
universe (Laubenbacher 16). According to his theory of general relativity, a material
object cannot move faster than the speed of light. Light has a velocity of approximately
300,000 kilometers per second. According to his theory, a ray of light will be bent if it
passes through a major gravitational field. The field surrounding a large star is one
example of such a gravitational field. In 1919, Einstein’s theory passed an important test,
and this helped to satisfy the scientists who were previously skeptical of it. A total eclipse
of the sun occurred on May 29, 1919. Scientists went to Brazil and Africa’s west coast
and photographed the eclipse. The same region of the sky was photographed
approximately six months later. The locations of the stars on the two sets of pictures were
compared. The scientists determined that the positions of the stars had shifted by the
exact amount that Einstein’s theory predicted. Most scientists now regard Einstein’s
general theory of relativity as the best theory of the nature of the universe (Maor 133).
12. Special Categories of Numbers.
A real number r is algebraic if it satisfies an equation of the form,
anxn + an-1x
n-1 + an-2x
n-2 + + a1x + a0 = 0, with an, an-1, an-2, … , a1, and a0 representing
integers (Laubenbacher 57). A real number is transcendental if it is not algebraic. In
Batchelor 45
1873, Charles Hermite showed that e is a transcendental number. In 1882, F. Lindemann
showed that π is a transcendental number (Laubenbacher 58). In 1844, Joseph Liouville
provided the following example of a transcendental number:
10-1!
+ 10-2!
+ 10-3!
+ 10-4!
+ = 0.110001000000000000000001000. When Liouville
discovered this number, mathematicians regarded transcendental numbers as
mathematical oddities. However, in 1874, Cantor proved that most real numbers are
transcendental numbers (Maor 53). The fact that π is a transcendental number is
particularly important. Mathematicians have long wondered if it is possible to square a
circle, or to use a compass and straightedge to construct a square with an area equal to the
area of a specific circle. The fact that π is a transcendental number implies that this
construction is not possible (Maor 53).
Cantor proved that the set of all algebraic numbers is a countable set. This means
that the set of all algebraic numbers has the same cardinality as the set of natural
numbers. In 1851, Joseph Liouville proved a theorem stating that every interval of real
numbers must contain an infinite number of transcendental numbers. Cantor expanded on
Liouville’s work when he proved that all intervals of real numbers have cardinality
strictly larger than the cardinality of the natural numbers. Therefore, every interval of real
numbers must contain an infinite number of transcendental numbers (Laubenbacher 58).
This discovery led to the important distinction between the set of real numbers, which is a
continuous set, and the set of integers, which is a discrete set.
Amicable numbers are pairs of numbers such that the sum of the proper divisors
of the first number equals the second number, and the sum of the proper divisors of the
second number equals the first number. The numbers 220 and 284 are an example of a
Batchelor 46
pair of amicable numbers. Mathematicians have not determined whether there are an
infinite number of pairs of amicable numbers (Moews 2). A perfect whole number is a
whole number that equals the sum of its proper divisors. For example, 6 is perfect, since
6 = 1 + 2 + 3 (Laubenbacher 159). Ancient Greek mathematicians were familiar with four
perfect numbers. These numbers were 6, 28, 496, and 8128. Nicomachus was an early
Greek mathematician. He recognized the rarity of perfect numbers. He stated that the
perfect numbers were remarkable, “even as fair and excellent things are few … while
ugly and evil ones are widespread” (qtd. in Dunham, Master 2). Euclid studied the
perfect numbers extensively. He stated the following proposition as Proposition 36 of
Book IX of the Elements:
Proposition. If as many numbers as we please beginning from a unit be set out
continuously in double proportion, until the sum of all becomes prime, and if the sum
multiplied into the last make some number, the product will be perfect (Dunham,
Master 3).
The following theorem is equivalent to Euclid’s proposition:
Theorem. If 2k – 1 is a prime number and if N = 2
k-1(2
k – 1), then N is a perfect number
(Laubenbacher 159).
Proof. Let p = 2k – 1 denote a prime number. Let N = 2
k-1(2
k – 1) = 2
k-1p. The number N
can be factored into prime numbers in exactly one way. Therefore, the proper divisors of
N will only contain the prime numbers p and 2. One can express the sum of these proper
divisors as follows:
Let N′ = Sum of all proper divisors of N, where N = 2k-1
p.
N′ = 1 + 2 + 4 + 8 + + 2k-1
+ p + 2p + 4p + 8p + 2k-2
p
Batchelor 47
N′ = (1 + 2 + 4 + 8 + + 2k-1
) + p(1 + 2 + 4 + 8 + + 2k-2
)
N′ = (2k – 1) + p(2
k-1 – 1)
N′ = p + p2k-1
– p
N′ = p2k-1
N′ = N
Therefore, the number N is equal to the sum of all of the proper divisors of N. N is a
perfect number (Dunham, Master 4). One can apply this theorem in the following way.
Let k = 3. The number 23 – 1 = 8 – 1 = 7 is a prime number. Then, the number
22(2
3 – 1) = 4*7 = 28 is a perfect number. Prime numbers with the form p = 2
k – 1 are
called Mersenne primes (Laubenbacher 160). Mathematicians have devoted a great deal
of effort to the search for Mersenne primes. In 1772, Euler told Daniel Bernoulli that he
had proved that 231
– 1 is a prime number. The fact that 231
– 1 is prime implies that
230
(231
– 1) = 2,305,843,008,139,952,128 is a perfect number. During the early part of the
1800s, a mathematician stated that this number was “… the greatest [perfect number] that
will ever be discovered, for, as they are merely curious without being useful, it is not
likely that any person will attempt to find one beyond it” (qtd. in Dunham, Master 5).
However, mathematicians have discovered larger Mersenne primes. Computers have
helped with the search for larger Mersenne primes and perfect numbers. In 1998,
mathematicians discovered that the number 23021377
– 1 is a prime number. Euclid’s
theorem then implies that 23021376
(23021377
– 1) represents a perfect number. This perfect
number contains over 1,800,000 digits (Dunham, Master 5).
The question of whether or not there are any odd perfect numbers represents an
important unsolved problem in mathematics (Laubenbacher 160). Carolus Bovillus
Batchelor 48
provided a proof that all perfect numbers are even in 1509. However, his proof was
flawed; he had assumed that all perfect numbers must have the structure, 2k-1
(2k – 1).
This assumption is not valid, since no one has successfully proved that every perfect
number must have this structure. In 1747, Euler stated, “Whether … there are any odd
perfect numbers is a most difficult question” (qtd. in Dunham, Master 13). While no odd
perfect numbers have ever been discovered, no one has ever proved that they cannot
exist. Mathematicians have proved several properties that any odd perfect number must
have. If there is an odd perfect number, it must be greater than 10300
(Dunham, Master
15). In addition, any odd perfect number has to contain a minimum of eight different
prime factors. J.J. Sylvester was very skeptical about the likelihood that any odd perfect
numbers actually exist. In 1888, he made the following statement: “… a prolonged
meditation on the subject has satisfied me that the existence of any one such – its escape,
so to say, from the complex web of conditions which hem it in on all sides – would be
little short of a miracle” (qtd. in Dunham, Master 16).
13. Non-Euclidean Geometry.
The concept of non-Euclidean geometry has caused mathematicians and scientists
to reexamine their understanding of the physical world. This concept was “inspired” by
the question of what properties parallel lines have at great distances from the observer.
Euclidean geometry is based on a set of ten axioms that Euclid accepted as true without
proof. The following postulate is Euclid’s Parallel Postulate, and it is Euclid’s most
controversial postulate (Hartshorne 296).
Parallel Postulate. If a straight line falling on two straight lines makes the interior angles
on the same side less than two right angles, the two straight lines, if produced
Batchelor 49
indefinitely, meet on that side on which the two angles are less than two right angles
(Maor 119).
This was Euclid’s original statement of the Parallel Postulate. The following is a modern
way of stating the Parallel Postulate.
Postulate. Given a line ℓ and a point P that is not located on ℓ, there exists exactly one
line m, located in the plane of P and ℓ, which is parallel to the line ℓ (Maor 119).
Some mathematicians did not believe that the Parallel Postulate was as self-evident as
Euclid’s other axioms. Proclus stated, “This [Parallel Postulate] ought even to be struck
out of the Postulates altogether; for it is a theorem involving many difficulties, which
Ptolemy, in a certain book, set himself to solve, and it requires for the demonstration of it
a number of definitions as well as theorems” (qtd. in Laubenbacher 3). During the 1600s,
some mathematicians believed that Euclid’s Parallel Postulate was actually a theorem
(Hall 108). Many mathematicians made unsuccessful attempts to prove the Parallel
Postulate using Euclid’s other axioms. Girolano Saccheri, an Italian priest, tried to prove
the Parallel Postulate using the method of proof by contradiction.
Carl Friedrich Gauss suspected that the Parallel Postulate was independent of
Euclid’s other axioms. Since Gauss did not publish this conclusion, Nicolai Lobachevsky
and Janos Bolyai took credit for discovering non-Euclidean geometry (Laubenbacher 13).
These mathematicians assumed that two or more parallels to a given line can pass
through a point that is not located on that line. This directly implies that given a point P
not located on a line ℓ, there exist infinitely many lines through P that are parallel to ℓ.
Hyperbolic geometry is based on this new postulate, along with Euclid’s other nine
axioms (Hartshorne 373). In hyperbolic geometry, the angles of a triangle have a sum less
Batchelor 50
than 180°, and this sum depends on the triangle’s size (Maor 125). Euclid’s second axiom
states that one can extend a line indefinitely in either direction. Bernhard Riemann
replaced this with an axiom stating that a straight line is not bounded. Riemann
emphasized the importance of distinguishing between infinite extent and unboundedness.
Riemann’s theory of geometry states that through a point not on a line, there exist no
parallels to the line. Riemann’s theory of geometry is called elliptic geometry. In elliptic
geometry, the angles of a triangle have a sum greater than 180°. This sum depends on the
triangle’s size. In addition, if two triangles are similar, then they are congruent
(Laubenbacher 16).
A great circle is one that divides a sphere into two hemispheres that are equal. For
example, the equator is a great circle if the earth is assumed to be a sphere. A sphere can
serve as a model of elliptic geometry (Hall 103). The tractroid represents a model of
hyperbolic geometry (Maor 130). Gauss stated, “Finite man cannot claim to be able to
regard the infinite as something to be grasped by means of ordinary methods of
observation” (qtd. in Maor 131). He was concerned with the implications of non-
Euclidean geometry with regard to physics. He conducted an experiment in which he
measured a large triangle’s angles, but he found the sum to be equal to 180°. In 1868,
Eugenio Beltrami proved that non-Euclidean geometry is as logically acceptable as
Euclidean geometry (Dunham, Genius 56). In 1916, Albert Einstein published a theory of
gravitation in which space and time represent a single, four-dimensional entity. This
entity is called the space-time continuum. He defined a straight line as the path traveled
by a ray of light. He believed that rays of light are bent when a strong gravitational field
is present. Einstein’s general theory of relativity is consistent with Riemann’s theory of
Batchelor 51
geometry. Although Einstein’s theory was originally met with skepticism, it has
withstood many tests. Most scientists accept Einstein’s theory as the best modern
scientific theory of the universe (Laubenbacher 16).
14. Religion and Infinity.
After Georg Cantor became disillusioned with his study of mathematics, he began
to devote most of his energy to the study of theology and philosophy. The Catholic
Church objected to Cantor’s theory of the actual infinite. Many theologians believed that
Cantor’s concept of the actual infinite was not congruent with their concept of God.
However, Pope Leo XIII believed that natural science could help to further the goals of
the Catholic Church (Dauben 141). He tried to reconcile scientific discoveries with
religious scripture. In addition, his enthusiasm for Cantor’s work helped Cantor to
overcome his setbacks and to become interested in mathematics again (Dauben 142). The
Jewish kabbalists believed that God had a transcendental nature, and their belief in God
led them to the study of the infinite (Maor 179). In 1886, Gutberlet stated that he feared
that mathematical infinity could represent a challenge to the absolute infinity associated
with God’s existence. However, Cantor believed that the transfinite numbers did not
diminish the extent of God’s greatness. Rather, he believed that the existence of these
numbers enhanced it (Dauben 143). Several theologians and philosophers in the Catholic
Church expressed an interest in Cantor’s work. Gutberlet presented Cantor’s work in
order to support his own ideas against the opposition of some theologians. Thomas Esser
participated in a complex study of the implications of Cantor’s work with regard to
theology. Cardinal Johannes Franzelin initially objected to Cantor’s views regarding
infinity. However, he later accepted Cantor’s theory. He reassured Cantor that his theory
Batchelor 52
did not represent a theological threat to religious beliefs. Cantor believed that God
inspired him to complete his work (Dauben 147).
The Kabbalah states that God does not reveal Himself directly, but only through
His deeds and virtues. In 1638, Galileo Galilei stated, “Infinities and indivisibles
transcend our finite understanding, the former on account of their magnitude, the latter
because of their smallness; Imagine what they are when combined” (qtd. in Maor 179).
Followers of Judaism believed that infinity represented their search for the divine spirit.
Followers of Buddhism and Hinduism believed in eternity and the infinite reincarnation
of the human soul. The Christian concept of the resurrection led to the building of large
cathedrals and churches (Dauben 146). In 1711, Sir Christopher Wren completed
building St. Paul’s Cathedral (Maor 182). The Gothic cathedral provided visitors with the
illusion that they could reach infinite heights. William Wordsworth referred to the Gothic
cathedral’s “spires whose silent finger points to heaven” (qtd. in Maor 181).
15. Astronomy and Infinity.
Throughout history, people have wondered whether the universe is finite. Edmond
Halley once stated, “… I have heard urged that if the number of Fixed Stars were more
than finite, the whole superficies of their apparent Sphere would be luminous …” (qtd. in
Maor 204). Many people have wondered whether the universe has an outer boundary.
Albert Einstein said, “We never cease to stand like curious children before the great
Mystery into which we are born” (qtd. in Maor 185). The early Greeks transformed
astronomy into a science. Early astronomical models placed the earth at the center of the
universe. Aristotle’s model was complex, consisting of 56 spheres which surrounded the
earth. Democritus first suggested that the Milky Way could be a large collection of stars,
Batchelor 53
not a continuous band of light. Hipparchus developed the science of trigonometry, which
he used to calculate the distance between the earth and the moon. In the Almagest,
Ptolemy asserted that the earth was located at the universe’s center. The Roman Catholic
Church officially accepted this theory (Dunham, Genius 106). Many astronomers were
afraid to publicize their theories because they were afraid of how the Catholic Church
might react to them. Giordano Bruno was killed in 1600 after he insisted that the universe
is infinite (Maor 198). Galileo Galilei was under house arrest for the last years of his life,
and the Catholic Church forced him to recant his theory (Maor 200). In 1600, William
Gilbert stated, “How immeasurable then must be the space which stretches to those
remotest of the fixed stars! How vast and immense the depth of that imaginary sphere!
How far removed from the earth must the most widely separated stars be and at a distance
transcending all sight, all skill and thought!” (qtd. in Maor 190). Nicolaus of Cusa
believed that the universe was infinite and did not have a center. He was fascinated with
the concept of mathematical infinity.
In 1473, Nicolaus Copernicus was born in Poland (Maor 192). He began his study
of astronomy at the University of Cracow. In 1507, he returned to Poland and became
canon of the church of Frauenburg. Copernicus believed that the sun was located at the
universe’s center, and the planets revolved around the sun. His fourth proposition stated,
“The heavens are immense in comparison with the earth.” Copernicus was afraid to
publish his work, apparently fearing the reaction of the Catholic Church. In 1610, Galileo
Galilei discovered four moons of Jupiter. He discovered that Venus exhibited phases
similar to those of the moon. This served as evidence that Venus orbited the sun, not the
earth. Galileo stated, “The book of Nature is … written in mathematical characters” (qtd.
Batchelor 54
in Laubenbacher 101). Kepler stated his three laws of planetary motion. These laws
describe the motion of the planets around the sun. The Cosmological Principle states that
the universe is isotropic and homogeneous. This means that the basic laws of physics are
the same at all locations in the universe (Maor 218).
In 1755, Immanuel Kant, a philosopher, suggested that the universe is composed
of many galaxies. He identified Cepheid variables, which are stars that can be used to
determine a galaxy’s distance from the earth. Lucretius stated, “The universe is not
bounded in any direction. If it were, it would necessarily have a limit somewhere. But
clearly a thing cannot have a limit unless there is something outside to limit it … In all
dimensions alike, on this side or that, upward or downward through the universe, there is
no end” (qtd. in Maor 216). Based on Albert Einstein’s general theory of relativity, it
appears that the universe is finite but unbounded (Laubenbacher 16). Since the discovery
of subatomic particles, scientists have searched for the ultimate particle from which all
matter is created. The question of whether there really is an ultimate particle represents
an important unsolved scientific problem. People have often wondered if intelligent life
exists somewhere else in the universe. Scientists have used radio telescopes to search for
intelligent messages from other solar systems (Maor 229). A child once asked Einstein
how long the earth will exist. Einstein replied, “There has been an earth for a little more
than a billion years. As for the question of the end of it I advise: Wait and see” (qtd. in
Maor 182).
Batchelor 55
16. Infinity and Art.
Maurits Cornelis Escher was born in Leeuwarden, Holland in 1898. He began his
artistic career as a landscape painter. Escher painted many pictures of small Italian and
Spanish towns. Later in his career, his art depicted mathematical concepts such as
infinity, reflections, and inversions. M.C. Escher’s depictions of infinity fall into three
main categories: limits, endless cycles, and regular divisions of a plane. Escher’s pictures
were not always consistent with scientific laws. He stated, “I cannot help mocking our
unwavering certainties. It is, for example, great fun deliberately to confuse two and three
dimensions, the plane and space, or to poke fun at gravity” (qtd. in Maor 166). Escher did
not usually depict abstract images. His paintings depicted objects such as horses, people,
and ants. He stated, “The Moors were masters in the filling of a surface with congruent
figures … What a pity it was that Islam forbade the making of images. In their
tessellations they restricted themselves to figures with abstracted geometrical shapes … I
find this restriction all the more unacceptable because it is the recognizability of the
components of my own patterns that is the reason for my never-ceasing interest in this
domain” (qtd. in Maor 168). Escher’s pictures displayed a great deal of symmetry
(Laubenbacher 53). His patterns included examples of reflections, translations, and
rotations. His pictures represent at least 13 of the 17 symmetry groups of the plane.
Escher made the following statement about the relationship between mathematics and art:
“By keenly confronting the enigmas that surround us, and by considering and analyzing
the observations that I had made, I ended up in the domain of mathematics. Although I
am absolutely without training in the exact sciences, I often seem to have more in
common with mathematicians than with my fellow-artists” (qtd. in Maor 164).
Batchelor 56
Several other artists depicted infinity through their art. Max Bill created a
sculpture of a Möbius strip, which he called Endless Ribbon. He did not realize that the
Möbius strip had already been discovered. When he learned this, he stated, “Sometime
later I was informed that my creation, which I thought I had discovered or invented, was
only an artistic interpretation of the so-called Möbius strip, and theoretically identical to
it … I was shocked by the fact that I was not the first one to discover this object. I
therefore stopped all further research in this direction for a while” (qtd. in Maor 140).
One can use mirrors to produce infinitely repeating images. Sir David Brewster, a
Scottish physicist, invented the kaleidoscope in 1816. In a kaleidoscope, the tube contains
two mirrors, which are usually placed 60 degrees apart. If four mirrors are placed to form
a square, the images will tile the plane with squares. When artists fill a plane with a
repeating design, they can use three kinds of transformations: rotations, reflections, and
translations. The ornamental band is the simplest infinite pattern. It consists of infinite
repetitions of one figure along a one-dimensional strip. Group theory is a branch of
algebra that involves analyzing infinite patterns (Laubenbacher 254). Johann Sebastian
Bach’s music showed that the composer had a great deal of mathematical talent. His
music displayed a tremendous amount of symmetry. Joan Miró, a Spanish artist, depicted
an endless void in his work, Towards the Infinite. Leonardo da Vinci was an expert in the
art of perspective. An artist using this technique tries to depict what a viewer’s eye would
actually see when viewing a scene (Maor 109).
Projective geometry is the branch of geometry that focuses on the properties of
figures that do not change under a projection. A projection is the collection of light rays
that come from an object and converge at the viewer’s eye. The properties of parallel
Batchelor 57
lines change under a projection. In projective geometry, parallel lines are defined to meet
at the point at infinity. Projective geometry allows mathematicians to make the
generalization that any two lines in a plane meet at exactly one point. Every family of
parallel lines must have a unique point where all of the lines intersect. The line at infinity
consists of the collection of all points at infinity (Maor 109). The following principle is
an important part of the theory of projective geometry.
Principle of Duality. Every true statement regarding the mutual relationship among lines
and points will remain true if the words line and point are interchanged everywhere in
the statement (Maor 112).
Artists apply projective geometry when they depict visual scenes in their pictures.
Maurits C. Escher is an example of a great artist who understood the concepts of
projective geometry. While he had little formal mathematical training, he displayed an
excellent understanding of the properties of mathematics.
17. Paradoxes and Antinomies.
Several significant paradoxes and antinomies are related to the study of the
infinite. Russell’s Paradox is one such paradox. Bertrand Russell, a British
mathematician, first described this paradox in 1902 (Miller 1). Some sets belong to
themselves. For example, let N denote the set of all objects whose descriptions do not
include the last letter of the alphabet. Since the description of set N does not include the
letter z, the set N belongs to itself. Next, let P denote the collection of all sets that do not
contain themselves as elements. Consider the question of whether the set P belongs to
itself. Suppose that P belongs to itself. This contradicts that definition of the set P.
Suppose that P does not belong to itself. Then the set P must belong to itself, since P
Batchelor 58
contains every set that does not contain itself as an element. This is a contradiction. This
logical paradox is called Russell’s Paradox, and many mathematicians refer to it as the
Barber’s Paradox (Dauben 262).
Bernhard Bolzano practiced a philosophical approach to the study of
mathematics. He studied mathematics at the University of Prague. His most famous work
is a book called Paradoxes of the Infinite. He argued that the best way to approach the
infinite is to use a mathematical approach (Dauben 124). He stated that infinite sets are
not all equal in terms of their multiplicity. One infinite set can be a portion of another
infinite set. In other words, some infinite sets are larger than others. Consider the ray
ACDB , where AC, CD, and DB represent finite, positive lengths of line segments. The
ray with endpoint D in the direction of B has infinite length. The ray with endpoint C in
the direction of B also has infinite length, but it can be thought of as having a length that
is greater by the length of the segment CD . The line ACDB , which extends infinitely in
both directions, can be considered even larger, by the length of the ray with endpoint C in
the direction of A; this ray also has infinite length (Laubenbacher 72). There are several
indeterminate concepts, which do not have distinct values. The only way to evaluate
them is to use limits. The most well known indeterminate concepts are ∞/∞, 0/0, ∞*0, 00,
1∞, ∞
0, and ∞ – ∞ (Maor 9).
The paradox of the universal set is an important paradox related to the study of
infinity. Let U denote the set of all sets. Since U contains every set, one cannot enlarge it.
Cantor’s Theorem asserts that U < P[U] . Therefore, P[U] contains more elements than U.
This is a contradiction. This paradox served as evidence that mathematicians needed to
formalize the system of axioms of set theory (Dunham, Genius 281). Zeno was a
Batchelor 59
philosopher who discovered several famous paradoxes. The runner’s paradox is a
particularly noteworthy paradox. Suppose that a runner wishes to travel from one location
to another location. The runner must first travel half of the distance between the
locations. Then, she/he must run half of the distance that remains. This process repeats
itself infinitely, so the runner must take infinitely many steps in order to reach her/his
destination. Therefore, Zeno claimed that the runner would never travel the complete
distance to the destination. This represented a paradox, since one can demonstrate that a
runner can cover a finite distance in a finite amount of time. One can represent the
runner’s paradox using the infinite sum, 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + (Maor 4). This
limit of this sum is 1 as the number of terms approaches infinity. The Gabriel’s Horn
Paradox is an example of a geometric paradox. The graph of the function y = 1/x is a
hyperbola. Consider the graph of this function for x 1. If one revolves this graph about
the x-axis, the result is a figure called a hyperboloid of revolution. This solid has an
infinite surface area. However, the volume of the solid is finite (Maor 85). Now suppose
that someone states, “I am lying.” The question of whether or not she/he is telling the
truth represents a logical paradox.
18. Unsolved Problems Regarding Infinity.
Goldbach’s Conjecture is an important unsolved problem in mathematics (Lam
1). Goldbach’s Conjecture states that one can express every even number greater than 2
as the sum of two prime numbers (Dunham, Genius 82). While mathematicians have
never succeeded in proving this conjecture, no one has disproved it. Mathematicians have
used modern computers to aid in their work with Goldbach’s Conjecture, but they have
still been unable to prove it (Lam 2). The question of the infinitude of twin primes is
Batchelor 60
another significant unsolved problem. Twin primes are consecutive prime numbers that
differ by exactly 2. For example, 11 and 13 represent a pair of twin primes.
Mathematicians know that there exist an infinite number of prime numbers, but no one
has proved that there are an infinite number of pairs of twin primes. The numbers
1,159,142,985 * 22304
1 represent a pair of 703-digit twin primes (Maor 23). Euclid
proved that every number with the form, 2n(1 + 2 + 4 + 8 + 16 + + 2
n) is a perfect
number. Mathematicians are not certain whether there are any perfect numbers besides
those that are represented by Euclid’s formula. Euler showed that every even perfect
number must be of the form that Euclid specified. However, no one has proven that odd
perfect numbers must have this form or that such numbers do not exist (Dunham, Genius
82).
Mathematicians have not been able to prove whether there are any odd perfect
numbers. If there is an odd perfect number, it must contain at least eight different prime
factors. In addition, the smallest odd perfect number must exceed 10300
(Dunham, Master
15). While mathematicians doubt whether any odd perfect numbers exist, they have been
unable to prove that they cannot exist. J.J. Sylvester strongly doubted the existence of
odd perfect numbers. He stated, “… a prolonged meditation on the subject has satisfied
me that the existence of any one such – its escape, so to say, from the complex web of
conditions which hem it in on all sides – would be little short of a miracle” (qtd. in
Dunham, Master 16). The symbol γ is used to denote Euler’s constant, which has an
approximate value of 0.5772156649 (Savard 1). The Nth
partial sum of the harmonic
series always has a value between ln(N) and 1 + ln(N), where ln(N) denotes the natural
logarithm of N. Euler’s constant γ denotes the limit of the expression,
Batchelor 61
(1 + 1/2 + 1/3 + 1/4 + + 1/N) – ln(N), as N approaches infinity (Savard 2).
Mathematicians have not determined whether γ is a rational number (Maor 28).
Batchelor 62
Afterword.
Infinity is a truly fascinating concept. The study of infinity has grown and evolved
since the early Greeks first studied it in a formal setting. Mathematicians have persevered
in the study of infinity despite objections from other mathematicians and the Catholic
Church. Euclid, Euler, Cantor, and Gauss made major contributions to the study of the
infinite. The concepts of cardinal numbers and one-to-one correspondence have been
important tools in the formal study of infinity. The study of infinite sets has led to some
interesting questions about topics such as amicable and perfect numbers and non-
Euclidean geometry. Several paradoxes or antinomies forced mathematicians to develop
more rigorous definitions and systems of axioms. Many unsolved problems remain, such
as Goldbach’s Conjecture, the infinitude of twin primes, and the search for an odd perfect
number (Lam 3). Mathematicians continue to make new discoveries about the infinite,
such as Andrew Wiles’s 1993 proof of Fermat’s Last Theorem (Laubenbacher 156).
As a student of mathematics, I have enjoyed this intellectual adventure. I feel that
I have learned a great deal about the nature of infinite sets. I particularly liked the parts
about set theory, logic and reasoning, and the unsolved problems. In addition, I have been
fascinated with the information about how infinity relates to art, astronomy, and religion.
I am an amateur astronomy enthusiast, and I enjoy studying the major theories of the
universe. M.C. Escher’s drawings of infinite patterns were fun to view. I am confident
that mathematicians will continue to make major discoveries related to infinity during the
coming years, and I look forward to hearing about these discoveries. I hope to complete
additional research regarding infinite sets in the future.
John H. Batchelor
Batchelor 63
Appendices.
I. Timeline of Major Events.
550 B.C. – Early Greeks first acknowledged the concept of infinity.
387 B.C. – Plato founded the Academy in Athens.
332 B.C. – Alexander the Great established the city of Alexandria in Egypt.
300 B.C. – Euclid went to Alexandria, intending to establish a school of mathematics.
1473 – Nicolaus Copernicus was born in Poland.
1507 – Copernicus returned to Poland and became canon of the church at Frauenburg.
1569 – Gerhardus Mercator published his map, which represents the earth using a
rectangular grid.
1600 – Giordano Bruno was killed after he insisted that the universe is infinite.
1610 – Galileo Galilei discovered four moons of Jupiter.
1631 – Pierre de Fermat received a degree in law.
1638 – Galileo discovered that the number of perfect squares equals the number of
natural numbers. Since not every natural number is a perfect square, this
appeared to represent a paradox.
1711 – Sir Christopher Wren completed St. Paul’s Cathedral.
1729 – Goldbach informed Euler of Fermat’s conjecture that every number of the form,
22n + 1, is prime.
1732 – Euler disproved Fermat’s conjecture by showing that 225 + 1 = (641)(6,700,417).
1736 – Euler proved Fermat’s Little Theorem.
1737 – Euler studied the sum of the reciprocals of the prime numbers. He proved that the
series diverges.
Batchelor 64
1747 – Euler stated, “Whether … there are any odd perfect numbers is a most difficult
question.”
1748 – Euler discovered the formula, eπi
+ 1 = 0.
1748 – Euler discovered the formula, e = 1 + 1/1 + 1/(1*2) +1/(1*2*3)+1/(1*2*3*4)+ .
1752 – Euler discovered that simple polyhedrons must satisfy the equation V-E+F=2,
where V = number of vertices, E = number of edges, and F = number of faces.
1755 – Immanuel Kant, a philosopher, suggested that the universe is composed of many
galaxies.
1772 – Euler wrote to Daniel Bernoulli and stated that he had proved that 231
–1 is prime.
1774 – Euler completed a paper called On a Table of Prime Numbers Up to a Million and
Beyond.
1775 – Euler studied series consisting of reciprocals of odd prime numbers, with positive
signs preceding 4k-1 primes and negative signs before 4k+1 primes. The series
converges to an approximate value of 0.3349816.
1777 – Carl Friedrich Gauss was born in Brunswick.
1796 – Gauss showed that a straightedge and compass could be used to construct a
17-sided regular polygon.
1799 – Gauss received doctoral degree from University of Helmstadt.
1801 – Gauss published his Disquisitiones Arithmeticae.
1807 – Gauss learned Sophie Germain’s true identity.
1816 – Sir David Brewster, a Scottish physicist, invented the kaleidoscope.
1831 – Gauss told Schumacher that he did not believe in the actual infinite.
Batchelor 65
1837 – Peter Gustav Lejeune-Dirichlet proved that every arithmetic progression with the
form, a, a + b, a + 2b, a + 3b, , a + kb, , must include an infinite number of
prime numbers, as long as a and b are relatively prime.
1844 – Joseph Liouville provided the following example of a transcendental number:
10-1!
+ 10-2!
+ 10-3!
+ 10-4!
+ = 0.110001000000000000000001000.
1845 – Georg Cantor was born in Russia.
1851 – Joseph Liouville proved a theorem stating that every interval of real numbers
must contain an infinite number of transcendental numbers.
1868 – Eugenio Beltrami proved that non-Euclidean geometry is just as logically valid as
Euclidean geometry.
1871 – Cantor began to consider exceptional sets.
1873 – Charles Hermite showed that e is a transcendental number.
1874 – Cantor showed that most real numbers are transcendental numbers.
1874 – Cantor showed that no interval of real numbers can be placed in a one-to-one
correspondence with the set of natural numbers.
1874 – Cantor proved the existence of infinite sets that have different and distinct
magnitudes.
1877 – Cantor discovered that a one-to-one correspondence exists between the square and
the interval. He exclaimed, “I see it, but I do not believe it!”
1878 – Cantor proved that one could restrict the analysis of continuity to the domain of
the real line.
1879 – Cantor published the first of a group of six papers about infinite linear sets of
points.
Batchelor 66
1879 – Paul du Bois-Reymond stated that he preferred the term pantachisch instead of
everywhere-dense.
1880 – Cantor published a short paper about transfinite symbols and derived sets.
1882 – F. Lindemann showed that π is a transcendental number.
1883 – Cantor published a paper about set theory developed to address the mathematical
and philosophical issues associated with transfinite set theory.
1884 – Cantor resumed his effort to prove the Continuum Hypothesis.
1884 – Cantor told Mittag-Leffler that the Continuum Hypothesis was a complete failure.
1885 – Gösta Mittag-Leffler wrote to Cantor, cautioning him against publishing his new
work.
1886 – Constantin Gutberlet stated that he feared that mathematical infinity could
represent a challenge to the absolute infinity associated with God’s existence.
1888 – Gutberlet founded the journal Philosophisches Jahrbuch der Görres-Gesellschaft.
1888 – J.J. Sylvester stated that he was skeptical about the existence of an odd perfect
number.
1888 – Cantor wrote to Jeiler and expressed confidence in the theory of transfinite
numbers.
1891 – Cantor wrote a paper entitled On an Elementary Question in the Theory of Sets.
1894 – Cantor told Charles Hermite, a French mathematician, that he was glad to have
had the chance to serve the Roman Catholic Church.
1896 – Jacques Hadamard and de la Vallée Poussin proved the Prime Number Theorem.
1896 – Cantor told Esser, “From me, Christian philosophy will be offered for the first
time the true theory of the infinite.”
Batchelor 67
1898 – Maurits Cornelis Escher was born in Leeuwarden, Holland.
1900 – David Hilbert emphasized the importance of Cantor’s Continuum Hypothesis at
the Second International Congress of Mathematicians in Paris.
1902 – Bertrand Russell criticized Euclid’s work.
1902 – Bertrand Russell first described Russell’s Paradox.
1910 – Bertrand Russell stated, “The solution of the difficulties which formerly
surrounded the mathematical infinite is probably the greatest achievement of
which our age has to boast.”
1916 – Albert Einstein published his theory of general relativity.
1919 – Einstein’s theory of general relativity passed an important test in which scientists
photographed a region of the sky before and after a total solar eclipse.
1963 – Paul Cohen showed that the Axiom of Choice is independent of the other set
theory axioms.
1963 – Kurt Gödel and Paul Cohen showed that the Continuum Hypothesis is
independent of the other axioms in the Zermelo-Fraenkel system.
1993 – Andrew Wiles proved Fermat’s Last Theorem.
1998 – Mathematicians discovered that the number 23021377
– 1 is a prime number.
Batchelor 68
II. Proof That 2 Is an Irrational Number.
Theorem. 2 is an irrational number.
Proof. The method of proof by contradiction will be used. Suppose that 2 is a rational
number. Then, one can express 2 as the quotient of two integers, denoted a and b:
2 = a/b.
Squaring both sides of the equation, we obtain the result, 2 = (a2)/(b
2). Multiplying both
sides by b2, we obtain the result, 2b
2 = a
2. This can be expressed as
a2 = 2b
2.
The variables a and b represent integers, as stated above. Therefore, a and b can be
decomposed into their prime factors in only one way. Now let a = c1c2c3cm, and
b = d1d2d3dn. Substituting into the equation, a2 = 2b
2, we obtain the result,
(c1c2c3cm)2 = 2(d1d2d3dn)
2.
This is equivalent to the following equation:
c1c1c2c2c3c3cmcm = 2d1d1d2d2d3d3dndn.
It is possible that the prime number 2 will occur among the prime numbers ci and di. This
will happen if a or b is even. Suppose that the number 2 occurs among these prime
numbers. In this case, the number 2 will occur an even number of times on the equation’s
left side. This is true because each of the prime numbers on the left side occurs twice.
The number 2 will appear an odd number of times on the equation’s right side. This is
true because the number 2 already occurs once on the right side, in addition to the
numbers represented by the di variables. Suppose that the number 2 does not occur
among the prime numbers ci or di. Then, the number 2 will not occur on the equation’s
left side, and it will appear one time on the equation’s right side. In all cases, the prime
Batchelor 69
number 2 occurs an even number of times on the equation’s left side and an odd number
of times on the equation’s right side. However, the prime number 2 must appear the same
number of times on both sides of the equation, due to the unique decomposition into
prime numbers. This is a contradiction. The initial assumption that 2 is rational is false.
Therefore, 2 is an irrational number. This completes the proof (Maor 236-237).
Batchelor 70
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Batchelor 71
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