Chapter 11: The NonDenumerability of the
Continuum
Presented by Lori Joyce and
Becky-Anne Taylor
The 1800’s and early 1900’s
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1807 - Slavery was abolished in Canada
1812 - War of 1812
1830 – Underground Railroad is established
1861 – American Civil War
1862 – First female student accepted into Mount Allison!
1867 – Canadian Confederation
1876 – Alexander Graham Bell invents the telephone
1879 – Thomas Edison invents the light bulb
The 1800’s and early 1900’s
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1891 – Whitcomb Judson invents the zipper
1896 – Gold Rush in Alaska and Canadian Northwest
1900 – First wireless radio broadcast by Reginald Fesseden
1903 – Wright Brothers first flight
1912 – Sinking of the Titanic
1914 – Canada enters WWI
1917 – US enters WWI
The 1800’s and early 1900’s
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1799 – Napoleon seizes control of France
1804-1815 – Napoleonic Wars
1818 – Karl Marx is born
1858 – Max Planck is born
1859 – Charles Darwin publishes the Origin of Species
The 1800’s and early 1900’s
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1871 – Germany removes the clergy from controlling
education
1879 – Albert Einstein is born
1889 – Adolf Hitler is born
1892 – Rudolf Diesel invents the diesel engine
1914 – Britain declares war on Germany; begins WWI
Mathematics at the Time
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1800’s: century of abstraction and generalization
Deeper analysis of the theories of Newton, Leibniz, and Euler
Decisive steps were made in the creation of non-Euclidean
geometry
Carl Friedrich Gauss, Janos Bolyai and Nikolai Lobachevsky
published works on hyperbolic geometry in the 1820’s
Eugenio Beltrami established that non-Euclidean geometry
was as logically consistent as Euclid’s
Which geometry system was “real”?
Both systems continued to be developed
Mathematics was undergoing a freedom from dependence on
reality
Similar phenomena was taking place in the world of art
Abstract Art
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Art was becoming more abstract
Paul Cezanne, Paul Gaugin, and Vincent Van Gogh began
the concept
Their art was more than what the eye saw
The Abstract
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Mathematicians carried the subject further from contact with
the real world
Constructs of these mathematical abstract ideas neared the
point of being unrecognizable by physicists
Same issues with expansion of abstract ideas arose in
calculus
The Abstract
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Underlying problem of early calculus: the use of “infinitely
large” and “infinitely small” quantities
Concept of “limit” needed tuning
In 1821, Augustin-Louis Cauchy proposed the following
definition:
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“When the values successively attributed to a particular variable
approach indefinitely a fixed value, so as to end by differing from it by as
little as one wishes, the latter is called the limit of all the others.”
Cauchy’s definition removed the philosophical aspect of
having to predict what happened at the moment of reaching
the limit
Augustin-Louis Cauchy
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French Mathematician born in 1789,
passed away in 1857
An engineer who renounced from the
profession to pursue mathematics
First to prove Fermat polygonal number
theorem
Created the residue theorem in complex
analysis
First to define complex numbers as pairs
of real numbers
His definition of limit needed revision; too
wordy
Karl Weierstrass
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German mathematician born in 1815, passed away in 1897
“Father of Modern Analysis”
Studied mathematics at the University of Münster, where he
was later certified as a teacher
Provided sound and concise definition for limit
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Limit: for any ε > 0, there exists a δ > 0 so that, if
0 < | x-a | < δ, then | f(x)-L | < ε
After Weierstrass
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Distinction between rational and irrational numbers
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Between any two rational numbers, there lie infinitely many irrationals,
and conversely, between any two irrationals there are found infinitely
many rationals
 Concluded that the real numbers must be divided into two large and
roughly equivalent families of rationals and irrationals
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Mathematics in the 19th century suggested that these two
classes of numbers were not equal in size
Mathematicians were coming to realize that some of the
fundamental questions of calculus rested upon the properties
of sets
This obstacle was tackled with the development of set theory
by Georg Ferdinand Ludwig Philip Cantor
Georg Ferdinand Ludwig Philip Cantor
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Born in Saint Petersburg Russia in 1845.
Moved to Frankfurt, Germany in 1857 at
age of 12.
Founded set theory, introduced concept of
infinite numbers and advanced the study
of trigonometric series
He was also interested in theology and
had artistic abilities.
Completed his doctorate at the University
of Berlin in 1867.
Cantor was hired at the University of Halle
and spent his entire career here.
In 1874, Cantor married Vally Guttmann
and they had six children.
Passed away January 16, 1918.
How to Compare the Size of Sets?
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You could just count the number of
members in each set and compare.
BUT what if you couldn’t count that
high?
Assume only being able to count to
three. How would you compare the
number of fingers on the left hand to
the number on the right?
You could place each of your fingers
from one hand together with a finger
from the other. If there were no left
over fingers you would know the two
sets of fingers were the same size.
Similar analogy for an audience and
auditorium seats.
How to Compare the Size of Sets?
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Definition: Two sets M and N are equivalent (have the same
cardinality) 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.
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Note this does not specify that M and N must be finite!
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“… I realize that in this undertaking I place myself in a certain
opposition to views widely held concerning the mathematical
infinite and to opinions frequently defended on the nature of
numbers.”
(Georg Cantor, Grundlagen)
Comparison with the Naturals
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Example One:
Let N = {1,2,3,…} be the set of all natural numbers, and let
E = {2,4,6,….} be the set of all even natural numbers.
One would expect there to be half as many elements in E as
in N, but…
1
2
3
…
n
2
4
6
…
2n
Same cardinality!
The Integers?
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Let Z = {…-2,-1,0,1,2,…} be the set of all integers.
N: 1
2
3
4
5 …
Z: 0
1
-1
f(n) = ¼ x (1+(-1)n(2n-1)
2
-2 …
Cantor defined N as having
number of elements.
N is the standard to which other sets are compared. Any set
with a one-to-one correspondence with the natural numbers is
denumerable or countably infinite.
Thus, cardinality of N = cardinality of E = cardinality of Z =
Try it Yourself:
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Prove there is a one-to-one correspondence between the set
of natural numbers and the following set S= {1/3,1/6, 1/12, 1/24,
1/ , . . . . . . }.
48
Try it Yourself:
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Prove there is a one-to-one correspondence between the set
of natural numbers and the following set S= {1/3,1/6, 1/12, 1/24,
1/ , . . . . . . }.
48
N= { 1, 2, 3, 4, 5, . . . . . . }
S= {1/3,1/6, 1/12, 1/24, 1/48, . . . . . . }
Try it Yourself:
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Prove there is a one-to-one correspondence between the set
of natural numbers and the following set S= {1/3,1/6, 1/12, 1/24,
1/ , . . . . . . }.
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Answer:
N
S
1/
1
3
1/ = 1/
2
6
(3 x 2)
1/
1
3
12 = /(3 x 2 x 2)
:
:
1/
n
(3 x 2(n-1))
f(n) = (3 x 2(n-1))-1
The Rationals?
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Is there a one-to-one relationship between N and Q?
The Rationals?
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Is there a one-to-one relationship between N and Q?
0
1
-1
2
-2
3
-3
…
1/
2
-1/2
2/
2
-2/2
3/
2
-3/2
…
1/
3
-1/3
2/
3
-2/3
3/
3
-3/3
…
1/
4
-1/4
:
2/
4
-2/4
:
3/
4
-3/4
:
…
:
:
:
The Rationals?
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As shown above we have the following:
N
1
2
3
4
5
6
…
Q
0
1
½
-1
2
-½
…
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Note that the numbers 1 =2/2 = 3/3 = … are only counted once.
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We now have established that the cardinality is the same for
N, E, Z and Q.
Great Theorem: The NonDenumerability of the Continuum
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Continuum – an interval of real numbers. Ex: (a,b) = set of all
real numbers such that a<x<b.
Theorem:
The interval of all real numbers between 0 and 1 is
denumerable.
Great Theorem: The NonDenumerability of the Continuum
Proof:
Assume N and (0,1) have a one-to-one correspondance.
1  x1 = 0.371652…
2
 x2 = 0.500000…
3
 x3 = 0.142678...
:
:
n
 xn = .a1a2a3…an…
:
:
If N and (0,1) are one-to-one, then each natural number on
the right will be matched with a particular number on the left.
Great Theorem: The NonDenumerability of the Continuum
Proof:
Pick a number b = 0.b1b2b3…bn…
Choose a number b1 to be any digit other than the first digit of
x1 and not equal to 0 or 9.
Choose a number b2 to be any digit other than the second
digit of x2 and not equal to 0 or 9.
Choose a number b3 to be any digit other than the third digit
of x3 and not equal to 0 or 9.
:
Choose a number bn to be any digit other than the nth digit of
xn and not equal to 0 or 9.
Great Theorem: The NonDenumerability of the Continuum
Now we observe two things:
1) b is a real number because it is an infinite decimal. With the
restrictions we cannot have 0.000…=0 or 0.999…=1 (or
0.499… = 0.5, etc.) so be must fall strictly between 0 and 1.
Thus, it must fall somewhere in (0,1) on the left side of the
list.
2) b cannot appear anywhere along the numbers x1, x2,
x3…xn… for b≠x1 because b1≠x1, b≠x2 because b2 ≠ x2,…b
≠ xn because bn ≠ xn…
We have reached a contradiction. Thus, our original
assumption must be false.
Great Theorem: The NonDenumerability of the Continuum
Theorem:
The interval of all real numbers between 0 and 1 is
denumerable.
Proof:
By contradiction, we have proved this to be false. Therefore
the interval of real numbers between 0 and 1 is NOT
denumerable.
Some skeptics may argue that, although b may not have
existed in the left-hand column, it would be easy to simply add
it to the list, making it one-to-one. However, you could then
use the same method to find a number b1 that fits this proof,
and continue on like so indefinitely.
What came next?
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Now there were suddenly different levels of infinity.
Real numbers can be divided into two groups; relatively
scarce rationals and relatively abundant irrationals
Real numbers can be split into algebraic and transcendental
numbers
Very few transcendental numbers were identified at this time
After proving all real numbers to be denumerable, Cantor
considered transcendental numbers to be denumerable as
well
Transcendental Numbers
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Consider the arbitrary interval (a,b)
Cantor already proved the algebraic numbers within this set
were denumerable
He knew there were far more real numbers in (a,b) than could
be accounted for by the collection of algebraic numbers
The remainder of the numbers must be transcendental, which
abundantly exceed the number of algebraic numbers
Cantor developed this idea in 1874 without a single concrete
example of a transcendental number!
e was proven to be a transcendental number in 1873
Transcendental Numbers
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e was proven to be a transcendental number in 1873 by
Charles Hermite
π was proven to be transcendental in 1882 by Ferdinand von
Lindemann
Cantor continued his work in set theory, which will be
discussed in chapter 12.
Discussion
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1. In the beginning of chapter 11 it discusses how math began
to move away from physical realities, leaving that to the
physicists. It came under a lot of criticism with people
questioning the validity of a pursuit with no practical
application. Does this still hold true today? Do you believe that
knowledge for knowledge’s sake is a worthwhile endeavor or
do you think that only research which applies to real life is
important?
Discussion
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2. We have seen in each chapter how external influences and
beliefs of the time period have affected the field of
mathematics. For example, symmetry with the Greeks, or in
this case, the Christian religion influencing Cantor. Looking
around today, can you identify any connections with what is
going on in the world and the direction mathematics is taking?
Discussion
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3. Lastly, we have seen how important it can be to have the
support of respected mathematicians. For example, Sophie
Germain was encouraged by Gauss. Cantor, on the other
hand, did not have strong support from the math community
for his work on infinity and was often criticized or ridiculed for
his ideas. What kind of an effect do you think this had, if any,
on his work? On his reputation as a mathematician?
Discussion