Chapter 11: The Non-Denumerability of the Continuum

Presented by Lori Joyce and Becky-Anne Taylor

The 1800’s and early 1900’s

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

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

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

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 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 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

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

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: “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

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

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

Limit: for any ε > 0, there exists a δ > 0 so that, if 0 < | x-a | < δ, then | f(x)-L | < ε

After Weierstrass

Distinction between rational and irrational numbers 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

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 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? 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.

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.

Note this does not specify that M and N must be finite!

“… 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)

How to Compare the Size of Sets?

Comparison with the Naturals 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? Let Z = {…-2,-1,0,1,2,…} be the set of all integers.

N: 1 2 3 4 5 …

Z: 0 1 -1 2 -2 … f(n) = ¼ x (1+(-1)n(2n-1)

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: 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: 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:

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, . . . . . . }.

Answer: N S

1 1/3

2 1/6 = 1/(3 x 2)

3 1/12 = 1/(3 x 2 x 2)

: :

n 1/(3 x 2(n-1))

f(n) = (3 x 2(n-1))-1

The Rationals? Is there a one-to-one relationship between N and Q?

The Rationals?

0

1 -1 2 -2 3 -3 …

1/2 -1/2 2/2 -2/2 3/2 -3/2 …

1/3 -1/32/3 -2/3 3/3 -3/3 …

1/4 -1/4 2/4 -2/4 3/4 -3/4 …

: : : : : :

Is there a one-to-one relationship between N and Q?

As shown above we have the following:

N 1 2 3 4 5 6 …

Q 0 1 ½ -1 2 -½ …

Note that the numbers 1 =2/2 = 3/3 = … are only counted once.

We now have established that the cardinality is the same for N, E, Z and Q.

The Rationals?

Great Theorem: The Non-Denumerability of the Continuum

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.

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 Non-Denumerability 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 Non-Denumerability 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 Non-Denumerability 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.

Great Theorem: The Non-Denumerability of the Continuum

What came next?

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

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

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 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 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 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