Chapter 11: The Non- Denumerability of the Continuum Presented by Lori Joyce and Becky-Anne Taylor

The 1800’s and early 1900’s Slavery was abolished in Canada War of – 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 – 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 19 th 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 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 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 23…n 2 46…2n Same cardinality!

The Integers? Let Z = {…-2,-1,0,1,2,…} be the set of all integers. N:12345 … Z: … 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: NS 1 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? … 1 / / 2 2 / / 2 3 / / 2 … 1 / / 3 2 / / 3 3 / / 3 … 1 / / 4 2 / / 4 3 / / 4 … : : : : : : Is there a one-to-one relationship between N and Q?

As shown above we have the following: N … Q01½ -12 -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  x 1 = … 2  x 2 = … 3  x 3 = : n  x n =.a 1 a 2 a 3 …a n … : 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.b 1 b 2 b 3 …b n … Choose a number b 1 to be any digit other than the first digit of x 1 and not equal to 0 or 9. Choose a number b 2 to be any digit other than the second digit of x 2 and not equal to 0 or 9. Choose a number b 3 to be any digit other than the third digit of x 3 and not equal to 0 or 9. : Choose a number b n to be any digit other than the n th digit of x n 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 x 1, x 2, x 3 …x n … for b≠x 1 because b 1 ≠x 1, b≠x 2 because b 2 ≠ x2,…b ≠ x n because b n ≠ x n … 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 b 1 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?

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