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Georg Cantor Born: March 3, 1845 Died: January 6, 1918
Georg Cantor lived at the end of the 19th century and early 20th century. This is a time period in both mathematics and the world that is referred to as "the age of abstraction". Ideas and philosophies were changing from the concrete to the abstract. This could be seen in many fields along with mathematics. In economics abstract notions of different types of economies such as communism were described Marx And Engle and capitalism was described by Adam Smith. The world of art was changing to a more abstract form. Artists moved from being a "camera" that could reproduce what the human eye could see to having an abstract eye. For example the works of Cezanne, Van Gogh and Gauguin differed greatly from the works of Monet. Mathematicians began to cross the gap of what visual or physical reality would dictate, such as the innovation of Bolyai and Lobachevski concerning non- Euclidean geometries.[1 p 246] Georg Cantor was born in Denmark and grew up with a deep appreciation for culture and the arts which was instilled in him by his mother who had considerable musical talent as a violinist. In terms of religion Georg's mother and father were a mixed marriage, his father was a Protestant who had converted from Judaism and his mother Roman Catholic. Georg was raised as a Protestant. Georg Cantor's father was a successful merchant and stock broker in St. Petersburg whose wealth enabled him to afford a private tutor for Georg's early education. The family moved to Germany because his father's health required a warmer climate. When the family first moved to Germany young Cantor lived at home and studied at the Gymnasium in Wiesbaden. Later the family moved to Frankfurt where he went away to boarding school at the Realschule in Darmstadt. In 1860 he graduated with an excellent academic record with exceptional skills in mathematics, and in particular trigonometry. He attended the HoherenGewerbeschule in Darmstadt for two years to study engineering and then entered the Polytechnic of Zurich in 1862. In that year, Cantor sought his father's permission to study mathematics and was overjoyed when his father gave his consent. His studies were cut short by the death of his father in the next year. [2]
Cantor moved to the University of Berlin where he had instructors such as Weierstrass, Kummer and Kronecker. Cantor would occasionally travel to Göttingen to study. He would complete his dissertation at the University of Berlin in the area of number theory in 1867. In Berlin he was involved with the Mathematical Society and would become its president in 1864-1865.[2] Cantor accepted a position teaching at girl's school in Berlin. He joined the Schellbach Seminar for mathematics teachers while completing his habilitation degree in 1869. Cantor received an appointment at Halle and the focus of his research changed from number theory to analysis. This is because one of his colleagues challenged him to prove an open problem concerning the representation of functions as trigonometric functions, in particular sines and cosines. This was a famous problem that had been attempted by Dirchlet, Lipschitz and Riemann. Cantor solved the problem in April 1870 and his solution clearly reflected the teaching of Weierstrass.[2] In 1872 Cantor was promoted to Extraordinary Professor at Halle. He published a paper that year in which he characterized irrational numbers as convergent series of rational numbers. Cantor had started a friendship with Dedekind (they met on holiday in Switzerland) who referred to Cantor's result in famous characterization of the real numbers consisting as "Dedekind Cuts".[2] Cantor's work with series of trigonometric functions and the convergence of series led him to consider intrinsic differences among various sets of numbers. In particular this meant devising a means for comparing the size of sets that did not rely on the concept of counting. Cantor used the idea of "equinumerosity" to characterize if two set contain the same number of elements. This is a notion where sets are compared by an element by element pairing to see if all
elements in one set could be paired with all elements in a second set. If such a pairing exists we call the sets equinumerous or in modern terms we say the sets are in one-to-one correspondence.[1 p. 253] • In Cantor's own words: • 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.[1 p. 253] • Cantor was moving to a completely new concept of characterizing the infinite. The mathematicians who preceded Cantor objected to the idea that a process that considered an infinite set as being able to be "completed" or a process that referenced them as finished was not sound reasoning. Gauss once commented: • ...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...[1 p 254] • At this time (1874) Cantor was engaged to Vally Guttmann a friend of his sisters, who introduced her to Cantor because she feared he was spending to much time on his professional activities. They married on August 9, 1874 they spent their honeymoon in Interlaken in Switzerland where Cantor knew Dedekind was on vacation. Cantor spent much of his honeymoon in mathematical discussions with Dedekind.[2] • Cantor's research led him to a discovery that many different types of infinity exist. In fact there
are an infinite number of different types of infinity. Along the way to this discovery he was able to find sets of points that were eqinumerous that he did not expect. He was able to prove that the set of points in the unit interval was equinumerous with the points in the unit solid in n-dimensional space. This surprised and amazed Cantor so much he is attributed with a famous quote: • I see it, but I don't believe it! [2] • In 1881 Heine died leaving open a very important position at Halle. Cantor was asked to recommend a replacement for Heine. Cantor asked his good friend Dedekind to fill the place and he was turned down. Weber and Mertens, his second and third choices also turned him down. Cantor realized that his work was not widely accepted and people (even his best friend) did not want there work associated with him. • Cantor states clearly the opposition to his ideas: • ...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. [2] • Cantor became very depressed. In 1884 he had his first recorded attack of depression that he recovered from after a few weeks. His experience made him lose confidence and fear the treatment he was subject to. The treatment for mental health disorders at this time was confinement in a sanatoria which was very unpleasant.
Cantor began to work on the Continuum Hypothesis, but was not able to make much progress. The Continuum Hypothesis was a theory that stated that the cardinality of the real numbers was next in order after the natural numbers. The inability to resolve this worsened his mental state. It was improved and his depression kept in check with his family and personal life. In 1886 he bought a new house and his wife gave birth to the last of his six children. Cantor published a strange paper in 1894 that showed Golbach's conjecture was true for all even numbers up to 1000. This had already been done for all numbers up to 10000 forty years before. This gives more evidence of his state of mind and wanting to be accepted again by the mathematical community. Cantor suffered from periods of depression from 1899 on, following the death of his youngest son. He was in and out of the sanatoria several times between 1902 and 1908. He had to take leave from his teaching responsibilities during many winter semesters. When Cantor suffered from periods of depression he turned away from mathematics and toward his family, philosophy and Shakespeare. He proporteda theory that Francis Bacon had wrote Shakespeare's plays. In 1911 Cantor was invited to the University of St. Andrews as a distinguished foreign scholar. Cantor had hoped to meet with Bertrand Russell who had just published Principia Mathematica to discuss his theory on sets, but news his son was ill made him return to Germany. He retired in 1913 and spent his final years ill and starving because of the small amount of food available to German citizens due to the war. A major celebration of his 70th birthday was planned at Halle but was canceled due to the war. In 1917 he entered the sanitarium for the last time were he would continually write his wife asking to go home. He died mentally ill, scared, starving and penniless of a heart attack.[2]
Hilbert said the following of Cantor's work: • ...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.[2] • [1] Dunham, William. Journey Through Genius The Great Theorems of Mathematics. New York: John Wiley & Sons 1990. • [2] Web site: MacTutor History. http://turnbull.mcs.st-and.ac.uk/~history(9/2000). • Cantor's work begins with his notion of "equinumerosity". In modern times we would say that two sets are "equinumerous" if there exists a one-to-one correspondence between the two sets. A one-to-one correspondence is a function from one set to another that is both one-to-one and onto. A cardinal number is what is used to denote the class of all sets that can be put into one-to-one correspondence with each other. • That is to say a if there is a function f:A→B that is both one-to-one and onto, then we say the set A is in one-to-one correspondence with the set B.
Sets and One-to-One Correspondences An important tool the mathematicians use to compare the size of sets is called a one-to-one correspondence. This concept is a way of saying two sets are the same size without counting the numbers in them. We call two sets equivalent if they have the same number of elements. Equivalent sets can be put into one-to-one correspondence with each other by showing how all the elements of one set exactly match with all the elements of another set. You can represent different one-to-one correspondences by drawing arrows between the sets. January Larry January Larry June Curly June Curly July Moe July Moe Each of the illustrations above shows a one-to-one correspondence between the sets {January, June , July} and {Larry, Curly, Moe}. These two sets would be considered equivalent but not equal.
Can the set {January, June, July} be put into one-to-one correspondence with the set {Red, Green, Blue, Orange}? Sets that are equal have exactly the same elements in them. Sets that are equivalent need only have the same number of elements in them. The sets {January, June, July} and {Red, Green, Blue} are equivalent but not equal. The sets {January, June, July} and {July, June, January} are both equal and equivalent. NO ! Often times it is useful to draw or picture the one-to-one correspondence in row format. For example a one-to-one correspondence between the sets {January, June, July} and {Red, Green, Blue} can be illustrated as in the figures below: As a Diagram In Function form f(January) = Red f(June) = Green f(July) = Blue
Reference Sets A reference set for a number is any set that has that number of elements in it. For example all of the sets listed below are reference sets for the number 3. {red, green, blue} {,,} {Larry, Curly, Moe} Cardinal Numbers A cardinal number is the collection of all sets that are equivalent to a particular reference set. Below is a table of cardinal numbers and a reference set for each one. The symbol 0 (pronounced aleph null it is the first letter of the Hebrew alphabet) is used to represent the number of elements in a set equivalent to N the natural numbers. A set with cardinality 0 is called denumerable or countably infinite. If a number has a finite reference set that number is called a finite cardinal. If a reference set for a cardinal number is infinite that number is called a transfinite cardinal.
A set is denumerable if there is a one-to-one correspondence with the natural numbers N. This implies a set S is denumerable if there is a function f:N→S that is one-to-one and onto. We call any function f:N→T a sequence. and use the following notation. x1=f(1) x2=f(2) x3=f(3) x4=f(4) x5=f(5) xn=f(n) or in in the case of a denumerable set: S={x1,x2,x3,x4,...} The phrase Cantor used was to say that denumerable set could be "exhausted" by a sequence. The table below shows how a sequence could exhaust various sets.
Sets in One-to-One Correspondence with N We will see there are many different sets that are in one-to-one correspondnece with the set of natural numbers N. One of the most famous examples is the set of integers Z. This seems very counter intuitive since there seems like there should be "twice" as many integers as natural numbers, but this is not the case. There are just as many integers as natural numbers! We show the one-to-one correspondence below by "interweaving" the positive and negative numbers. 34 ,35 46 86 17 43 ,-43 For a negative number in Z look just ahead of it, find the positive and multiply by 2. For a positive number in Z multiply by 2. For an even number in N divide by 2. For an odd number in N divide look at the even ahead divide by 2 and the next one will be negative.
Putting a plus (+) or a minus sign above a set means that you are looking at just the positive (+) or negative (-) numbers in that set. The number 0 is not considered to be either positive or negative. • = Positive Integers = {1,2,3,4, ….} (i.e. this is another name for ) • = Negative integers = {-1,-2,-3,-4,…} • = Positive Rational Numbers = Positive Fractions • = Negative Rational Numbers = Negative Fractions • One of the surprising results we will talk about is that there are just as many positive Integers as there are positive Rational numbers (i.e. the sets and are equivalent). • What is difficult here is to show how the sets correspond. For some sets the function that describes the sequence xn can be very complicated to write down or does not exist in terms of products, quotients, powers or other combinations of known functions. It can be described (such as the case with prime numbers) by shown the order that the numbers would appear in the sequence. • This will enable us to show that the sets and are also equivalent by "interweaving" the positives and negative like we did for and .
We line up the fractions with the same numerators going across and the same denominators going down. We then serpentine back and forth skipping over any unreduced fraction.
Rational Numbers and Decimals • The decimal expansion of any rational number (integer divided by a natural number will have a decimal that will look one of two ways. • 1. The decimal will terminate: • ¼ = .25 ½ = .5 ¾ = .75 ⅜ = .375 • 2. The decimal will repeat: The reason that fractions must terminate or repeat is because there are only a finite number of remainders you can get when you do long division. If any remainder is zero the decimal terminates, if any remainder repeats itself so will the digits generated by long division. The decimal for a fraction must terminate or repeat in no more digits than what the denominator is. ¾ terminates after 2 digits < 4 repeates after 2 digits < 33
We have seen any fraction can be turned into either a terminating or repeating decimal by long division. It is also the case that a terminating or repeating decimal can be turned back into a fraction. • Decimals that Terminate • If the decimal terminates look at the place value of the last digit in the number. That place value becomes your denominator and the digits you see are your numerator. .357 2.9063 1000's place 10,000's place • Decimals that Repeat • In order to change a decimal that repeats back into a fraction there are a few terms associated with a repeating decimal that we refer to. • The repeat block are the digits that are repeated. The repeat block is marked with a line over the repeated digits. The size of the repeat block are the number of digits making up the repeat block. • The delay are the digits in the decimal after the decimal point that do not repeat.
The digits 217 make up the repeat block for this number. The size of the repeat block is 3. There is no delay for this number. 1st Repeat Block 2nd Repeat Block The digits 82 make up the repeat block for this number. The size of the repeat block is 2. The digits 346 are the delay for this number. 1st Repeat Block 2nd Repeat Block The procedure for changing a repeating decimal back into a fraction makes use of algebra and the fact that multiplication by powers of 10 "shift" the decimal point. The idea is to make the repeating parts "line-up" so they cancel out when you subtract. 1. Set decimal equal to x. 2. Multiply x by the power of 10 to move decimal point to the end of the first repeat block. 3. Multiply x by the power of 10 to move decimal point to the beginning of the first repeat block. 4. Subtract (the repeating parts should cancel) and solve for x.
Change .37 to a fraction. This means that the fraction for the decimal number .37 is : This was a bit of a special case since there was no delay for the number you can always subtract the original x. When the decimal delays (i.e. has digits before the repeat block) you will need to multiply by another power of 10 to get the decimal point to the beginning of the first repeat block.
Change .26194 to a fraction. Change .7495 to a fraction.
Irrational Numbers • There are numbers whose decimal does not repeat in a block or terminate. We call these numbers irrational. Any irrational number can not be converted into a fraction. Here are some examples of irrational numbers. • - Pronounced Pie and written Pi is one of the most famous examples. • .47447444744447… - The repeating block keeps growing. • - Taking square roots of numbers that are not perfect squares.
A number is called algebraic if it is the root of a non-zero polynomial with integer coefficients. Any rational number is algebraic since if r is ration then r=m/n and is the root of the linear non-zero polynomial nx-m=0. Almost all numbers you know are algebraic. For example the square root of a prime p is algebraic because it is the root of the non-zero polynomial x2-p=0. It can be proven that a root, sum, product, or quotient of algebraic numbers is another algebraic number. A number that is not algebraic is called transcendental. Very few transcendental numbers are known for example and e are transcendental. The set of algebraic numbers is denumerable Proof A polynomial of degree n has at most n roots. For each natural number n we will count the number of roots a polynomial of degree n could have whose coefficients do not exceed n in absolute value. Let A1={a : a is a root of a polynomial of degree 1 whose coefficients have absolute value less than 1} Each polynomial is of the form c1x+c0=0 where c1,c0 are in {-1,0,1}. There are 3 choices for c1 and 3 choices for c2 giving 9 total polynomials. Since each polynomial has at most 1 root, thus there are at most 19=9 possible roots (there are actually less since some of the polynomials have no roots like 1=0 and some have the same root such as x-1=0 and -x+1=0). The number of elements in A1 is finite. Let A1={a11,a12,...,a1m1}.
Let A2={a : a is a root of a polynomial of degree 2 whose coefficients have absolut value less than 2 and a is not in A1} Each polynomial is of the form c2x2+c1x+c0=0 where c2,c1,c0 are in {-2,-1,0,1,2}. There are 5 choices for c1,c2 and c3 giving 125 total polynomials. Since each polynomial has at most 2 roots, thus there are at most 2125=250 possible roots (there are actually less). The number in A2 is finite. Let A2={a21,a22,...,a2m2}. In general, Let An={a : a is a root of a polynomial of degree n whose coefficients have absolute value less than n and a is not in A1 or A2 or...or An-1} Each polynomial is of the form cnxn+cn-1xn-1+...+c0=0 where cn,cn-1,...,c0 are in {-n,-(n-1),...,-1,0,1,...,n-1,n}. There are 2n+1 choices for cn,cn-1,...,c0 giving (2n+1)(n+1) polynomials. Each polynomial has at most n roots, thus there are at most n(2n+1)(n+1) possible roots (there are actually less). The number in An is finite. Let An={an1,an2,...,anmn} All of the elements in all of the sets An can be exhausted by the following sequence. If a number x is algebraic then x is a root of cnxn+cn-1xn-1+...+c0=0. Let M=Max{n,|cn|,|cn-1|,...,|c0|} then x is in AM or AM-1 or...or A1. This x = aij = xk for some k. The sequence (xk) exhausts the algebraic numbers. QED
A Different Infinity The closed interval [0,1] (i.e. every number that can be written as a decimal between 0 and 1) is called the unit interval. The unit interval [0,1] is not equivalent to N. In other words the infinity represented by the natural numbers is a different type of infinity that is represented by the unit interval [0,1]. The reasoning for this is very ingenious. Suppose the unit interval [0,1] has a one-to-one correspondence with N. We don't know what numbers in [0,1] correspond to N so we call them x1, x2, x3,…. If we knew the numbers arrange them this way. We can always create a number that is not in this list by changing the digit in red to a 4 if it is not a 4 and to a 5 if it is a 4. In this case the new number would be: 0.4544…
Closed Intervals If we start with any closed interval [a,b] we can show it has just as many points (i.e. can be put into one-to-one correspondence) with the unit interval [0,1]. This can be visualized as making the endpoints match up from a common point. For example if we want to show the closed interval [3,7] is equivalent to the closed unit interval [0,1] we show how the points correspond. To locate the points that correspond to x and y on the other interval we first locate point P We then draw a straight line connecting P and x or y. Where that line hits the other segment is the corresponding point. P 0 1 y 3 7 x This could also be calculated directly if you knew one of the numbers by using a proportion. P 0 1 y 6 3 7 3 4
Equivalent Shapes Just like two segments of different size represent the same infinity so can different shapes. For example the are just as many points on the small circle below as there are on the large triangle. To find the points that correspond to the orange, green and blue points draw a line from the black point. Where it hits the other shape is the corresponding point. It is also a well know fact that the unit segment [0,1] is equivalent to the unit square [0,1] [0,1]. This is an amazing fact because these two shapes are of different dimension. A line segment is 1 dimensional where the unit square is two dimensional. These two sets are equivalent!
Equivalence of [0,1] and [0,1] [0,1] The points on the unit interval corresponds to the point on the unit square by "interweaving" their corresponding decimals. That is if the ordered pair (x,y) is on the unit square with x and y have the following decimal expansions: x = 0.x1x2x3x4x5… and y = 0.y1y2y3y4y5… The corresponding point on the unit segment is given by: 0.x1y1x2y2x3y3x4y4x5y5… ↔ (0.x1x2x3x4x5… , 0.y1y2y3y4y5…) With this scheme decimals that terminate have the decimal digits after the last one all zero. (0.218, 0.46) 0.24168
Power Sets The power set of a set A (sometimes denoted P(A) or S) is the set of all subsets of the set A. The table below shows some sets with different cardinality along with their power sets and the number in their power set. What you want to notice is that the cardinality of a set is always smaller than the cardinality of its power set. Taking the power set gets you a bigger cardinal number than what you started with.
Cantor's Theorem • The set operation of cross product () does not change the cardinality of a set for sets that are infinite. Here are some examples: • [0,1] [0,1] has the same cardinality as [0,1] • has the same cardinality as • has the same cardinality as • Cantor's Theorem states that the cardinality of the set of subsets of a set A (Power Set of A) is always greater than A (regardless if A is infinite or not). (the power set of A) • Since the set of subsets of a set is another set we can create all of its subsets, thus creating a lager number than what we started with. This leads to the following conclusion: • There are an infinite number of different sizes of infinity!
The way Cantor argued that the cardinality of the power set is always larger is very ingenious. He started with the natural numbers and showed why they could not have the same cardinality as their power set. He assumed if there were a one-to-one correspondence line them up next to each other. If a number is not in the set it is paired with color it red. Some of the numbers will be red and some black. Take all the numbers that are red and call that set R. The set R is paired with a number x. The question Cantor asked himself is if x is red or black? If x is red that means that x is not in R, but the elements of R are red numbers which means that x is in R. This is illogical so x must not be red. If x is black that means x is in R, but the elements of R are all the red numbers which means x is not in R. This is illogical so x must not be black. Since x must be one of the two colors it must be impossible to do this.
Cardinal numbers can be ordered according to the following definition: A cardinal number c1 is less than a cardinal number c2 if A is a reference set for c1 and B is a reference set for c2, then A can be put in one-to-one correspondence with a proper subset of B and the set A can not be put into one-to-one correspondence with the set B. To denote this we write c1<c2. Here is a more formal argument for why the cardinality of the P(A) is always greater than the cardinality of the set A. Proof: Let A be a non-empty set. (The case of the empty set is trivial.) Define f:A→P(A) by f(x)={x}. This function is a one-to-one correspondence with a proper subset of P(A) since the empty set will not be paired with anything. To show the cardinality is less we must show that A can not be put into one-to-one correspondence with P(A). Use contradiction and assume A can be put into one-to-one correspondence with P(A).
This implies there exists a function f:A→P(A) that is a one-to-one correspondence. For each x in A there is a corresponding f(x) which is a subset of A. The element x is either in f(x) or it is not. Define the following set S. S = { x in A : x is not in the corresponding set f(x)} The set S is a subset of A. That implies there is an element y in A that that corresponds to it (i.e. f(y) = S). The element y is either in S or it is not in S. Consider each of the following cases. Case 1: y is in S By the definition of S these are all elements x such that x is not in f(x). Since y is in S and f(y)=S then y is in f(y), therefore y is not in S. This implies Case 1 is impossible. Case 2: y is not in S Because f(y) = S then y is not in f(y). This means y is one of the members x such that x is not in f(x). By the definition of the set S this means y is in S. This implies Case 2 is impossible. Because both cases are impossible this is a contradiction. It must be true no such function (one-to-one correspondence) exists. QED
This gives us a way to create an infinite sequence of set with different cardinality. Let 0 = cardinality of . Let 1 = cardinality of Let 2 = cardinality of . Let 3 = cardinality of . . . . Then 0<1<2<3<4<.... All the above sets are transfinite cardinal numbers and they are all different.
