m370 Notes Cantor1 1

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Georg Cantor

Born: March 3, 1845

Died: January 6, 1918


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Georg Cantor lived at the end of the 19th century and early 20th century. This is a time period inboth mathematics and the world that is referred to as "the age of abstraction". Ideas andphilosophies were changing from the concrete to the abstract. This could be seen in many fieldsalong with mathematics. In economics abstract notions of different types of economies such ascommunism 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" thatcould reproduce what the human eye could see to having an abstract eye. For example theworks 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 asthe 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 aProtestant who had converted from Judaism and his mother Roman Catholic. Georg was raisedas 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 inWiesbaden. Later the family moved to Frankfurt where he went away to boarding school at theRealschule in Darmstadt. In 1860 he graduated with an excellent academic record withexceptional skills in mathematics, and in particular trigonometry. He attended the HoherenGewerbeschule in Darmstadt for two years to study engineering and then entered thePolytechnic of Zurich in 1862. In that year, Cantor sought his father's permission to studymathematics 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]


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Cantor moved to the University of Berlin where he had instructors such as Weierstrass, Kummer

and Kronecker. Cantor would occasionally travel to Gttingen 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 Seminarfor 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 "DedekindCuts".[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


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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. Themathematicians 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


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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 choicesalso 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 thenature 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.


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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 birthto 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 proported a 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 Mathematicato 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]


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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 numberis what is used to

denote the class of all sets that can be put into one-to-one correspondence witheach other.

That is to say a if there is a functionf:AB that is both one-to-one and onto, then we

say the setA is in one-to-one correspondence with the setB.
http://turnbull.mcs.st-and.ac.uk/~history/http://turnbull.mcs.st-and.ac.uk/~history/http://turnbull.mcs.st-and.ac.uk/~history/http://turnbull.mcs.st-and.ac.uk/~history/http://turnbull.mcs.st-and.ac.uk/~history/


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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 havethe 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

June

July

Larry

Curly

Moe

January

June

July

Larry

Curly

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.


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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 rowformat. 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:

{January, June, July}

{Red, Green, Blue}

As a Diagram In Function form

f(January) = Red

f(June) = Green

f(July) = Blue


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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 number3.

{red, green, blue} {,,} {Larry, Curly, Moe}

Cardinal

Number

Reference Set

0

1 {a}

2 {one, won}

3 {,,}

4 {,,,}

0 N= {1,2,3,}

Cardinal Numbers

A cardinal numberis the collection of all sets

that are equivalent to a particular reference set.

Below is a table of cardinal numbers and areference 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 withcardinality 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.


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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 functionf:NS that is one-to-one and onto.

We call any functionf:NT 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.

Numbers Sequence Set

Even Numbers xn = 2n {2,4,6,8,10,}

Powers of 2 xn = 2n {2,4,8,16,32,}

Perfect squares xn = n2 {1,4,9,16,25,}

Integers {0,1,-1,2,-2,3,-3,}

Primes No known formula {2,3,5,7,11,13,17,}

4

1211

n

n

n

x


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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.

N = {1, 2, 3, 4, 5, 6, 7, , x, , y, , 87, }

Z = {0 1, -1, 2, -2, 3, -3, , -17, , 23, , z, }

17

34

For a negative number in Z look just ahead of it, find the positive and multiply by 2.

,35 46 86

For a positive number in Z multiply by 2.

43,-43

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.


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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 manypositive 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 sequencexn can be very complicated to write down or

does not exist in terms of products, quotients, powers or other combinations ofknown 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 .


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1

1

2

1

1

2

1

3

1

4

1

5

1

6

1

7

2

2

3

1

4

1

5

1

6

1

7

1

3

2

4

2

5

2

6

2

7

2

2

3

2

4

3

3

4

3

5

3

6

3

7

3

3

4

4

4

5

4

6

4

7

4

2

5

2

6

3

5

4

5

5

5

6

5

7

5

3

6

4

6

5

6

6

6

7

6

2

7

3

7

4

7

5

7

6

7

7

7

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.

= {

+ = {

1,

1,

2,

,

3,

2,

4,

3,

5,

,

6,

,

}

}

= {

= {

1,

0,

2,

1,

3,

1,

4,

,

5,

,

6,

2,

}

}


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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:

= .25 = .5 = .75 = .375

2. The decimal will repeat:

3.333333.

3

1 12.1212121212.

33

4 285714.142857142857.

7

2

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 decimalterminates, 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.

.75

4 3.00

2 8

20

20

0

.12

33 4.00

3 3

70

66

4

terminates

after 2 digits < 4

repeates after

2 digits < 33

33

4


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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

1000's place1000

357357. 2.9063

10,000's place10000

290639063.2

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.


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217217217.4217.4

1st Repeat Block

2nd Repeat Block

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.

346828282.082346.0

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.

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 yousubtract.

1. Set decimal equal tox.

2. Multiplyx by the power of 10 to move decimal point to the end of the first repeat block.

3. Multiplyx 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 forx.


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Change .37 to a fraction.

Setx =.37 writing out two repeat blocks. x = .3737

Multiply by 100. (You want to move it 2 decimal places.) 100x = 37.37

Subtractx. (The decimal point forx is already at the beginning

of the first repeat block in this case.)

-x = -.3737

Solve forx. 99x = 37

99

37x

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 originalx. 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.

99

37


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Multiply by 100000. (Move it 5 decimal places.) 100000x = 26194.194

Multiply by 100. (Move to beginning of repeat block.) 100x = 26.194

Subtract 100000x-100x. 99900x = 26168

Solve forx.99900

26168x



Multiply by 10000. (Move it 4 decimal places.) 10000x = 7495.5

Multiply by 1000. (Move to beginning of repeat block.) 1000x = 749.5

Subtract 10000x-1000x. 9000x = 6746

Solve forx. 9000

6746

x


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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.5


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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 primep is algebraic because it is the root of the non-zero polynomialx2-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 transcendentalnumbers 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 numbern 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 forc1and 3 choices forc2 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 asx-1=0 and -x+1=0). The number

of elements in A1 is finite.

Let A1={a11,a12,...,a1m1}.


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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 forc1,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 inA2 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-1x

n-1+...+c0=0 where cn,cn-1,...,c0 are in {-n,-(n-1),...,-

1,0,1,...,n-1,n}. There are 2n+1 choices forcn,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 numberx is algebraic thenx is a root ofcnxn+cn-1x

n-1+...+c0=0. Let M=Max{n,|cn|,|cn-1|,...,|c0|}

thenx is in AM or AM-1 or...or A1. Thisx = aij =xk for some k. The sequence (xk) exhausts the

algebraic numbers.

QED

,,,,,,,,,,,,,,,,4321212222111211 21

xxxxaaaaaaaaan

nmnnmm


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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 notequivalent 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 themx1,x2,x3,.

N = {1, 2, 3, 4, 5, }

[0,1] = {x1, x2, x3, x4, x5, }

N [0,1]

1 0.132786

2 0.345802

3 0.035211

4 0.250000

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


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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. Forexample 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.

0 1

3 7x

y

To locate the points that correspond tox and

y on the other interval we first locate point P

We then draw a straight line connecting Pandx ory. Where that line hits the other

segment is the corresponding point.

P

0 1

3 7

y

P This could also be calculated directly if you

knew one of the numbers by using a

proportion.

6

4

3

4

3

4

3

1

y

y


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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 unitsquare [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.

-0.5 0.5 1 1.5 2

-0.5

0.5

1

1.5

2

-0.5 0 0.5 1 1.5 2

These two sets are equivalent!


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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 withx andy 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.

[0,1] [0,1] [0,1]

0.24168 (0.218, 0.46)

52.0 2.0,5.0

76314.0 73.0,614.0

2038.0 230.0,0238.0

502.0 5.0,2.0


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Power Sets

The power set of a setA (sometimes denotedP(A) orS) is the set of all subsets of

the setA. The table below shows some sets with different cardinality along with

their power sets and the number in their power set.Cardinal

NumberReference

Set

Subsets (Power Set) Cardinalityof Power

Set

0 1

1 {} ,{} 2

2 {,} ,{},{},{,} 4

3 {,,} ,{},{},{},{,},{,},{,},{,,} 8

4 {,,,} ,{},{},{},{},{,},{,},{,},{,},

{,},{,},{,,},{,,},{,,},

{,,},{,,,}

16

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.


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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 isinfinite 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 followingconclusion:

There are an inf ini te number of different sizes of inf ini ty!


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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 setR is paired with a numberx.

Number Subset

1 {2, 3, 5}

2 E = Even Nos

3 D = Odd Nos

4 {8,9,10,}

5 {5,10,15,}

x R

The question Cantor asked himself is ifx is red or black?

Ifx is red that means thatx is not inR, but the elements ofR are red numbers

which means thatx is inR. This is illogical sox must not be red.

Ifx is black that meansx is inR, but the elements ofR are all the red numbers

which meansx is not inR. This is illogical sox must not be black.

Sincex must be one of the two colors it must be impossible to do this.


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Cardinal numbers can be ordered according to the following definition:

A cardinal numberc1 is less than a cardinal numberc2 ifA is a reference set forc1

andB is a reference set forc2, thenA can be put in one-to-one correspondence

with a proper subset ofB and the setA can not be put into one-to-onecorrespondence with the setB. To denote this we write c1


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This implies there exists a functionf:AP(A) that is a one-to-one correspondence.

For eachx inA there is a correspondingf(x) which is a subset ofA. The elementx is

either inf(x) or it is not. Define the following set S.

S = {x inA :x is not in the corresponding setf(x)}

The set S is a subset ofA. That implies there is an elementy inA that that

corresponds to it (i.e.f(y) = S). The elementy is either in S or it is not in S. Consider

each of the following cases.

Case 1:y is in SBy the definition of S these are all elementsx such thatx is not inf(x). Sincey is in S

andf(y)=S theny is inf(y), thereforey is not in S. This implies Case 1 is impossible.

Case 2:y is not in S

Becausef(y) = S theny is not inf(y). This meansy is one of the membersx such that

x is not inf(x). By the definition of the set S this meansy is in S. This implies Case 2is impossible.

Because both cases are impossible this is a contradiction. It must be true no such

function (one-to-one correspondence) exists.

QED


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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

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