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The Mathematics of Infinity

- Georg Cantors Theory of Sets

To see the world in a grain of sand. And heaven

in a wildflower Hold infinity in the palm of

your hand, And eternity in an hour. --Willia

m Blake

From the paradise created for us by Cantor no

one will drive us out. David Hilbert

Georg Cantor

An Introduction to Sets

- A 2, 3, 5, 7, 11, 13, 17, 19
- N 1, 2, 3, 4, N is the set of Natural

numbers or the set of counting numbers - Z -3, -2, -1, 0, 1, 2, 3, Z is the

set of Integers - Q a/b a, b ? Z, b ? 0 Q is the set of

Rational Numbers - A ? N, N ? Z, and Z ? Q

Equal and Equivalent Sets

- Two sets are Equal if they contain exactly the

same elements.A 2,4,6, B 4,2,6 so

A B - Two sets are Equivalent if there exists a 1-1

correspondence between their elements - A and B above are certainly equivalent, but C

1,2,3 is also equivalent to each of them.

What does Infinity mean?

- The known is finite, the unknown infinite

intellectually we stand on an island in the midst

of an illimitable ocean of inexplicability. Our

business in every generation is to reclaim a

little more land. - --Thomas H Huxley

Infinite Sets

- A set A is finite if it is empty or if there is

a natural number n such that A is equivalent

to 1,2,3, ,n - Thus a set is infinite if its elements cannot be

counted completely. - An alternate but equivalent definitionA set is

infinite if it is equivalent to a proper subset

of itself.

Question

- Are all infinite sets equivalent?
- Or, in other words, can two infinite sets always

be put in 1-1 correspondence? - Consider the Natural numbers and the even

positive integers

A 1-1 correspondence is shown so these are

equivalent sets.

1, 2, 3, 4, , n, ...

2, 4, 6, 8, , 2n, ...

N is equivalent to Z

1, 2, 3, 4, 5, 6, 7, 8, 9,

Here is a 1-1 correspondence between the Natural

numbers and the Integers.

0, 1, -1, 2, -2, 3, -3, 4, -4,

Every even Natural number, n, is paired with n/2

Every odd Natural number, n, is paired with (1

n)/2.

It is even possible to show a 1-1 correspondence

between the Natural numbers and Q, the rational

numbers.

So N is equivalent to Q as well.

(No Transcript)

More about Rational Numbers

- By definition, any number which may be expressed

as a fraction is a Rational number. - It may also be shown that any decimal which

terminates or repeats may be made into a fraction

and is thus a Rational number.

The Irrationals and the Reals

- The Irrational numbers comprise all decimal

numbers which are not rational, i.e., cannot be

made into fractions. - The Irrational numbers together with the Rational

numbers include all possible decimals and form

the Real numbers R. - The Real numbers may be shown to be in 1-1

correspondence with the points on a number line.

Because of this they are sometimes called the

Continuum.

An Answer and More Questions

- N is not equivalent to R.
- Proof by contradictionAssume N is equivalent

to R. Then there must be a 1-1 correspondence

between them. - This correspondence will pair each Natural number

with a Real number. - We will show that this assumption leads to a

contradiction by constructing a Real number which

has not been included in the correspondence.

A possible correspondence

- N 0,1
- 1 0 . 3 0 1 2 5 9 4
- 2 0 . 1 6 6 5 2 1 8
- 3 0 . 4 1 1 2 1 0 7
- 4 0 . 2 0 5 0 9 6 3
- . .
- . .
- . .

Constructing a new number(not in the

correspondence)

N 0,1 1 0 .(3)0 1 2 5 9 4 2 0 . 1(6)6 5

2 1 8 3 0 . 4 1(1)2 1 0 7 4 0 . 2 0 5(0)9

6 3 . . . . . .

Construct a number w which will consist of

digits di where the ith digit will be either a 1

(if the ith digit in row i is not a 1) or a 2

(if the ith digit in row i is a 1)

Constructing a new number(not in the

correspondence)

N 0,1 1 0 .(3)0 1 2 5 9 4 2 0 . 1(6)6 5

2 1 8 3 0 . 4 1(1)2 1 0 7 4 0 . 2 0 5(0)9

6 3 . . . . . .

So, for this correspondence, the number w,

would start w 0.1121......

A Contradiction!!

- So w is different from every Real number in the

original correspondence. - Since w is not part of the proposed

correspondence or any other possible

correspondence, no correspondence is possible. - Thus, N is not equivalent to R.
- This means that there are at least two sizes or

classes of infinite sets.

So now, we have at least two sizes of infinity

- Sets equivalent to N, and
- Sets equivalent to R.
- We say that sets equivalent to N have cardinality

?0 (read aleph null). - The collection of all sets equivalent to R are

said to have cardinality c where c stands for

the cardinality of the continuum. - Are there others?

Sizes of Infinity

- Just as the cardinal number 3 is used to describe

the size of any set equivalent to a, b, c, - ?0 is used to describe the size of any infinite

set equivalent to N. - And c is used to describe the size of any

infinite set equivalent to R. - If there are other sizes of infinite sets, then

we need other numbers to describe the cardinality

of these sets.

I could be bounded in a nutshell, and count

myself a king of infinite space.

--Shakespeare

- It is also reasonable to assume that if these

sizes are truly different, there must be some

sort of ordering of them. - e.g. ?0 lt c.
- If we use the symbols ?1, ?2, to represent the

cardinality of these sets we might assume that ?0

lt ?1 lt ?2 lt

When is a set bigger than another?

- With finite (countable) sets, it is easy to

determine if the cardinality of one set is larger

than the cardinality of another. - For infinite sets we will use the definitionIf

A and B are sets and A and B represent the

cardinality of sets A and B respectively, then we

say A gt B if1. there exists a 1-1

correspondence between all of set B and a proper

subset of A. and2. there does not exist a 1-1

correspondence between B and all of A.

Two Questions

- Are there a finite number(perhaps only the two we

have seen) or an infinite number of these sizes

of infinity? - Where does c, the cardinality of R fit into

this scheme? We know it is larger than ?0, but

is it larger than ?1, etc.?

Power Sets

- The Power Set of set A, denoted by P(A) is the

set of all the subsets of A. i.e. P(A)

X X ? A - It is fairly obvious that for all finite sets A,

P(A) gt A . - For example If A p,q,r then the Power

Set of A, P(A) p,q,r,p,q,p,r,q,

r,p,q,r,? - and P(A) 8 gt A 3.

Cantors Theorem

- Cantor proved that for infinite sets, also, the

set of all subsets of a set is larger than the

original set. - Let A be an infinite set with cardinality A
- Let S be the set of all the subsets of set A.
- Cantor showed that A lt S .

I can see it, but I dont believe it. Georg

Cantor

Let A be any set with cardinality A. Let S

be the set of all subsets of A and have

cardinality S.

- To prove that A lt S we must show that
- A can be put in 1-1 correspondence with a proper

subset of S. - A cannot be put in 1-1 correspondence with all of

S. - The first step is easy since A can be put in a

1-1 correspondence with all the single element

subsets of A. i.e. if x ? A then x ? x ? S.

Now assume that there is a 1-1 correspondence

between A and S. i.e. every element of A is

paired with a subset of A found in S.

Now with this matching, it makes sense to ask if,

for each pairing, the element from A is

contained in the subset from S with which it is

matched. If this is not the case then put all

such elements of A in a set and call it

W. Clearly W is a subset of A. So W must be

paired with some element, say z of A. Now

either z is in W or it is not. Lets look at

both possibilities.

In an Indirect Proof, we assume the negation of

what we wish to prove and look for a

contradiction. If such is found, it means the

assumption was wrong.

If z is in W, then z is contained within the

set with which it is matched and by the

definition of W cannot be in W, which is a

contradiction. If z is not in W, then it is

not contained within the set with which it is

matched and again by the definition of W, z

must be contained in W. This, also gives a

contradiction. Thus we are forced to conclude

that no 1-1 correspondence was possible between

A and S.

There is no smallest among the small and no

largest among the large but always something

still smaller and something still

larger. --Anaxagoras

So we have answered one of the questions. This

result means that there are an infinite number of

sizes of infinity.Now for the second

question. Where does c, the cardinality of the

Real numbers fit into this limitless progression

of larger and larger infinite sets? ?0 , ?1 , ?2

, ... Cantor showed that c is the size of the

set of all the subsets of the Natural numbers and

he guessed that c ?1 , i.e. that c was the

next smallest size.

The Continuum Hypothesis

- But Cantor was never able to prove that there was

no size of infinity between ?0 and c. - Thus was born the Continuum HypothesisIf a set

has size ?0 , then the set of all its subsets has

size ?1 .And more generally, if a set has size

?n , then the set of all its subsets has size

?n1. - Many tried to prove this hypothesis, but none

succeeded.

Contradictions??

- In 1940, Kurt Gödel proved that the Continuum

Hypothesis was consistent with the axioms of set

theory, i.e. that it will not lead to any

contradictions. - In 1963, Paul Cohen proved that assuming the

Continuum Hypothesis false will not lead to any

contradictions within set theory.

A new Axiom

- This means the Continuum Hypothesis can neither

be proven true or false within set theory. So we

may treat it as a separate axiom. Thus, much

like Euclids 5th postulate, we get one kind of

set theory if we assume it true and another if we

assume it false.

Thank You

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