∞∞Exploring Infinity ∞Exploring Infinity ∞

By Christopher ImmBy Christopher ImmJohnson County Community CollegeJohnson County Community College

1. The “Z mod n”, group, . This is the set formed by the remainders of the integers divided by a positive integer n.

2. The set of all Natural numbers, , {1,2,3,4,…}.

3. The set of all positive rational numbers, , the ratio of two integers in simplified form.

4. The set of real numbers,

What are the sizes of the What are the sizes of the following sets?following sets?

n

, on 0,1 .

Example 1Example 1

0,1,2,3, 1n

The “Z mod n”, , group is formed by the remainders of the integers divided by a positive integer n. This is intuitive by asking how many elements would be in the set.

Counting the number of elements in the set, the size of this is set is n.

What are the possible remainders upon division by n?

n

Example 2Example 2

What is the size of the set, , {1,2,3,4,…}?

This may be a bit harder to visualize, our first question may be, is infinity allowed to represent the size of a set?

If so, how do we represent this?

Example 3Example 3

What is the size of the set of , the positive ratio oftwo integers in simplified form?Again the answer seems to be infinity, however, this Again the answer seems to be infinity, however, this

setsetseems a bit “larger” than the one in example 2.seems a bit “larger” than the one in example 2.

Why? Because you can think of many positive rationalWhy? Because you can think of many positive rationalnumbers between the first two natural numbers…numbers between the first two natural numbers…

For example the sequence:For example the sequence:

This means there are an infinite number ofThis means there are an infinite number ofnumbers between the first two numbers of anothernumbers between the first two numbers of anotherinfinite set, the natural numbers.infinite set, the natural numbers.

1 1 1 1, , , ,

2 3 4 5

Example 4Example 4What is the size of the set of the real numbers on the interval [0,1]?

, on 0,1

The answer is infinity, again, however, the underlying question is, how can we compare these infinite sets?

Introducing Georg CantorIntroducing Georg Cantor

A German mathematician born in St. Petersburg, Russia in 1845.

Cantor introduced different Cantor introduced different “sizes” of infinity“sizes” of infinity• Cantor devised a system from the Cantor devised a system from the

Hebrew letter aleph, called Hebrew letter aleph, called aleph aleph numbersnumbers..

• These numbers were also called These numbers were also called Cardinal Cardinal numbersnumbers or or CardinalsCardinals, for short., for short.

• The “smallest” infinite set was described The “smallest” infinite set was described as Cardinal as Cardinal aleph-naught aleph-naught or or aleph-nullaleph-null. .

• It was denoted asIt was denoted as

• The set of the natural numbers, , as The set of the natural numbers, , as mentioned in Examplementioned in Example 2, have Cardinal 2, have Cardinal

0.

0.

Discerning between sizes of Discerning between sizes of infinityinfinity

• Cantor used the idea of a Cantor used the idea of a bijectionbijection between a set and the natural numbers, between a set and the natural numbers, , to describe all sets Cardinality , to describe all sets Cardinality

• A A bijectionbijection is a one to one, onto mapping is a one to one, onto mapping between two sets.between two sets.

• Sets of this type were sometimes called Sets of this type were sometimes called countably infinite countably infinite or or countablecountable..

• The positive rational numbers, , in The positive rational numbers, , in example 3, are another example example 3, are another example Cardinality Cardinality

0.

0.

There is a bijection between There is a bijection between and and

The idea of the 1-1, onto correspondence follows in the diagram below.

Cont.Cont.

If any fractions not in reduced form are If any fractions not in reduced form are eliminated and we follow the arrow, the set eliminated and we follow the arrow, the set of of is in one to one and onto is in one to one and onto correspondence with . Therefore is correspondence with . Therefore is countable.countable.

and two good examples of sets of and two good examples of sets of Cardinality Cardinality

0.

The Song The Song 0To the tune 99 bottles of beer on the wall.

♪ Aleph-null cups of coffee on the wall,

Aleph-null cups of coffee.

Take one down, pass it around,

Aleph-null cups of coffee on the wall. ♫

(And repeat…)

Larger sizes of infinity.Larger sizes of infinity.

• Cantor realized that there were sets larger than Cantor realized that there were sets larger than aleph- naught. The next Cardinal number he aleph- naught. The next Cardinal number he defined was defined was aleph-onealeph-one..

• These sets were called the These sets were called the Cardinality of the Cardinality of the continuumcontinuum represented by the real numbers, represented by the real numbers,

• It was denoted as or It was denoted as or cc, for the continuum.for the continuum.

• The set of real numbers, , or the subset on the The set of real numbers, , or the subset on the interval [0,1] as mentioned in Exampleinterval [0,1] as mentioned in Example 4, have 4, have CardinalityCardinality

• Sets of this size are referred to as Sets of this size are referred to as uncountableuncountable. .

1.

1.

Cantor’s diagonalization Cantor’s diagonalization argumentargument

5 if 6 and 5 if 5 iiiiii dpdp

Prove that the set S = is uncountable. , on 0,1

Cont.Cont.

• Cantor’s original diagonal argument Cantor’s original diagonal argument was done with a binary was done with a binary representation of the real numbers in representation of the real numbers in decimal form. Thus, the new decimal decimal form. Thus, the new decimal representation was chosen to be the representation was chosen to be the complement of each diagonal complement of each diagonal element, forming a new number not element, forming a new number not in the set. in the set.

The SongThe Song1

To the tune 99 bottles of beer on the wall.

♪Aleph-one cups of coffee on the wall,Aleph-one cups of coffee.Take infinity down, pass infinity around,Aleph-one cups of coffee on the wall. ♫

(And repeat…)

The Cantor SetThe Cantor Set

• To create the Cantor set, take the interval To create the Cantor set, take the interval [0,1] on the real number line, call this the [0,1] on the real number line, call this the initial stage or Cinitial stage or C00..

• In the first stage, CIn the first stage, C11, we remove the middle , we remove the middle third of the segment.third of the segment.

• For each additional stage continue to For each additional stage continue to remove the middle third of each segment, remove the middle third of each segment, call the nth stage Ccall the nth stage Cnn..

• The Cantor set is C where, The Cantor set is C where, lim .nn

C C

Graphical Representation of Graphical Representation of CCThe first few “stages” below, CThe first few “stages” below, C00, C, C11, C, C22::

What is the “length” of C?What is the “length” of C?

To figure this out, consider these To figure this out, consider these questions:questions:

• How many segments are taken away in How many segments are taken away in each stage?each stage?

• What is the length of each segment What is the length of each segment taken away in each stage?taken away in each stage?

• How can we represent the sum of all the How can we represent the sum of all the segments taken away in each stage?segments taken away in each stage?

• What is the limit of this sum as What is the limit of this sum as n-n->∞.

Cont.Cont.• The number of segments taken away at The number of segments taken away at

each stage can be represented by each stage can be represented by

• The length of each segment taken away at The length of each segment taken away at each stage can be represented byeach stage can be represented by

• The total length taken away at each stage The total length taken away at each stage can be represented by the series can be represented by the series

• The total length taken away is given by: The total length taken away is given by:

12 , 0.n n

1, 0.

3

n

n

1

1

2, 0.

3

kn

n kk

L n

1

1 1

2 1 2 1 2 / 3 1lim 2 1.

3 2 3 2 1 (2 / 3) 2

kkn

knk k

L

ConclusionConclusion

• Since the Cantor set is constructed Since the Cantor set is constructed by a set of length one and the sum of by a set of length one and the sum of the segments taken away is one, the segments taken away is one, the the Cantor set has a length of zeroCantor set has a length of zero..

• Length of sets are referred to as Length of sets are referred to as measuremeasure..

• Thus, the Cantor set has measure 0.Thus, the Cantor set has measure 0.

What is the Cardinality of the What is the Cardinality of the Cantor set?Cantor set?

To discover this, ask some other questions.To discover this, ask some other questions.

• What are some elements remaining in the What are some elements remaining in the Cantor set?Cantor set?

• Is there a convenient representation for Is there a convenient representation for the entire Cantor set?the entire Cantor set?

• Is there a bijection between the Cantor set Is there a bijection between the Cantor set and the natural numbers, ?and the natural numbers, ?

What are some elements What are some elements remaining in the Cantor set?remaining in the Cantor set?

• All the endpoints of the remaining intervals.All the endpoints of the remaining intervals.

• For example: 0, 1/3, 2/3, 1, 1/9, 2/9, 7/9, For example: 0, 1/3, 2/3, 1, 1/9, 2/9, 7/9, 8/9, and so on…8/9, and so on…

Is there a convenient Is there a convenient representation for the entire representation for the entire Cantor set?Cantor set?

Notice that all elements in the Cantor set Notice that all elements in the Cantor set are powers of 1/3. A unique way to are powers of 1/3. A unique way to represent all elements is to use base 3 or represent all elements is to use base 3 or the the ternaryternary representation. representation.

Exs: 0=03, 1/3=0.13, 2/3=0.23, 1=0.222…3,

1/9=0.013, 2/9=0.023,7/9=0.213,8/9=0.223,and so on…

• The answer to this question is no, by viewing The answer to this question is no, by viewing the ternary representation of the Cantor set, C.the ternary representation of the Cantor set, C.

• What about the real numbers,What about the real numbers,

• Consider the real numbers on the interval Consider the real numbers on the interval [0,1]. [0,1].

• Try to find a Try to find a surjective (onto)surjective (onto) mapping, mapping, f f , from , from C to the real numbers on [0,1].C to the real numbers on [0,1].

• To this end, represent the real numbers on To this end, represent the real numbers on [0,1] in base 2 or binary.[0,1] in base 2 or binary.

Is there a bijection between Is there a bijection between the Cantor set and the natural the Cantor set and the natural numbers, ?numbers, ?

?

Find the function Find the function f f :C->[0,1]:C->[0,1]

We can express every number in C in its ternaryWe can express every number in C in its ternary

representation only consisting of 0’s and 2’srepresentation only consisting of 0’s and 2’s

(repeating).(repeating).

For example for the first few stages:For example for the first few stages:0=03, 1=0.222…3

1/3=0.13=0.0222…3, 2/3=0.23

1/9=0.013=0.00222…3, 2/9=0.023,

7/9=0.213=0.20222…3, 8/9=0.223,

and so on…

Cont.Cont.

Similarly, we can express all reals on the Similarly, we can express all reals on the interval [0,1] in their binary representation, interval [0,1] in their binary representation, from from 0=02, to 1=0.111…2.

Thus, replacing all 2’s in the numbers in C by 1’s, creates a surjective (onto) mapping from C to the real numbers in [0,1].

Define f as follows: 1 13 2

1 1

/ 2 .k k

k kk k

f a a

ConclusionConclusion• Since, C is surjective to the reals on [0,1],Since, C is surjective to the reals on [0,1],

C must have at least the cardinality of C must have at least the cardinality of cc or or

Aleph-1. However since C is a subset of the Aleph-1. However since C is a subset of the reals on [0,1], it must be at most that reals on [0,1], it must be at most that Cardinality as well. Thus C has Cardinality of Cardinality as well. Thus C has Cardinality of cc or Aleph-1.or Aleph-1.

• It is worth noting, It is worth noting, ff is is notnot bijective (1-1). For bijective (1-1). For example, example,

• Hence, Hence, ff (7/9)= (7/9)=ff (8/9), however 7/9≠8/9. (8/9), however 7/9≠8/9.

70.20222... 0.10111... 0.11,

9f f

80.22 0.11

9f f

Summary of the Cantor SetSummary of the Cantor Set

• The Cantor set has measure 0.The Cantor set has measure 0.• The Cantor set is uncountably infinite, with The Cantor set is uncountably infinite, with

Cardinality of Cardinality of c c or Aleph-1.or Aleph-1.• The Cantor set is an example of a set The Cantor set is an example of a set

which you can take an uncountably infinite which you can take an uncountably infinite number of elements away from an number of elements away from an uncountable set and still have an uncountable set and still have an uncountably infinite set.uncountably infinite set.

• The Cantor set has the added property of The Cantor set has the added property of being closed and bounded.being closed and bounded.

Enter Waclaw SierpinskiEnter Waclaw Sierpinski

A Polish mathematician born in Warsaw, Poland in 1882.

Sierpinski’s CarpetSierpinski’s Carpet

The process:The process:

• To build Sierpinski's Carpet, S, start with a To build Sierpinski's Carpet, S, start with a square with side length 1 unit, completely square with side length 1 unit, completely shaded. (Iteration 0, or the initiator)shaded. (Iteration 0, or the initiator)

• Divide each square into nine equal Divide each square into nine equal squares and cut out the middle one. (the squares and cut out the middle one. (the generator)generator)

• Repeat this process on all shaded squares.Repeat this process on all shaded squares.

Graphical representation of Graphical representation of SS

Interactive Interactive Sierpinski'sSierpinski's Carpet Carpet

The size of S.The size of S.

• Start with an area of 1 square unit.Start with an area of 1 square unit.

• The number of squares taken away at The number of squares taken away at iteration iteration nn is is

• The size of each square taken away at The size of each square taken away at iteration iteration n n isis

• The total area taken away at iteration The total area taken away at iteration n n is is

• The total area taken away from S isThe total area taken away from S is

18 , 0.n n

2

21 1 13 93

, 0.n n n

n

1

1

8, 0.

9

kn

n kk

A n

1

1 1

8 1 8 1 8 / 9 1lim 8 1.

9 8 9 8 1 (8 / 9) 8

kkn

knk k

A

ConclusionConclusion

The area of Sierpinski’s Carpet is 0.The area of Sierpinski’s Carpet is 0.

The Menger SpongeThe Menger Sponge

The process:The process:

• To build the Menger Sponge, M, start To build the Menger Sponge, M, start with a cube edge 1 unit. (the initiator)with a cube edge 1 unit. (the initiator)

• Divide the cube into twenty-seven equal Divide the cube into twenty-seven equal cubes and cut out the middle one. (the cubes and cut out the middle one. (the generator)generator)

• Repeat this process on all remaining Repeat this process on all remaining cubes.cubes.

Graphical representation of Graphical representation of MM• Visualizing the Visualizing the

MengerMenger Sponge Sponge

The size of MThe size of M

• Start with an cube of volume 1 cubic unit.Start with an cube of volume 1 cubic unit.

• The number of cubes at iteration The number of cubes at iteration nn is is

• The volume of each cube at iteration The volume of each cube at iteration n n isis

• The total volume at iteration The total volume at iteration n n is is

• The total volume of M is The total volume of M is

20 .n

3

31 1 13 273

.n n n

20 20.

27 27

nn

n nV

20lim 0.

27

n

nM V

Extensions to higher Extensions to higher dimensionsdimensionsThe procedure then for creating an N dimensional The procedure then for creating an N dimensional

pyramid can be summarized by the following rules. pyramid can be summarized by the following rules. • Start with an N-1 dimensional cube centered at the Start with an N-1 dimensional cube centered at the

origin. origin. • Pull the midpoint of the cube (origin) into the Nth Pull the midpoint of the cube (origin) into the Nth

dimension. dimension. • Make edges from the midpoint to each vertex of Make edges from the midpoint to each vertex of

the N-1 cube. the N-1 cube. • Make faces using the midpoint and each edge of Make faces using the midpoint and each edge of

the N-1 cube. the N-1 cube. • Using these rules the 4D pyramid (hyper-pyramid) Using these rules the 4D pyramid (hyper-pyramid)

is constructed by taking a 3D cube and pulling its is constructed by taking a 3D cube and pulling its midpoint into the 4th dimension. midpoint into the 4th dimension.

A “4-D” Hyper-GasketA “4-D” Hyper-Gasket

The process can be continued The process can be continued by forming another pyramid by forming another pyramid with the hyper-cube, and so with the hyper-cube, and so on… on…