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How Big Is Infinity?Some Mathematical Surprises

Roger HouseScientific Buzz Café

French Garden Restaurant & BistroSebastopol, CA

2013 July 25Copyright © 2010 Roger House

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What we're going to talk about

Some infinities are bigger than others. Sounds weird, but soon it will make sense.

It goes on forever. There is no “biggest” infinity.

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Georg Cantor (1845-1918)

• Born in St. Petersburg, Russia.• His background: Danish, Catholic,

Jewish, Lutheran.• At the age of 11 Cantor and his family

moved to Germany where he stayed for the rest of his life.

• His entire career was at the Univ. of Halle.

• He invented set theory, the basis of math.

• All the ideas we discuss tonight are his.

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Finite Sets

• Infinite sets are too hard to think about, so let's start with finite sets.

• But, to make it interesting, imagine we don't know how to count.

• This is not as artificial as it might seem.• Charles Seife says this in Zero: The

Biography of a Dangerous Idea:• “The Siriona Indians of Bolivia and the

Brazilian Yanoama people don't have words for anything larger than three; instead, these two tribes use the words for 'many' or 'much.'”

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Counting Sheep

• Joe has a herd of sheep which he lets out of a pen each morning into a pasture.

• Each evening the sheep are returned to the pen.

• Joe wants to know if all the sheep which left in the morning return in the evening.

• How can he tell?• Count them.• But what if Joe is a Siriona and cannot

count higher than three?

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Joe's Solution• As each sheep leaves the pen, Joe puts

a pebble in a bag.• As each sheep returns, Joe removes a

pebble from the bag.• What are the possible outcomes?

– The bag is empty.– There are one or more pebbles in the

bag.– Any other possibilities?

• The bag becomes empty before all the sheep have entered the pen!

• Joe quickly adds pebbles to the bag.

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We Can Use Joe's Solution• Joe has discovered something very

important.• He can keep track of the size of his herd

without knowing how to count.• This is rather astounding; not at all trivial.• We may be able to count sheep, but we

don't know how to count infinite sets.• We're in the same situation as Joe when

it comes to infinite sets.• So, let's use Joe's idea and apply it to

infinite sets to see “how big” they are.

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One-to-One Correspondence• What Joe invented is called a one-to-

one correspondence (or a bijection):• Given two sets S and T, and a mapping

of elements of S to elements of T, we call the mapping a one-to-one correspondence if these conditions hold:– Every element of S is mapped to some

element of T.– Distinct elements of S are mapped to

distinct elements of T.– Every element of T has some element of

S mapped to it.

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Example of a One-to-One Correspondence

S T

1

2

3

4

A

B

C

D

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The Mapping Is In Both Directions

S T

1

2

3

4

A

B

C

D

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Another Correspondence

S: 1 2 3 4 5 6 7 8 9 ...

T: 2 4 6 8 10 12 14 16 18 ...

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Yet Another One

S: 1 2 3 4 5 6 7 8 9 …

T: 1 3 5 7 9 11 13 15 17 ...

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Perhaps More Interesting

S: 1 2 3 4 5 6 7 8 ...

T: 1 4 9 16 25 36 49 64 ...

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What's This One?

S: 1 2 3 4 5 6 7 8 ...

T: 1 8 27 64 125 216 343 512 ...

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What About This One?

S: 1 2 3 4 5 6 7 8 9 ...

T: 3 1 4 1 5 9 2 6 5 ...

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Cardinality

Did you notice that we've been looking at correspondences between infinite sets?

We're going to use the idea of a one-to-one correspondence to compare the sizes of sets, infinite as well as finite.

If a one-to-one correspondence exists between two sets S and T then we say that S and T have the same cardinality, denoted S ~ T.

Informally, the cardinality of a set is the size of the set.

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Cardinality of Finite Sets

Do {1, 2, 3, 4} and {A, B, C, D, E} have the same cardinality?

What about {1, 2, 3, 4} and {A, B, C}? {1, 2, 3, 4} and {A, B, C, D}? For any finite set, just count the

elements to determine the cardinality. If there are 7 elements, then, when you

finish counting, you have a one-to-one correspondence between the elements of the set and the elements of {1, 2, 3, 4, 5, 6, 7}.

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The Natural Numbers

Our favorite infinite set is = {1, 2, 3, ...}, the set of natural numbers, also called:— counting numbers— positive whole numbers— positive integers

• We use the natural numbers to investigate the size of infinite sets.

• By the way, are we sure that is infinite?

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Countable Sets

If a set S is finite, or if it has the same size as (i.e., there is a one-to-one correspondence between S and ), then the set is said to be countable.

In other words, you can count the mem-bers of a countable set.

The term denumerable is also used; it means the same as countable.

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Same-Size Sets

So far, we've seen that all the following sets have the same cardinality (i.e., the same “size”) as the natural numbers :– even natural numbers– odd natural numbers– squares of natural numbers– cubes of natural numbers

All of the above are subsets of ; let's go for something bigger than .

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~

What does this correspondence show?

… 8 6 4 2 1 3 5 7 9 ...

… -4 -3 -2 -1 0 1 2 3 4 …

This is a one-to-one correspondence between the natural numbers and the integers , so is countable.

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The Rational Numbers

Consider numbers of the form n/d, where both n and d are integers, i.e., whole numbers.

Some examples:— 1/2, 2/3, 4/1, 1001/57, -3/2, 1000000/18— 1/1, 2/1, 3/1, 4/1, 5/1, ...

These numbers are called rational because they are ratios of integers.

Note that is a subset of , and is a subset of .

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Counting Numbers

Consider the interval [-2, 2]:

--|------------|------------|------------|------------|-- -2 -1 0 1 2

How many integers are in this interval? How many natural numbers? How many rational numbers?

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The Rationals Are Dense

Consider the interval [0, 1]:

--|--------------------------------------------------|-- 0 1

How many rationals are in this interval? Lots and lots and lots. The rationals are dense: Between any

two of them there is another one.

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Are the Rationals Countable?

Claim: The set of rational numbers is countable.

Do you believe this? Any ideas on how to show that it is true

or not true? To prove it true, we need to come up

with a one-to-one correspondence between the rationals and the natural numbers.

Does this mean we need to find a smallest rational number?

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Any Positive Rationals Missing?

1 2 3 4 5 6 71: 1/1 ½ 1/3 ¼ 1/5 1/6 1/7 ...2: 2/1 2/2 2/3 2/4 2/5 2/6 2/7 ...3: 3/1 3/2 3/3 3/4 3/5 3/6 3/7 ...4: 4/1 4/2 4/3 4/4 4/5 4/6 4/7 ...5: 5/1 5/2 5/3 5/4 5/5 5/6 5/7 ...6: 6/1 6/2 6/3 6/4 6/5 6/6 6/7 ...7: 7/1 7/2 7/3 7/4 7/5 7/6 7/7 … … … … … … … … … …

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Which Numbers Are Gone?

1 2 3 4 5 6 71: 1/1 ½ 1/3 ¼ 1/5 1/6 1/7 ...2: 2/1 2/3 2/5 2/7 ...3: 3/1 3/2 3/4 3/5 3/7 ...4: 4/1 4/3 4/5 4/7 ...5: 5/1 5/2 5/3 5/4 5/6 5/7 ...6: 6/1 6/5 6/7 ...7: 7/1 7/2 7/3 7/4 7/5 7/6 … … … … … … … … … …

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The Rationals Are Countable

1 2 3 4 5 6 71: 1/1 ½ 1/3 ¼ 1/5 1/6 1/7 ...2: 2/1 2/3 2/5 2/7 ...3: 3/1 3/2 3/4 3/5 3/7 ...4: 4/1 4/3 4/5 4/7 ...5: 5/1 5/2 5/3 5/4 5/6 5/7 ...6: 6/1 6/5 6/7 ...7: 7/1 7/2 7/3 7/4 7/5 7/6 … … … … … … … … … …

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The Rationals Are Countable

1 – 1 11 – 1/5 2 – 1/2 12 – 1/6 3 – 2 13 – 2/5 4 – 3 14 – 3/4 5 – 1/3 15 – 4/3 6 – 1/4 16 – 5/2 7 – 2/3 17 – 6 8 – 3/2 18 – 7 9 – 4 19 – 5/3 10 – 5 20 – 3/5

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The Rationals Are Countable

At this point you should be stunned. The rationals are dense; between any two

distinct rationals there is another rational. In fact, between any two distinct rationals

there are an infinite number of rationals. The rationals seem to completely fill the

number line; the integers appear with gaps between them.

And yet, there are no more rationals than there are integers.

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Maybe All Sets Are Countable?

is countable. Every subset of is countable. The superset of is countable. The superset of is countable. What else is there? Might a set exist which is “bigger” than

?

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The Power Set of

Let S be the set of all subsets of . Technical term: S is the power set of . Note that the members of S are not

natural numbers. The members of S are sets of natural

numbers. Name a few members of S.

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Some Members of S

The set of even natural numbers. The set of odd natural numbers. The set of squares of natural numbers. The set of cubes of natural numbers. The sets {1}, {2}, {3}, ..., {n}, … The sets {1,2}, {2,3}, {3,4}, ..., {n,n+1}, … The sets {1,2,3}, {2,3,4}, {3,4,5}, …

{n,n+1,n+2}, … Are there any more?

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Naming Members of S

: 1 2 3 4 5 6 7 8 9 10 11 12 13 …

S1: 0 1 0 1 0 1 0 1 0 1 0 1 0 …

S2: 1 0 1 0 1 0 1 0 1 0 1 0 1 …

S3: 1 0 0 1 0 0 0 0 1 0 0 0 0 …

S4: 1 0 0 0 0 0 0 1 0 0 0 0 0 …

S5: 1 0 0 0 0 0 0 0 0 0 0 0 0 …

S6: 0 1 0 0 0 0 0 0 0 0 0 0 0 …

S7: 0 0 1 0 0 0 0 0 0 0 0 0 0 …

S8: 1 1 0 0 0 0 0 0 0 0 0 0 0 …

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It's all 0's and 1's

Note that every sequence of 0's and 1's describes a unique subset of .

Note that every subset of can be described uniquely as a sequence of 0's and 1's.

So, what have we got? A one-to-one correspondence between

the set of subsets of and the set of sequences of 0's and 1's.

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Let's Make an Assumption

So, the set of all subsets of and the set of all sequences of 0's and 1's have the same cardinality; these two sets are the same size.

Can these sets be put in a one-to-one correspondence with ?

Assume this is the case, and consider any one-to-one correspondence between and the set of sequences of 0's and 1's.

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A 1-to-1 Correspondence

1: 0 1 0 1 0 1 0 1 0 1 0 1 0 …

2: 1 0 1 0 1 0 1 0 1 0 1 0 1 …

3: 1 0 1 1 0 0 0 0 1 0 0 0 0 …

4: 1 0 0 0 0 0 0 1 0 0 0 0 0 …

5: 1 0 0 0 1 0 0 0 0 0 0 0 0 …

6: 0 1 0 0 0 0 0 0 0 0 0 0 0 …

…

Note that this list goes on forever.

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So, What Do We Have?

• For every natural number n we have a unique sequence of 0's and 1's.

• In principle, we can list all the sequences.

• Also, there are no sequences of 0's and 1's which are not in the list because we have a one-to-one correspondence.

• However, there is a little problem ...

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Another Member of S

1: 0 1 0 1 0 1 0 1 0 1 0 1 0 …

2: 1 0 1 0 1 0 1 0 1 0 1 0 1 …

3: 1 0 1 1 0 0 0 0 1 0 0 0 0 …

4: 1 0 0 0 0 0 0 1 0 0 0 0 0 …

5: 1 0 0 0 1 0 0 0 0 0 0 0 0 …

6: 0 1 0 0 0 0 0 0 0 0 0 0 0 …

x: 1 1 0 1 0 1 - - - - - - - …

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Prelude to a Contradiction

Is the sequence x a member of the set of all sequences of 0's and 1's?

Of course. Look at it. It's a sequence of 0's and 1's, so it must be in the set of all sequences of 0's and 1's.

Is there some natural number n that corresponds to x in the given one-to-one correspondence?

There must be, because otherwise we wouldn't have a one-to-one correspondence.

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A Contradiction

We have a little problem here: Which natural number n corresponds to x?

We defined x in such a way that it differs from EVERY sequence in the correspondence in at least one place.

So we have arrived at a contradiction: x must correspond to some natural number, but x doesn't correspond to any natural number.

What's wrong? How did we get into this mess?

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A Way Out of the Mess

We made an assumption, namely that a one-to-one correspondence exists between the set of all sequences of 0's and 1's and the set .

This assumption led to a contradiction. Which means the assumption is false. So there does not exist a one-to-one

correspondence between and the set of all sequences of 0's and 1's.

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The Grand Conclusion Is ...

There is always at least one sequence of 0's and 1's which does not correspond to any natural number.

So the set of sequences of 0's and 1's is bigger than .

Recall that the set of all 0's and 1's is the same size as S, the set of all subsets of .

So S is bigger than .

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Are You Stunned Yet?

If you weren't stunned to see that the rational numbers are countable, it's now time to be stunned.

What you have just seen is a proof that some infinities are bigger than others.

There are more sets of natural numbers than there are natural numbers.

In fact, the cardinality of S is *way* bigger than the cardinality of .

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Don't Stop Now

Let T be the set of all subsets of S. Remember that members of S are

subsets of . A member of T:

– { {odd numbers}, {square numbers} } Another member of T:

– { {1}, {2, 10}, {primes} } How does the size of T compare to the

size of S?

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, S, T, U, ...

The size of T is way bigger than the size of S.

Remember that the size of S is way bigger than the size of .

So, T is way, way bigger than . Now let U be the set of all subsets of T. What is your guess about the size of U

compared to the size of T?

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It Just Keeps Going On and On and On ...

You guessed right: The size of U is bigger than the size of T.

Here's the scary part: It goes on forever. Given any set X, let Y be the set of all

subsets of X. Then Y is a bigger set than X.

There is no largest infinity. Given any infinite set we can easily

construct a larger infinite set. It just keeps going on and on and on ...

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What we have seen

The two main points of this talk:— Some infinities are bigger than others.— There is no biggest infinity.

Don't feel bad if these results leave you with an uneasy feeling.

Here is what Cantor had to say about one of his own results:— "I see it, but I don't believe it!"

Mathematics is full of surprises.

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David Hilbert On Cantor's Work

David Hilbert (1862-1943) on Cantor's work on set theory:

"The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity."

“No one will expel us from this paradise Cantor has created for us.”