Mathematical Ideas that Shaped the World

An infinity of infinities

Plan for this class

Does infinity really exist? If so, what are its rules? How do we compare the sizes of

different infinite sets? Is the number of even numbers less

than the number of all whole numbers? Who were Cantor and Gödel, and what

ideas made them go mad? Can mathematics ever contradict itself?

A history of infinity

For most of history, infinity has been a philosophical concept.

Attempts to use infinity in maths led to paradoxes and nonsense. (e.g. Zeno!)

Infinities in physical theories are still a sure sign that something is wrong.

If anything, infinity was equated with the idea of God: something unknowable and all-powerful.

lemniscate

An infinite universe?

Giordano Bruno (1548 – 1600), an Italian mathematician and astronomer, believed that the universe was infinite in size.

He was burned at the stake by the Catholic church, since they believed that the only thing which was infinite was God.

Does infinity exist?

Even up until the middle of the 19th century, people continued to avoid infinity.

Questions about infinity were turned into questions about limits, which only spoke of finite quantities.

By mathematicians, infinity was thought of as a process – like the act of counting without stopping.

What is infinity?

One day people started asking

What if we thought of infinity as an actual number?

How would it interact with other numbers?

Can we write down a set of laws for infinity to follow?

The pioneers

Two men set out to understand infinity and include it in the very foundations of mathematics:

Hilbert and Cantor

One man ended up in an insane asylum and the other died with his dream shattered.

Hilbert’s Hotel Hilbert’s hotel has infinitely many

rooms: one for each natural number 1, 2, 3, 4, etc.

All of the rooms are full.1 2 3 4 5 6

Puzzle 1

One new guest arrives looking for a room. Can you work out how to fit him in?

Making one more room

1 2 3 4 5 6

Conclusion

+ 1 =

Puzzle 2

Our previous guest is now happy, but then a bus containing infinitely many people arrives at the hotel. Can we fit them all in?

Making infinitely many rooms

1 2 3 4 5 6

Conclusion

+ =

Puzzle 3 Just when the hotel manager thought

they were safe, news comes that infinitely many buses, each carrying infinitely many people, is heading their way.

Is there anything that can be done to keep everyone happy?

Finding a solution (there are many!)1. Make all the odd-numbered rooms

free like before.2. Each passenger comes with a pair of

numbers: bus number and seat number. E.g. the man on bus 7, seat 3 is (7,3).

3. Draw a grid and make a path that goes through each passenger once and doesn’t miss any out…

A grid of passengers

(1,1) (1,2) (1,3) (1,4) (1,5) ….

(2,1) (2,2) (2,3) (2,4) (2,5) ….

(3,1) (3,2) (3,3) (3,4) (3,5) ….

(4,1) (4,2) (4,3) (4,4) (4,5) ….

(5,1) (5,2) (5,3) (5,4) (5,5) ….

…. …. …. …. …. ….

Conclusion

=

Rules for infinity

Hilbert’s hotel shows us that + 1 = 2 = + = = - = ?

Cantor (1845 – 1918)

Born in St Petersburg and obtained his PhD from the University of Berlin.

Became a full professor at the University of Halle at the age of 34.

Had 6 children and enjoyed going walking in the Alps.

Set theory

Cantor is best known for his creation of set theory, a cornerstone of modern mathematics.

A set is simply a collection of objects.

Cantor was the first person to study the properties of infinite sets.

Sizes of things

Question: How do we decide whether two sets of objects have the same size?

Answer: we pair off objects, one from each set, and see if there are any left over.

Sizes of things

!!

Sizes of things

When we “count”, we are pairing objects with numbers.

1 2 3

How many even numbers are there? Contrary to your intuition, we can show

that there are the same number of even numbers as of natural numbers.

This is because we can pair them up exactly:1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

Can you find a way of pairing all the positive and negative whole numbers with the natural numbers?

-5 -4 -3 -2 -1 0 1 2 3 4 5

How many integers are there?

1 23 45 67 89 1011

How many fractions are there? We are going to look at the set of

fractions where numerator and denominator are whole numbers, e.g. 65/341.

Are there as many of these as of whole numbers, or are there more?

We want to make a list of them in such a way that we don’t miss any out…

Counting the fractions

1/1 1/2 1/3 1/4 1/5 ….

2/1 2/2 2/3 2/4 2/5 ….

3/1 3/2 3/3 3/4 3/5 ….

4/1 4/2 4/3 4/4 4/5 ….

5/1 5/2 5/3 5/4 5/5 ….

…. …. …. …. …. ….

Finally, the decimals!

How many decimal numbers are there? That is, numbers like 5.9678401746283… ?

Can you make a list of them so that none are missed out?

Amazingly, the answer is NO! Cantor proved that if we ever try to make a list of decimals then we will always miss one out.

Why we can’t list the decimal numbers Suppose we can list all the decimals.

1) 0.100000… 2) 0.120000… 3) 0.146000… 4) 0.2235600…

…. But then we can write down a number

which is different from every number in this list: E.g. 0.2376…

Bigger infinities!

This argument is called Cantor’s diagonal argument.

It proves that there are more decimal numbers than whole numbers!

The infinity of the whole numbers is called “countable”, while the infinity of the real numbers is called “uncountable”.

In fact, there are infinitely many sizes of infinity!

Examples

Countable infinities

Whole numbers Fractions Prime numbers All possible words

you could make out of the English alphabet

Uncountable infinities

Irrational numbers Decimal numbers

between any two numbers, e.g. between 0 and 1

Points on a line Points inside a

square or a cube

Objections to the proof

Not everybody accepted Cantor’s diagonal argument at first.

Some mathematicians didn’t believe in the existence of infinite sets.

Others argued on religious grounds: God is infinite and there is only one God, so therefore there can be only one infinity.

Criticism

One loud critic was Kronecker, a maths professor at the University of Berlin. He opposed the publication of Cantor’s work and called him“a corrupter of youth”

and “a scientific charlatan”

Kronecker claimed

“I don’t know what pre-dominates in Cantor’s theory, philosophy or theology, but I am sure there is no mathematics

there.”

He never gave Cantor the job he sought at the prestigious University of Berlin.

Criticism

The great geometer Poincaré wrote“later generations will regard [Cantor’s

work] as a disease from which they have recovered”

while the philosopher Wittgenstein thought that set theory was

“utter nonsense” and “laughable”

Criticism

Even his friends discouraged him from publishing, with one of them saying

“…it is 100 years too soon”

However, one staunch supporter was Hilbert:

“No one will drive us from the paradise which Cantor has created for us”

Cantor’s madness

By 1884, at the age of 39, Cantor was severely depressed and had no confidence to continue with his work.

He instead studied English Literature and tried to prove that Bacon had written Shakespeare’s plays.

Later went back to maths, but spent an increasing amount of time in a sanatorium.

We now think he had bi-polar disorder.

The Continuum Hypothesis

After Cantor’s proof of the uncountability of the decimals, people started wondering if there was an infinity in between that of the naturals and the decimals.

This problem is known as the continuum hypothesis.

The answer was to be more mind-boggling than anyone had anticipated…

David Hilbert (1862 – 1943)

Born in Königsberg (now Kaliningrad) and went to same school as Immanuel Kant.

Moved to Göttingen, where most of his colleagues were forced out in the Nazi purges.

Helped formulate relativity (with Einstein) and quantum mechanics.

Hilbert’s 23 problems

In 1900 Hilbert made a list of the 23 most important problems of the time.

These problems have influenced the direction of mathematics ever since.

Some of the more famous problems are 1) The Continuum Hypothesis 2) That the axioms of arithmetic are

consistent 8) The Riemann Hypothesis 18) The sphere packing problem

Hilbert’s second problem

Axioms are self-evident truths which we assume to be true and from which we derive all other statements.

The second of Hilbert’s 23 problems was to show that the axioms of arithmetic are consistent.

This means that we should never be able to get contradictions, like proving that a statement is both true and false.

Example: a theory of sheep

Our axioms are 1) That sheep are mammals 2) That sheep have a woolly coat 3) That sheep eat only grass

From these axioms we can deduce things like

Sheep are warm-blooded (from axiom 1) Sheep have 4 limbs (from axiom 1) Sheep are vegetarian (from axiom 3)

Example: a theory of sheep

If we had a 4th axiom which said 4) Sheep have a secret penchant for cake

Then we would be able to show Sheep don’t eat cake (axiom 3) Sheep do eat cake (axiom 4)

which contradict each other.

Axioms of arithmetic

Our axioms of arithmetic are things like 0 + n = n, for all numbers (a + b) = (b + a) for any two numbers a and

b. 1 x n = n, for all numbers (a x b) = (b x a) for any two numbers a and b. For every whole number n, there is a next

whole number n+1.

It is not obvious whether these axioms will ever produce a contradiction.

‘Self-evident’ truths?

Statements which sound ‘self-evident’ are often wrong in maths.

For example, the Greek mathematician Euclid had an axiom which said

The whole is greater than the part. We saw earlier that this is not true for

infinite sets!

Set theory paradoxes

Even our reasoning about collections of objects (sets) can run into problems.

How big is the set of all sets? It must surely be the biggest one, but by

Cantor’s work we know it is always possible to find a bigger one.

There is an analogue of the Barber paradox for sets: If a barber shaves every man who does not

shave himself, then who shaves the barber?

Hilbert’s tombstone

On Hilbert’s tombstone were carved the words

Wir müssen wissen.Wir werden wissen.

meaningWe must know.We will know.

Kurt Gödel (1906 – 1978) Born in Brno, which is

now in the Czech Republic.

Studied logic at the University of Vienna.

Escaped WWII by emigrating to the US – going the long way via Japan!

Became close friends with Einstein.

The incompleteness theorem

In 1931, Gödel proved that, in any system powerful enough to describe whole-number arithmetic,

If the system is consistent, it cannot be complete.

The consistency of the axioms cannot be proven within the system.

This means that there must be some statements in mathematics which are true but can neither be proved nor disproved.

Example: sheep again

Earlier we had some axioms about what makes a sheep: 1) That sheep are mammals 2) That sheep have a woolly coat 3) That sheep consume only grass and water

A statement such as Sheep are amazing at mental arithmetic

cannot be derived from these axioms. Whether it be true or false, it will never contradict anything else we know about sheep.

The incompleteness theorem

The incompleteness theorem was a great blow to Hilbert and to mathematics in general.

However, there was still a hope that such undecidable statements would never crop up in actual mathematics.

Gödel’s madness

In 1933, two years after his incompleteness theorem, Gödel suffered a nervous breakdown. He spent several months in a sanatorium recovering from depression.

Like Cantor, he had been trying to prove the Continuum Hypothesis…

Undecidable theorems

In 1940 Gödel proved that the Continuum Hypothesis was a statement that could neither be proved nor disproved.

The Axiom of Choice is another undecidable theorem. It states that, given any collection of sets, that we can choose one element from each set.

The Axiom of Choice

Most mathematicians use the axiom of choice in their work.

It sounds very intuitive, but it also leads to some very strange conclusions!

One of these is the Banach-Tarski paradox A solid ball can be broken up and re-

assembled to create two balls identical to the first.

Gödel’s madness

Had a fear of being poisoned and would only eat the food cooked for him by his wife.

This eventually led him to starve himself to death when she was no longer well enough to cook for him.

Lessons to take home

That the concept of infinity is more mind-boggling than you can imagine.

That thinking too hard about infinity will probably make you go mad.

That secret paradoxes lurk at the heart of mathematics.

That we can never know everything!