Mathematical Ideas that
Shaped the World
An infinity of infinities
Plan for this class
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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
lemniscate
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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 allpowerful.
An infinite universe?
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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?
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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?
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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
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Two men set out to understand infinity and
include it in the very foundations of
mathematics:
Hilbert and Cantor
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One man ended up in an insane asylum and
the other died with his dream shattered.
Hilbert’s Hotel
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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
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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
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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
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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.
2.
3.
Make all the odd-numbered rooms free like
before.
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).
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
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Hilbert’s hotel shows us that
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+1=
2=+=
  =
-=?
Cantor (1845 – 1918)
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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
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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
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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
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When we “count”, we are pairing objects with
numbers.
1
2
3
How many even numbers are there?
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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
How many integers are there?
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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
11 9
7
5
3
1
2
4
6
8
10
How many fractions are there?
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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!
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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
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Suppose we can list all the decimals.
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1)
2)
3)
4)
0.100000…
0.120000…
0.146000…
0.2235600…
….
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But then we can write down a number which
is different from every number in this list:
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E.g. 0.2376…
Bigger infinities!
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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
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Whole numbers
Fractions
Prime numbers
All possible words you
could make out of the
English alphabet
Uncountable infinities
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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
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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
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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”
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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.”
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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”
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while the philosopher Wittgenstein thought
that set theory was
“utter nonsense” and “laughable”
Criticism
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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If we had a 4th axiom which said
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4) Sheep have a secret penchant for cake
Then we would be able to show
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Sheep don’t eat cake (axiom 3)
Sheep do eat cake (axiom 4)
which contradict each other.
Axioms of arithmetic
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Our axioms of arithmetic are things like
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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?
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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
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Even our reasoning about collections of
objects (sets) can run into problems.
How big is the set of all sets?
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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:
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If a barber shaves every man who does not shave
himself, then who shaves the barber?
Hilbert’s tombstone
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On Hilbert’s tombstone were carved the
words
Wir müssen wissen.
Wir werden wissen.
meaning
We must know.
We will know.
Kurt Gödel (1906 – 1978)
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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,
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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
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Earlier we had some axioms about what makes a
sheep:
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1) That sheep are mammals
2) That sheep have a woolly coat
3) That sheep consume only grass and water
A statement such as
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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
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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
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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
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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
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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
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A solid ball can be broken up and re-assembled to
create two balls identical to the first.
Gödel’s madness
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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
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That the concept of infinity is more mindboggling 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!