Infinity and the Limits of Mathematics

PhilSoc @ Manchester High School for GirlsFriday 30th September 2011

Dr Richard Elwes, University of Leeds

Counting with Cantor

How many letters are there in ‘CANTOR’?

1 2 3 4 5 6

↕ ↕ ↕ ↕ ↕ ↕

C A N T O R

Counting with Cantor

So, two sets are the same size if there is a one-to-one correspondence between them.

In the late 19th century Georg Cantor had an amazing thought.

What would happen if he applied this idea to infinite sets?

The most familiar infinite set is the set of natural numbers:

{0,1,2,3,4,5,6,….}

Counting the Infinite

We call this set 0א (“aleph nought”).

How to count

Which infinite sets are the same size as 0א ?

0 1 2 3 4 … n …

↕ ↕ ↕ ↕ ↕ … ↕ …

0 2 4 6 8 … 2n …

Counting the Infinite

The even numbers:

How to count

Which infinite sets are the same size as 0א ?

0 1 2 3 4 … 2n-1 2n …

↕ ↕ ↕ ↕ ↕ … ↕ ↕ …

0 1 -1 2 -2 … n -n …

Counting the Infinite

The integers, or whole numbers (positive, negative and zero):

How to countCounting the InfiniteWhat about the rational numbers (fractions)?

1 1 1 1 1 …

1 2 3 4 5

2 2 2 2 2 …

1 2 3 4 5

3 3 3 3 3 …

1 2 3 4 5

4 4 4 4 4 …

1 2 3 4 5

5 5 5 5 5 …

1 2 3 4 5

⁞ ⁞ ⁞ ⁞ ⁞

2

3

4

5 6

7

1

8

9

10

11

How to countCounting the InfiniteThe sets of even numbers, prime numbers, whole numbers, and rational numbers are all countably infinite, meaning they are the same size as 0א .

So, are there any uncountably infinite sets?

Yes!

It is well known that the decimal expansion of π continues forever without ever stopping or getting stuck in a repetitive loop:3.1415926535897932384626433832795…

The Uncountably Infinite

How many other such numbers are there?

Infinitely many, of course, but…

The Uncountably Infinite

A real number is an infinite decimal string.

We’ll just focus on the ones between 0 and 1, which all begin ‘ 0. ’

Georg Cantor proved that this set is uncountable, meaning bigger than 0א .

The Uncountably Infinite

He provided a famous proof, called Cantor’s diagonal argument.

The Uncountably Infinite

Imagine that there is a correspondence between 0א and the real numbers between 0 and 1.

The Uncountably Infinite

1 ↔ 0.111111111111…

2 ↔ 0.141592653589…

3 ↔ 0.000000000000…

4 ↔ 0.272727272727…

5 ↔ 0.718281828459…

6 ↔ 0.414213562373…

It might look like this:

The Uncountably Infinite

In general, a correspondence will look like this:

The Uncountably Infinite

1 ↔ 0. a1 a2 a3 a4 a5 …

2 ↔ 0. b1 b2 b3 b4 b5 …

3 ↔ 0. c1 c2 c3 c4 c5 …

4 ↔ 0. d1 d2 d3 d4 d5 …

5 ↔ 0. e1 e2 e3 e4 e5 …

⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞

Every real number between 0 and 1 must be somewhere on this list….

which was missed out, it can’t have been a genuine correspondence after all.

…so if Cantor could find just one number

The Uncountably Infinite

To find a new number not on the list…

The Uncountably Infinite

1 ↔ 0. a1 a2 a3 a4 a5 …

2 ↔ 0. b1 b2 b3 b4 b5 …

3 ↔ 0. c1 c2 c3 c4 c5 …

4 ↔ 0. d1 d2 d3 d4 d5 …

5 ↔ 0. e1 e2 e3 e4 e5 …

⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞

…Cantor specified every one of its decimal places…

…and made sure that it disagreed with a1, b2,

c3, d4, and so on.

Cantor defined a new real numberx = 0. x1 x2 x3 x4 x5 ….

with this rule:• if a1=3, then x1=7

• if a1≠3, then x1=3 (This guarantees x≠a.)

• if b2=3, then x2=7

• if b2≠3, then x2=3 (This guarantees x≠b.)

• And so on.

The Uncountably Infinite

• Cantor’s diagonal argument proved that the set of real numbers is bigger than 0א .

• This set is known as the continuum, or 2 0א .

• Cantor wanted to know whether there was a level between 0א and the continuum.

• He thought there wasn’t...• …but he couldn’t prove it.• This became known as the continuum hypothesis

• 1א= 0א 2 ?

The Continuum Hypothesis

• In 1963, Paul Cohen resolved the continuum hypothesis…

• …sort of.• He constructed two mathematical universes.• Almost everything looks the same in each.• But in one, the continuum hypothesis is true…• …and in the other, it isn’t!• The continuum hypothesis is formally

undecidable from the usual laws of maths.

The Continuum Hypothesis

• Is there any way to tell which the ‘correct’ mathematical universe is? And whether the continuum hypothesis can really be said to be ‘true’ or not?

• Some people think so. E.g. Hugh Woodin.• Others think not. E.g. Joel Hamkins.• What do you think?

• Thank You!

Beyond the Continuum Hypothesis