Infinity and the Limits of Mathematics Manchester High School for Girls Friday 30 th September 2011 Dr Richard Elwes, University of Leeds

Counting with Cantor How many letters are there in ‘CANTOR’? ↕↕↕↕↕↕ CANTOR

Counting with Cantor So, two sets are the same size if there is a one-to-one correspondence between them. In the late 19 th 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 א 0 (“aleph nought”).

How to count Which infinite sets are the same size as א 0 ? 01234…n… ↕↕↕↕↕…↕… 02468…2n… Counting the Infinite The even numbers:

How to count Which infinite sets are the same size as א 0 ? 01234…2n-12n… ↕↕↕↕↕…↕↕… 012-2…n-n… Counting the Infinite The integers, or whole numbers (positive, negative and zero):

How to count Counting the Infinite What about the rational numbers (fractions)? … … … … …

How to count Counting the Infinite The sets of even numbers, prime numbers, whole numbers, and rational numbers are all countably infinite, meaning they are the same size as א 0. 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: … 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 א 0. The Uncountably Infinite He provided a famous proof, called Cantor’s diagonal argument.

The Uncountably Infinite Imagine that there is a correspondence between א 0 and the real numbers between 0 and 1. The Uncountably Infinite 1↔ … 2↔ … 3↔ … 4↔ … 5↔ … 6↔ … It might look like this:

The Uncountably Infinite In general, a correspondence will look like this: The Uncountably Infinite 1↔0.a1a1 a2a2 a3a3 a4a4 a5a5 … 2↔ b1b1 b2b2 b3b3 b4b4 b5b5 … 3↔ c1c1 c2c2 c3c3 c4c4 c5c5 … 4↔ d1d1 d2d2 d3d3 d4d4 d5d5 … 5↔ e1e1 e2e2 e3e3 e4e4 e5e5 … 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.a1a1 a2a2 a3a3 a4a4 a5a5 … 2↔ b1b1 b2b2 b3b3 b4b4 b5b5 … 3↔ c1c1 c2c2 c3c3 c4c4 c5c5 … 4↔ d1d1 d2d2 d3d3 d4d4 d5d5 … 5↔ e1e1 e2e2 e3e3 e4e4 e5e5 … …Cantor specified every one of its decimal places… …and made sure that it disagreed with a 1, b 2, c 3, d 4, and so on.

Cantor defined a new real number x = 0. x 1 x 2 x 3 x 4 x 5 …. with this rule: if a 1 =3, then x 1 =7 if a 1 ≠3, then x 1 =3(This guarantees x≠a.) if b 2 =3, then x 2 =7 if b 2 ≠3, then x 2 =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 א 0. This set is known as the continuum, or 2 א 0. Cantor wanted to know whether there was a level between א 0 and the continuum. He thought there wasn’t... …but he couldn’t prove it. This became known as the continuum hypothesis 2 א 0 = א 1 ? 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