Infinity and the Limits of Mathematics

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

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

3. 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 infinitesets?

4. Counting the Infinite The most familiar infinite set is the set of natural numbers: {0,1,2,3,4,5,6,….} We call this set א0 (“aleph nought”).

5. Counting the Infinite How to count Which infinite sets are the same size as א0? The even numbers:

6. Counting the Infinite How to count Which infinite sets are the same size as א0? The integers, or whole numbers (positive, negative and zero):

7. Counting the Infinite How to count What about the rational numbers (fractions)? 1 2 5 6 11 3 7 8 4 9 10

8. Counting the Infinite How to count The sets of even numbers, prime numbers, whole numbers, and rational numbers are all countably infinite, meaning they are the same size as א0. So, are there any uncountably infinite sets? Yes!

9. The Uncountably Infinite It is well known that the decimal expansion of π continues forever without ever stopping or getting stuck in a repetitive loop: 3.1415926535897932384626433832795… How many other such numbers are there? Infinitely many, of course, but…

10. The Uncountably Infinite 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 א0. He provided a famous proof, called Cantor’s diagonal argument.

11. The Uncountably Infinite The Uncountably Infinite Imagine that there is a correspondence between א0 and the real numbers between 0 and 1. It might look like this:

12. The Uncountably Infinite The Uncountably Infinite In general, a correspondence will look like this: Every real number between 0 and 1 must be somewhere on this list…. …so if Cantor could find just one number which was missed out, it can’t have been a genuine correspondence after all.

13. The Uncountably Infinite The Uncountably Infinite To find a new number not on the list… …Cantor specified every one of its decimal places… …and made sure that it disagreed with a1, b2,c3, d4, and so on.

14. The Uncountably Infinite Cantor defined a new real number x = 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.

15. The Continuum Hypothesis • Cantor’s diagonal argument proved that the set of real numbers is bigger than א0 . • This set is known as the continuum, or 2א0. • Cantor wanted to know whether there was a level between א0 and the continuum. • He thought there wasn’t... • …but he couldn’t prove it. • This became known as the continuum hypothesis • 2א0 = א1?

16. 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.

17. Beyond 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!