Season 5 Episode 5
In this episode; surreal numbers! They're made out of sets, they're recursive, they're really weird. Manna is on the livestream to show us how they work.
The Empty Set
Nothing is not the same thing as the empty set, even though in the English language we might use the same sorts of words for both. Consider the following nonsense proof;
"Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness."
This joke relies on the ambiguous meanings of the word "nothing" in English. For more like that, see this play.
Set-theoretic definition of natural numbers
It might have been a good idea to show you this first, before we did surreal numbers. There’s a way to "build" a series of sets that resemble the natural numbers, starting with the empty set. Surreal numbers are playing in the same space, but with more complexity.
For a book, see this webpage. That link goes to a free copy of the first part of the book. Warning: this book is a bit weird? If you thought surreal numbers were weird on the livestream, this book won’t make you think that it’s less weird.
For a more traditional resource, see this webpage This resource is quite long. It explains addition and multiplication, and has proofs and examples on why they are just like the normal addition and multiplication. The last part of this talks about infinity (very large) and infinitesimal (very small).
For more on the game at the end of the livestream, see this webpage
This talks about rules of Hacken-games and how to assign a game a Surreal number in more detail.
For some more related and much harder games, see this webpage.
What do you call a knight who is really weird and abstract?
What do you call a knight who goes all the way around the Round Table?
(and so on and so on)
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk