Season 7 Episode 3

It's time to face the hydra; a mythical creature that regrows heads whenever you cut one off. In this episode, we're looking at some variant hydras and some associated number systems.

Further Reading

Hydra game on Wikipedia

See https://en.wikipedia.org/wiki/Hydra_game for more information about the hydra game.

Ordinals

In the livestream, we assigned ordinals to the various hydras we intended to defeat. The ordinals are a number system that goes beyond finite numbers. The key feature is that they are a "well-ordered set"; given any two ordinals, they're either equal, or one of them is larger than the other, and given any non-empty set of ordinals, there's a least element in the set. In fact, this property can be used to define ordinals; John von Neumann defines "each ordinal is the well-ordered set of all smaller ordinals".

Wikipedia has an amazing diagram of ordinals but the rest of that article uses some advanced ideas from logic and set theory. In the Oxford Mathematics course, this is covered in third-year courses. It's particularly important for Mathematics and Philosophy students, and they can opt to attend the Logic and Set Theory courses in their second year.

There's a video and a poster by Joel David Hamkins at this webpage. While I'm linking to resources by the same author, see recursive chess and infinite chess.

Kirby-Paris Hydra

See Googology for another interpretation of the Kirby-Paris hydra in terms of nested brackets.

Largest number game

We dealt with some large numbers on the stream; if you chop off the rightmost head each time, even a reasonably small hydra will take a very long time to defeat.

Scott Aaronson's article on large numbers is a classic; read about big numbers here.  

The largest number competition is an event at MIT where people compete to name the largest number.

This is where Rayo's number comes from, if you've heard of that particularly large number.

There's a match report, by Rayo, here. That link is worth a visit partly to see the poster design they came up with for the event! In some sense, this is the modern equivalent of the sort of competitive equation-solving that people like Nicolo Tartaglia got up to in the 1500s.