Season 11 Episode 8
By viewer request, an Oxford Online Maths Club episode about set theory! We'll see how to make something out of nothing, and then how to make everything out of something.
Further Reading
Introduction to University Mathematics
You can find out more about sets, relations, and functions in my lecture notes for Introduction to University Mathematics (an introductory course for Maths students at Oxford); the notes are freely available on the course website and the lecture videos I recorded are in a playlist on the Oxford Mathematics Plus YouTube channel (that's the same channel as the Maths Club!).
More generally, I recommend the book How to think like a mathematician by Kevin Houston if you’re looking ahead to university mathematics. If you do a Google search you might be able to find a copy at a local library.
Numbers are sets
Something that we didn’t talk about on the livestream, that would have got us full circle, is the concept that we can build numbers out of sets.
One way to do this is called "Von Neumann ordinals". The empty set represents the number zero, and the set containing the empty set represents the number one. Now to continue this, for each number \(n\), von Neumann defines \(n+1\) to be \(\lbrace n, \lbrace n\rbrace \rbrace\). This has the effect that each number is precisely the set of all smaller numbers. The power of this idea is that we can extend to infinite sets (without having infinitely deep nested brackets). Writing \(\omega\) for the set of natural numbers, von Neumann defines \(\omega +1\) to be \( \lbrace \omega, \lbrace \omega \rbrace \rbrace \) in exactly the same way. if you like, this is a formal definition for "infinity plus one".
Wikipedia has an incredible diagram of what happens next.
The idea here is that, with sets (just containing other sets, down to the empty set), we can go on to build all of mathematics. It's all sets. Always has been.
Russell’s paradox
Someone in chat asked why this is a paradox. In part, this comes down to how you think sets should behave, and how you feel about contradictions. That puts us near the territory marked “philosophy of mathematics” on my map, so I’m linking to Stanford Encylopedia of Philosophy again for more on Russell’s paradox.
If you'd prefer a YouTube video, how about this one? Russell's Paradox - a simple explanation of a profound problem | Jeffrey Kaplan | YouTube.
Axioms
If you'd like to see the formal axioms (the starting points) for set theory, you can see a list of them here on Wolfram Mathworld; axioms of set theory.
But if you want to learn about the axioms one-by-one and build from there (what I think of as “doing it properly”, compared to what I do in Introduction to University Mathematics), then you need something like B1.2 Set Theory, a third-year course at Oxford.
If you’d like a book, you might like “Classic Set Theory: For Guided Independent Study” by Derek Goldrei. Derek Goldrei was a mathematician at Oxford and was a big part of the Open University over many years. The book starts with a “proper” definition of the real numbers.
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.