# Numbers and Sets

Though concepts such as addition, multiplication, rational numbers and decimal expansions are familiar ground to anyone who didn’t utterly fail high school mathematics few stop to question how these concepts are formalised or properly constructed. For example, why are we allowed to add infinitely many terms after a decimal place? What does ‘infinitely’ even mean? Heck, what *is *a number even? Though it might seem a first glance such questions are perhaps pointless navel gazing they are actually very important to consider because from them flow a large amount of mathematics. After all, if you don’t understand what is actually meant by the integers, the rationals or the reals building more complicated concepts from them is a dangerous game to play.

Hence why one of the first undergrad courses taught at Cambridge (at least in my day, 2002) was Number and Sets, a crash course into the world of much more formal mathematics than encountered in secondary school. It includes the following;

- Basic axiomatic logic and proofs
- Introductory number theory on the integers
- Extending the integers to the rationals and reals
- Cardinality of infinite sets, including Cantor’s diagonal argument

Unfortunately the lecture notes capture only a small part of the lectures, Professor Gowers gave very good descriptions and explanations but only the blackboards were transcribed.

This is a 1st year lecture course.