0Introduction
IA Numbers and Sets
0 Introduction
According to the Faculty, this course is not aimed at teaching you any new
knowledge. In particular, the Faculty says
This course is concerned not so much with teaching you new parts
of mathematics . . .
Instead, this course is intended to teach you about how to do maths properly.
The objective of this course is to start from a few axioms, which are assumptions
about our system, and try to prove everything rigorously from the axioms.
This is different from how mathematics is usually done in secondary school.
In secondary school, we just accept that certain statements are true. For example,
probably no one rigorously proved to you that each natural number has a unique
prime factorization. In this course, nothing is handwaved. Everything will be
obtained as a logical consequence of the axioms and previous (proven) results.
In the course, you shouldn’t focus too much on the content itself. Instead,
the major takeaway should be how we do mathematics. In particular, how we
construct and present arguments.
The actual content of this course is rather diverse. As the name suggests,
the course touches on numbers and sets. The “numbers” part is mostly basic
number theory, which you may have come across if you have participated in
mathematical olympiads. We also study some properties of real numbers, but
the actual serious study of real numbers is done in IA Analysis I.
The “sets” part starts with some basic definitions of sets, functions and
relations, which are really important and will crop up in all courses you will
encounter. In some sense, these form the language in which mathematics is
written. At the end of the course, we will touch on countability, which tells us
how “big” a set is.