Welcome to the online course Foundations of Abstract Mathematics, created by lecturers and students of Stellenbosch University.
The aim of this course is to introduce some foundational topics of contemporary mathematics to a beginner. The material is based on well-known deep ideas, but the presentation is highly non-standard (among other things, some of the notation and terminology used appears here for the first time). The present version of the course is essentially an identical copy of the course delivered by Dr Janelidze in 2010-2011 at Stellenbosch University. So far, the presenters in the videos (Ms Chipfakacha, Ms Van Tonder, and Ms Cunliffe) are students from that class.
The material is divided in five parts, called "books".
The aim of Book I is to introduce the language of mathematics. We assume no prior knowledge of any specific area of mathematics, except very elementary arithmetic.
Book II begins by giving a precise mathematical interpretation of "abstraction", as a process of partitioning a given set of "concrete" objects. After introducing very basic set-theoretic notation and terminology, we define a correspondence from a set $X$ to a set $Y$ as an element of a particular partition of the set of all mathematical expressions about objects $x$ and $y$. Then a mapping is defined as a correspondence such that for each $x$ in $X$ there is a unique $y$ in $Y$ which corresponds to $x$. Thus, correspondences capture the idea of a relation between two sets, while mappings --- that of a function. Relations and functions will be formally introduced in Book IV, after developing little bit more of set-theoretic intuition in Book III. At the end of Book II we discuss basic relational properties --- reflexivity, symmetry, antisymmetry and transitivity.
Each chapter concludes with a list of questions for practice, for students of different levels: "exercises" are intended for a beginner, "problems" are intended for those who already have certain mathematical maturity, while questions labeled as "application" or "research" are intended for those students who have a substantial knowledge of mathematics --- often answers to such a question can be quite lengthy and can constitute either entire or part of a third year project in mathematics. Occasionally, there are also "computer projects" for students who are able to program.
The aim of Book III is to introduce an axiomatic approach to set theory. Notice however that we do not include all axioms of set theory: instead, we only discuss several of the most basic axioms and focus on practicing how to form and prove simple formulas in "the language of set theory". We also discuss the idea of a "universe of small sets" --- a world of sets within a world of sets. Some attention is paid to interpreting the language of set theory in a non set-theoretical context (specifically, we investigate the case when we interpret sets as natural numbers and "$x\in y$" is interpreted as "$x+1\leqslant y$").
In Book V we explore basic operations for sets, such as union, intersection, difference, product, exponentiation, and then we define and study relations and functions between sets, which model correspondences and mappings within set theory. The ideas introduced in Book IV are essential in every area of modern mathematics.
Book V begins with a study of special types of functions --- injections, surjections and bijections. These functions are then used to introduce the concept of infinity, and to discover that there can be infinities of different sizes. This naturally leads to cardinal numbers, which allow to "count" elements in both finite and infinite sets. Then, natural numbers are introduced as cardinal numbers of a special type, and the principle of mathematical induction is obtained. Finally, the universal property of the natural number system is established, with some applications given in the form of problems.