Many mathematics students have trouble understanding why set theory is important since their introduction to mathematics is usually based on arithmetic and algebra, rather than calculus. To understand why set theory is important, we should look at the fundamental question of what mathematics is. The engineer's viewpoint of mathematics is that it is a tool to solve problems, the scientist uses it to describe reality, the statistician uses it to build models, but what is mathematics to the mathematician?
Who Cares? edit
Descartes said it well when he said that "mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency." In mathematics, we have a fully formalized discipline - there is no subjectivity, because everything is based upon rules and definitions. While these rules are taught to children as the immutable "way that mathematics works," they are in fact only conveniences. Any defined set of rules is a valid construct in mathematics, but some (the ones that are studied) are more interesting than others. For instance, if we were to redefine equality of numbers so that any two numbers were equal, we would have a well defined system, but there are very few interesting things to say about it, since every proposition is true in such a system.
To answer our second question, then, mathematics is the building of interesting systems. We are still left with a confusion about why we would discuss this particularly in the context of set theory, which leads us to some history of the concept. In the middle of the 19th century, mathematics was formless and void—the Greek concept of rigorous proofs had been displaced by vagueness, and the rigor of a coherent set theory was still in the future. It was around this time that De Morgan and Boole began to write about logic, as a branch of mathematics. This was important to mathematics, because until this point, mathematical proofs were convincing arguments, but were not formalized.
Georg Cantor asked some very puzzling questions about infinity that undermined some very obvious notions. For instance, he started to claim that there was more than one infinity, and that they were different sizes. The furor and disputes, which were partially caused by ambiguousness in Newton's works concerning infinity and infinitesimals, led to a formalization of what numbers are. Because of this, all of modern mathematics was re-built using set theory, so that all of modern mathematics is in some sense just applications of the notion of a set.
What is a Set? edit
Before commencing our study of sets, we will comment on what a set is, and why they are used in mathematics. In many ways, a set is the simplest possible conceptual object, from which the rest of mathematics can be rigorously developed. It is conceptually easiest to start where set theory does. Everything in set theory is formed based on the empty set, denoted or . Other sets can contain the empty set, so that the set is a simple set containing only the empty set. We can build sets of varying complexity based on this idea; would be a set containing two items: the empty set, and the set containing the empty set.