Set Theory

In previous chapters we introduced mathematical structures, and we followed with a detailed description of basic number structures. Now it is time to look at structures in general. The classical number structures fit the definition: a set with a set of relations on it. But what about other structures? Are they all sets? Can a set of relations always be associated with them? Clearly not. Not everything in this world is a set. I am a structured living organism, but I am definitely not a set. Nevertheless, once a serious investigation of set theory got underway, it revealed a fantastically rich universe of sets, and it showed that, in a certain sense, every structure can be thought of a set with a set of relations on it. To explain how it is possible, we need to get a closer look at sets. As we saw in the previous chapter, the deceptively simple intuitive concept of set (collection) leads to unexpected consequences when we apply the well understood properties of finite collections to infinite sets. The role of axiomatic set theory is to provide basic and commonly accepted principles from which all other knowledge about infinity should follow in a rigorous fashion. There are many choices for such theories. In this chapter we will discuss the commonly used axioms of Zermelo and Fraenkel.