Set Theory

The mathematical logic that we’ve studied in the previous two chapters is foundational to all types of math. However, as demonstrated in the last few sections of Chapter 2, mathematicians rarely lay out all of the full, gory details when writing proofs. Instead, they rely on axioms, definitions, and previous theorems to work out the desired result. Occasionally, propositional logic may be used when doubts arise about the validity of a given proof, but that too is rare.

Something we’ve hinted at in the previous two chapters is that when we want to validate an argument, we start with some number of premises. Another way of stating this is that we start off with some initial collection, or set, of premises, and from those premises, we hopefully arrive at the desired conclusion. The group of premises we start off with may change from one argument to another, but in any case, we have a starting collection of premises no matter what argument we’re trying to build up.

The idea of a collection, or set, of objects underlies almost all of mathematics, whether that be a collection of premises (in mathematical logic), a collection of points (in geometry), a collection of possible outcomes for an experiment (in probability), or a collection of outputs for some given collection of inputs (mathematical relations). In this chapter, we start to define and work with these collections of objects, and what we can do with collections of objects in general. Even though mathematicians rarely describe all of the formal logic used in their arguments, any set-theoretic aspects are almost always explicitly laid out. Hence, getting a good understanding of Set Theory will be vital in learning to not only do mathematics, but to read mathematics as well.