You are probably already familiar with sets before you got involved with mathematics. Perhaps you collect Pokémon cards yourself or have a stamp collection. For a collector, duplicates do count in the collection. But in mathematics, we do not do this.

The everyday concept of a set allows for a collection to contain for instance both coins and stamps. You simply get a collection by taking a set of things together. In any case, to form a set, you first need to have objects, and they do not need to be related to each other: a monkey, a stamp, and an integer can certainly form the objects of a set. Within mathematics, it is more common to use objects 'of the same kind' or 'with similar values' in a set. Examples of this are the set of natural numbers, the set of integers, and the set of prime numbers.

Where a stamp collection or the collection of city names in England can only consist of a finite number of objects, this is not necessary in mathematics. The set of natural numbers, denoted by \(\mathbb{N}\), and the set of integers, denoted by \(\mathbb{Z}\), each consist of infinitely many objects. Since every natural number is also an integer, it follows that \(\mathbb{N}\) is contained in \(\mathbb{Z}\); we say that \(\mathbb{N}\) is a subset of \(\mathbb{Z}\).

If a set consists of only finitely many objects, then we can determine the size of the set, i.e., the number of distinguishable objects, by counting. But with infinite sets, there is a problem. Because \(\mathbb{N}\) is a subset of \(\mathbb{Z}\) that does not contain all objects from \(\mathbb{Z}\), intuitively \(\mathbb{N}\) seems smaller than \(\mathbb{Z}\). But intuition can be deceiving: in the table below, we number the integers with a natural number as a label.

In this setup, there is much to be said for the idea that \(\mathbb{N}\) and \(\mathbb{Z}\) are of equal size. The size of infinite sets is far from trivial and was systematically studied only at the end of the nineteenth century by Georg Cantor.

What the above introduction shows is that the concepts of object, set, and size of a set are not self-evident or easily defined. But even without precise definitions, we can manage with the concept of a set as a collection of objects, as long as we are careful enough. We engage in naive set theory, which usually works well in practice but also produces paradoxes. To truly avoid problems, we would need to address axiomatic set theory, developed by Zermelo and Fraenkel in the early twentieth century. But this falls outside this class.

Warning in advance: the course material introduces much terminology that is commonly used in mathematics.