Sets: finite and infinite

Abstract

Sets occur abundantly in mathematics and in daily life. But what is a set? Cantor (1845-1918) defined a set as a collection of all objects which have a certain property in common. Russell showed in 1902 that this assumption yields a contradiction, known as Russell’s paradox, and hence is untenable. In 1908 Zermelo (1871-1953) weakened Cantor’s postulate considerably and consequently had to add a number of additional axioms.We present the set theory of Zermelo-Fraenkel. Next we discuss relations and functions. We use the Hilbert hotel with as many rooms as there are natural numbers to illustrate a number of astonishing properties of sets which are equally large as the set N of the natural numbers. We shall discover that there are many sets which in a very precise sense are much larger than N. We shall even see that for any set V, finite or infinite, there is a larger set P(V), called the powerset of V. Amazingly, although all sets we experience in the world are finite, we are still able to imagine infinite sets like N and to see amazing properties of them. This reminds us of the statement by cardinal Cusanus (1400-1453) that in our pursuit of grasping the divine truths we may expect the strongest support of mathematics. Finally we point out that Kant was right that mathematical (true) propositions are not analytic, but synthetic, and that Russell and Frege’s logicism, stating that all of mathematics may be reduced to logic, is wrong. What may be true is that mathematics can be reduced to logic plus set theory.