Hausdorff paradox

In this diploma paper we will deal with the Hausdorff paradox, which says, that the sphere without a finite number of points is paradoxical. First we cover the concept of an infinite set. Then we focus on the axiom of choice, which is essential for prooving the Hausdorff paradox. The purpose of the next chapter on rotation groups is just a revision of knowledge, crucial for understanding further theorems and proofs. In Chapter 4 we talk about a congruence of figures based on the set theory. Before defining the paradoxical set, we revise the concept of a group action. We also deal with the free group of rank 2, which is required to prove the paradox. Chapter 6 is about paradoxes of the plane, in Chapter 7 we finally present a proof of the Hausdorff paradox