This thesis is divided into four chapters. The first chapter deals with finite and infinite countable sets in which we defined some notions and stated and proved some statements which represent the foundation for the further understanding. In the second chapter we are dealing with cardinal numbers. To define cardinal number, we are proving that set of all sets does not exist and we are introducing the notion of a class. As important statement of this chapter, we are proving the Cantor-Schröder-Berstein theorem and providing basic operations with cardinal numbers. In the third chapter we are defining some basic mathematical notions and proving important statements from the field of mathematical analysis which are of great importance for this chapter. In the end, the forth chapter starts with the definition of a binary relation. This chapter deals with well-ordered sets and comparability of cardinal numbers which is especially mentioned in the last important statement of this thesis.