Set Countability

U ovome radu promatramo skupove i njihovu prebrojivost, kroz naivnu teoriju skupova. Najprije se kratko bavimo istobrojnim, odnosno ekvipotentnim skupovima, a zatim prelazimo na beskonačne skupove. Uz pomoć beskonačnih skupova gradimo prebrojive i neprebrojive skupove te se njima bavimo do kraja rada. Uvodimo pojam kardinalnosti za proizvoljan skup, prolazimo kroz glavna pravila aritmetike kardinalnosti te pokazujemo neka karakteristična svojstva. Zatim navodimo Cantor-Schröder-Bernsteinov teorem i potkrepljujemo ga primjerima koji ilustriraju njegovu izravnu primjenu. Najveći fokus stavljen je na dokaze svojstava i pravila računanja s kardinalnostima te brojnu primjenu istih na skupovima iz različitih grana matematike. Sadržajno su postavljeni temelji za daljnji razvoj aksiomatske teorije skupova koja precizno potvrđuje i razjašnjava ono što ostaje "visjeti u zraku" u pristupu naivne teorije skupova.In this paper, we will observe sets and their countability through naive set theory. Firstly, we will briefly deal with equipotent sets, and then proceed with infinite sets. With the help of infinite sets, we will form countable and uncountable sets and focus on them until the end of the paper. We will also introduce the concept of cardinality for an arbitrary set, go through the main rules of cardinality arithmetic, and present the main features. Then we will state Cantor-Schröder-Bernstein’s theorem and corroborate it with examples which illustrate its direct application. The greatest focus is placed on the proofs of the properties, rules of computation with cardinals as well as on their various use on sets from different branches of mathematics. In terms of content, the foundations have been laid for the further development of axiomatic set theory, which accurately confirms and clarifies what remains "hanging in the air" in naive set theory