About the book
Sets theory provides the basis for the developments of all mathematics. In it, several concepts are established and provide the fundamental structure for the creation of a whole construction of mathematical theory. For example, the concept of infinity, which plays a fundamental role, has always been a source of controversy among mathematicians and philosophers. This type of problem, which causes divergences and raises questions, is responsible for the establishment of new theories and sets.
The concepts of sets emerges naturally in the human mind, being an abstract concept created by our human reasoning. An example is the set of natural numbers, which can be seen as an immediate consequence of the notion of sets.
However, when the human mind creates abstract concepts, such as numbers, which are often associated with concrete concepts; However, in some cases, it is impossible to count the number of elements of a set, even knowing that it has a finite amount of elements. The number of residences in the world or stars in the sky is a finite number, no matter how amazingly large. Thus, one of the fundamental questions of the sets theory arises that is "Does the order we use to count things not affect the result?".
There are many questionings of this kind that arise. Despite the protests of several lines of research, such as constructivists and intuitionists, the pillars of the theory of sets developed by Cantor, Zermelo, Frankel, and Von Neumann, served as the basis for the rationale of mathematics. The fundamental idea is to use sets to define all mathematical objects as sets. Everything is set!
This book aims to show new advances and representations in sets theory, asking questions still open and explaining complex axioms. Applications from sets theory to real-world representation problems can also be presented. Philosophical problems and new modeling can also be addressed.