Classical Set Theory: Theory of Sets and Classes
Taras Banakh
TL;DR
This work surveys Classical Set Theory through the lens of von Neumann–Bernays–Gödel (NBG), contrasting it with ZF/ZFC and emphasizing a finite axiomatization that accommodates both sets and classes. It develops foundations from naive notions and paradoxes (Berry, Russell) to a rigorous two-sorted framework, covering language, axioms, basic constructions, order theory, and transfinite dynamics, culminating in Gödel's constructible universe $\mathbf L$. Core contributions include the systematic treatment of class existence (Gödel's theorem), recursions and transfinite processes within a finite axiom system, and the construction of $\mathbf L$ as a robust model illustrating relative consistency phenomena. The practical impact lies in providing a coherent, class-friendly foundation for mathematics and category theory, with explicit links to ZF/ZFC and insights into constructibility and consistency arguments.
Abstract
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--Gödel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.