Set Theory Revisited: Related Theorems and Realms and Their Development

Abstract

The concept of set dates back to the beginning of counting, and logical concepts regarding classes have existed since the tree of Porphyry, which was created in the third century CE. In light of this, it is difficult to determine where the idea of a set came from in the first place. However, sets are not collections in that this word is commonly understood, nor are they classes in the sense that logicians understood them before the middle of the 19th century. This study further presents a comprehensive trail on the development of set theory by comparing and organizing the disciplines, theories, and hypotheses associated with this major mathematical invention. Additionally, the study also presents the research paradigms that are related to this invention. This result highlights the inherent limitations of formal axiomatic systems, which undermines the quest for a complete and self-contained foundation for mathematics solely based on set theory. Furthermore, the second incompleteness theorem proposed by Godel asserts that no consistent formal system, including set theory, is capable of proving its consistency of being consistent. Mathematicians have been prompted to investigate alternative foundational approaches and philosophical perspectives due to the incompleteness theorems proposed by Gödel. These theorems doubt the absolute certainty and completeness of mathematical knowledge derived from set theory.