Axiomatic Foundation of Set Theory and its Non-Standard Applications: A Systematic Review

Abstract

This systematic review explores the axiomatic foundations of set theory and examines its non-standard applications across various fields. Set theory, primarily based on the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC), serves as the cornerstone of modern mathematics. This review delves into the historical development of set theory, highlighting significant milestones and contributions from notable mathematicians. We investigate the robustness and limitations of ZFC, addressing paradoxes and alternative axiomatic systems such as Von Neumann-Bernays-Gödel (NBG) and New Foundations (NF). Furthermore, the review identifies and analyzes non-standard applications of set theory in areas including computer science, linguistics, and philosophy. Particular attention is given to how set-theoretic concepts underpin programming language theory, formal semantics, and the modelling of infinite processes. We also explore the implications of set theory in the foundations of mathematics and its philosophical interpretations. Through this comprehensive review, we aim to provide a deeper understanding of both the theoretical and practical dimensions of set theory, offering insights into its enduring significance and versatility in addressing complex problems beyond traditional mathematical boundaries.