Published Paper Details:

Mathematical Logic and Set Theory form the foundational pillars of modern mathematics, providing the necessary tools for rigorous reasoning, formal proof structures, and the development of advanced mathematical theories. This paper explores the historical evolution, fundamental concepts, and critical roles of these disciplines in shaping mathematics and its applications. We discuss their contributions to computer science, philosophy, linguistics, and theoretical physics, while also addressing contemporary challenges and extensions. Through a systematic review, this study highlights the indispensable nature of logic and set theory in both pure and applied mathematics. This paper delves into the profound and often understated significance of mathematical logic theory and set theory as the bedrock of modern mathematics. We explore their historical origins, fundamental concepts, and their pervasive influence across diverse mathematical disciplines and interdisciplinary applications. Beginning with the formalization of reasoning and the concept of collections, we demonstrate how these theories provide the necessary rigor, consistency, and universality for mathematical inquiry. From establishing the foundations of number systems and analysis to shaping the development of computer science and artificial intelligence, logic and set theory are shown to be not merely abstract constructs but essential tools for understanding and advancing scientific knowledge.