Axiomatic Set Theory and the Foundations of Large Cardinal Hypotheses

Since it provides the formal framework for a large portion of mathematical logic, topology, and abstract algebra, axiomatic set theory has been instrumental in the development of modern mathematics. Large cardinal hypotheses are strong extensions of standard set theory, especially when considering the hierarchy of infinities and the universe of sets. The axiomatic underpinnings of set theory are examined in this paper, along with an outline of its basic ideas. The role of large cardinal hypotheses in the investigation of the set-theoretic universe, their consequences for the structure of mathematical objects, and their relationships to set theory models are also examined.