Question

What is the application of set theory?

Answer 1

Set theory is a branch of mathematicallogic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

Answer 2

Set theory does not add anything to the problem-solving techniques. It is just a concept that makes exposition of other ideas easier, which require talking about collections and their properties thereof.

Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs , manifolds, rings and vector spaces can all be defined as sets satisfying various properties.
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second logic order.