Research
Cardinal Arithmetic
Cardinal arithmetic is a branch of set theory that extends the familiar arithmetic of natural numbers to the realm of infinite sets. It provides a framework for defining operations on cardinal numbers, representing the sizes of sets, and establishing relationships between them. Cardinal addition, multiplication, and exponentiation are the fundamental operations, each defined in terms of set-theoretic operations such as disjoint unions, Cartesian products, and sets of functions.
Forcing
Forcing is a powerful technique for constructing models of set theory and proving independence and consistency results. It provides a method for extending a given model of set theory by introducing a "generic" object, allowing for the creation of new models that satisfy certain properties not present in the original model. This technique was first used by Paul Cohen in 1963 to prove the independence of the Continuum Hypothesis and the Axiom of Choice from Zermelo-Fraenkel set theory (ZFC).
Set-theoretic Topology
Set-theoretic topology, a branch of mathematics, interweaves the concepts of set theory and general topology, exploring topological questions whose solutions extend beyond limitations of Zermelo-Fraenkel set theory (ZFC). Set-theoretic topology has delved into the study of diverse topological spaces, including Hausdorff spaces, compact spaces, and metrizable spaces, providing a rich framework for analyzing their properties and relationships.