Probability theory has long provided a rigorous framework for quantifying uncertainty, yet its extension to infinite sets introduces profound conceptual challenges and opportunities. Contemporary ...

Transfinite set theory encompasses the rigorous study of infinite hierarchies, particularly those structured by ordinals and cardinals. This field has been instrumental in deepening our comprehension ...

With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary. The boundary does not pass between ...

The original version of this story appeared in Quanta Magazine. All of modern mathematics is built on the foundation of set theory, the study of how to organize abstract collections of objects. But in ...

Mathematicians are finding inevitable structures in sufficiently large sets of integers. Despite that, Furstenberg’s paper left a lasting imprint on mathematics. His new argument contained a kernel of ...