Who developed set theory?
Mathematical logic
Set theory is the study of sets, which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization, due to Zermelo (1908b), was extended slightly to become Zermelo–Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics.
Paradoxes of set theory
Before set theory was introduced, the notion of the "size" of a set had been problematic. It had been discussed by Galileo Galilei and Bernard Bolzano, among others. Are there as many natural numbers as squares of natural numbers when measured by the method of enumeration?