The concept of a set is very basic and natural, and has been used in mathematical writings since ancient times. But the theory of abstract sets, as objects to be studied for their own interest, was originated largely by Georg Cantor (1845 - 1918). Cantor studied set theory by accident after studying trigonometric series of the form:

1871 | he realized that a certain operation on sets of real numbers could be iterated more than a finite number of times; starting with a set one could form
1873 | Cantor proved that the set of all real numbers could not be put into one-to-one correspondence with integers
1874 | Cantor published his results
1879 and subsequent years | Cantor published a series of papers setting forth the general concepts of abstract sets and "transfinite numbers"
1889 | the use of the symbol to denote membership was initiated by Giuseppe Peano
1893 and 1903 | Gottlob Frege published a two-volume work in which he indicated how mathematics could be developed from principles that he regarded as being principles of logic
- Bertrand Russell informed Frege about a paradox regarding this principle
- Russel's paradox had a tremendous impact on the ideas of that time regarding the foundation of mathematics
1897 | Cesare Burali-Forti had observed a paradoxical predicament in Cantor's theory of transfinite ordinal numbers
1908 | first axiomatization of set theory was introduced by Ernst Zermelo
1925 | an axiomatization adding proper classes as legitimate objects was formulated by von Neumann

Modern Research on Set Theory

Present day research in set theory fall into two branch:

one branch involves investigating the consequences of new and stronger axioms
other branch involves the "metamathematics" of set theory, which is the study not of sets but of the workings of set theory itself: its proofs, its theorems, and its nontheorems.

Elements of Set Theory: Historical Notes

Modern Research on Set Theory

Further readings: