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.
Further readings:
Disclaimer: this is a summary of section 1.6 from the book "Elements of Set Theory" by Herbert B. Enderton, the content apart from rephrasing is identical, most of the equations are from the book and the same examples are treated. All of the equation images were screenshot from generated latex form using [typora](https://typora.io/).
