1.15 Historical Note

NCERT Class 11 Mathematics for blind and visually impaired students made screen readable by Professor T K Bansal.

The modern theory of sets is considered to have been originated largely by the German mathematician Georg Cantor (1,845-1,918). His papers on set theory appeared sometimes during 1874 to 1897. His study of set theory came when he was studying trigonometric series of the form

\[a1 \sin x\ +\ a2 \sin 2x\ +\ a3\ \sin 3x\ +\ \dots\ \]

He published in a paper in 1874 that the set of real numbers could not be put into one-to-one correspondence with the integers. From 1879 onwards, he published several papers showing various properties of abstract sets.

Cantor’s work was well received by another famous mathematician Richard Dedekind (1,831-1,916). But Kronecker (1,810-1,893) castigated him for regarding infinite set the same way as finite sets. Another German mathematician Gottlob Frege, at the turn of the century, presented the set theory as principles of logic. Till then the entire set theory was based on the assumption of the existence of the set of all sets. It was the famous English Philosopher Bertand Russell (1,872-1,970 ) who showed in 1,902 that the assumption of existence of a set of all sets leads to a contradiction. This led to the famous Russell’s Paradox. Paul R. Halmos writes about it in his book ‘Naïve Set Theory’ that “nothing contains everything”.

The Russell’s Paradox was not the only one which arose in set theory. Many paradoxes were produced later by several mathematicians and logicians.

As a consequence of all these paradoxes, the first axiomatisation of set theory was published in 1908 by Ernst Zermelo. Another one was proposed by Abraham Fraenkel in 1922. John Von Neumann in 1925 introduced explicitly the axiom of regularity. Later in 1937 Paul Bernays gave a set of more satisfactory axiomatisation. A modification of these axioms was done by Kurt Gödel in his monograph in 1940. This was known as Von Neumann-Bernays (VNB) or Gödel- Bernays (GB) set theory.

Despite all these difficulties, Cantor’s set theory is used in present day mathematics. In fact, these days most of the concepts and results in mathematics are expressed in the set theoretic language.

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End of Chapter 1.