Cantor

Georg Ferdinand Ludwig Philipp Cantor

Born: 3 March 1845 in St Petersburg, Russia
Died: 6 Jan 1918 in Halle, Germany

Georg Cantor founded set theory and introduced the concept of infinite numbers with his discovery of cardinal numbers. He also advanced the study of trigonometric series.

Cantor attended the University of Zürich for a term in 1862 but then went to the University of Berlin where he attended lectures by Weierstrass, Kummer and Kronecker. He received his doctorate in 1867 from Berlin and accepted a position at the University of Halle in 1869, where he remained until he retired in 1913. In 1885 he built a house on Händelstrasse, a street named after the German composer Handel.

Cantor founded set theory and introduced the mathematically meaningful concept of infinite numbers with his discovery of transfinite numbers. He also advanced the study of trigonometric series and was the first to prove the nondenumerability of the real numbers.

His first papers (1870-1872) showed the influence of Weierstrass's teaching, dealing with trigonometric series. In 1872 he defined irrational numbers in terms of convergent sequences of rational numbers. In 1873 he proved the rational numbers countable, i.e. they may be placed in 1-1 correspondence with the natural numbers.

A transcendental number is an irrational number that is not a root of any polynomial equation with integer coefficients. Liouville established in 1851 that transcendental numbers exist. Twenty years later Cantor showed that in a certain sense 'almost all' numbers are transcendental.

Closely related to Cantor's work in transfinite set theory was his definition of the continuum.
Cantor's work was attacked by many mathematicians, the attack being led by Cantor's own teacher Kronecker. Cantor never doubted the absolute truth of his work despite the discovery of the paradoxes of set theory. He was supported by Dedekind, Weierstrass and Hilbert, Russell and Zermelo. Hilbert described Cantor's work as

the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity .

A major event planned in Halle to mark Cantor's 70 th birthday in 1915 had to be cancelled because of the war. Russell describes the eccentric Cantor vividly in his Autobiography, and letters from Cantor to Russell are reproduced there. Cantor died in a psychiatric clinic in Halle in 1918.

Cantor was given an honorary degree by the University of St Andrews in 1911.

References:

  1. Dictionary of Scientific Biography
  2. Biography in Encyclopaedia Britannica
  3. J W Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Cambridge, Mass, 1979).
  4. A Fraenkel, Georg Cantor, Jahresberichte der Deutschen Mathematiker vereinigung 39 (1930), 189-266.
  5. I Grattan-Guinness, Towards a biography of Georg Cantor, Ann. of Sci. 27 (1971), 345-391.
  6. J Ferreirós, On the Relations between Georg Cantor and Richard Dedekind, Historia Mathematica 20 (1993), 343-363.
  7. K Richter, Cantor, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).
  8. W Purkert and H J Ilgauds, Georg Cantor (Leipzig, 1985).
  9. H Meschkowski, Georg Cantor : Leben, Werk und Wirkung (Mannheim, 1983).
  10. W Purkert and H J Ilgauds, Georg Cantor 1845-1918 (Basel, 1987).
  11. A Kertesz, Georg Cantor, 1845-1918 : Schopfer der Mengenlehre, Acta historica leopoldina 15 (1983).
  12. D Stander, Makers of modern mathematics : Georg Cantor (1989).
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