George Cantor

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Georg Ferdinand Ludwig Philipp Cantor was born on March 3, 1845, in Saint Petersburg to Georg Waldemar Cantor and Maria Anna Bohm.

Cantor was brought up as a staunch Protestant and inherited the love for the arts from his parents.

He is said to have been an outstanding violinist himself.

Cantor’s father was German and his mother was Russian and a Roman Catholic.

Cantor had a private tutor from a very young age and attended primary school in St. Petersburg.

In 1856, when Cantor was eleven years old, his family moved to Germany, although Cantor was never at ease in this country.

Cantor studied at the Gymnasium here and graduated with an outstanding report in 1860.

His tutors began to notice that he was brilliant in mathematics, particularly in trigonometry.

After the Gymnasium, he entered the Polytechnique of Zurich in 1862, where he studied mathematics. With his parent’s approval, he studied there for a couple of years, till his study was cut short due to his father’s death in 1863.

After the death of his father, Cantor moved to the University of Berlin where he befriended Hermann Schwarz and attended lectures by Kronecker, Weierstrass, and Kummer.

He also studied at the University of Gottingen over summer and completed his first dissertation on the number theory named ‘De aequationibus secondi gradus indeterminatis’ in 1867.

He became the President of the society in 1865 and in 1868; he joined the Schellbach Seminar for mathematicians.

He was appointed at the University of Halle in 1869, and continued working on separate dissertations on the number theory and analysis.

It was here Cantor decided to probe further into the subject of trigonometry and began pondering over the uniqueness of the representation of a function of the trigonometric series, introduced to him by a senior, called Heine.

By 1870, Cantor solved the problem, proving the uniqueness of the representation, much to Heine’s astonishment.

In 1873, Cantor proved that rational numbers were countable and could be placed in correspondence to the natural numbers.

By the end of 1873, Cantor had proved that real and algebraic numbers were also countable.

He was promoted to Extraordinary Professor in 1872 and was appointed as a full professor in 1879. He was pleased with his achievement but he wanted the chair at a more prestigious university.

Throughout the 1880s and 1890s, he refined his set theory, defining well-ordered sets and power sets and introducing the concepts of ordinality and cardinality and the arithmetic of infinite sets.

What is now known as Cantor’s theorem states generally that, for any set A, the power set of A (i.e. the set of all subsets of A) has a strictly greater cardinality than A itself.

More specificially, the power set of a countably infinite set is uncountably infinite.

Cantor had few other mathematicians with whom he could discuss his ground-breaking work, and most were distinctly unnerved by his contemplation of the infinite.

During the 1880s, he encountered resistance, sometimes fierce resistance, from mathematical contemporaries such as his old professor Leopold Kronecker and Henri Poincaré, as well as from philosophers like Ludwig Wittgenstein and even from some Christian theologians, who saw Cantor’s work as a challenge to their view of the nature of God.

Cantor himself, a deeply religious man, noted some annoying paradoxes thrown up by his own work, but some went further and saw it as the wilful destruction of the comprehensible and logical base on which the whole of mathematics was based.