Georg Cantor


I. Georg Cantor

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 and was the first to prove the nondenumerability of the
real numbers. Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg,
Russia, on March 3, 1845. His family stayed in Russia for eleven years until the
father\'s sickly health forced them to move to the more acceptable environment of
Frankfurt, Germany, the place where Georg would spend the rest of his life.
Georg excelled in mathematics. His father saw this gift and tried to push his
son into the more profitable but less challenging field of engineering. Georg
was not at all happy about this idea but he lacked the courage to stand up to
his father and relented. However, after several years of training, he became so
fed up with the idea that he mustered up the courage to beg his father to become
a mathematician. Finally, just before entering college, his father let Georg
study mathematics. In 1862, Georg Cantor entered the University of Zurich only
to transfer the next year to the University of Berlin after his father\'s death.
At Berlin he studied mathematics, philosophy and physics. There he studied under
some of the greatest mathematicians of the day including Kronecker and
Weierstrass. After receiving his doctorate in 1867 from Berlin, he was unable to
find good employment and was forced to accept a position as an unpaid lecturer
and later as an assistant professor at the University of Halle in1869. In 1874,
he married and had six children. It was in that same year of 1874 that Cantor
published his first paper on the theory of sets. While studying a problem in
analysis, he had dug deeply into its foundations, especially sets and infinite
sets. What he found baffled him. In a series of papers from 1874 to 1897, he was
able to prove that the set of integers had an equal number of members as the set
of even numbers, squares, cubes, and roots to equations; that the number of
points in a line segment is equal to the number of points in an infinite line, a
plane and all mathematical space; and that the number of transcendental numbers,
values such as pi(3.14159) and e(2.71828) that can never be the solution to any
algebraic equation, were much larger than the number of integers. Before in
mathematics, infinity had been a sacred subject. Previously, Gauss had stated
that infinity should only be used as a way of speaking and not as a mathematical
value. Most mathematicians followed his advice and stayed away. However, Cantor
would not leave it alone. He considered infinite sets not as merely going on
forever but as completed entities, that is having an actual though infinite
number of members. He called these actual infinite numbers transfinite numbers.
By considering the infinite sets with a transfinite number of members, Cantor
was able to come up his amazing discoveries. For his work, he was promoted to
full professorship in 1879. However, his new ideas also gained him numerous
enemies. Many mathematicians just would not accept his groundbreaking ideas that
shattered their safe world of mathematics. One of these critics was Leopold
Kronecker. Kronecker was a firm believer that the only numbers were integers and
that negatives, fractions, imaginaries and especially irrational numbers had no
business in mathematics. He simply could not handle actual infinity. Using his
prestige as a professor at the University of Berlin, he did all he could to
suppress Cantor\'s ideas and ruin his life. Among other things, he delayed or
suppressed completely Cantor\'s and his followers\' publications, belittled his
ideas in front of his students and blocked Cantor\'s life ambition of gaining a
position at the prestigious University of Berlin. Not all mathematicians were
hostile to Cantor\'s ideas. Some greats such as Karl Weierstrass, and long-time
friend Richard Dedekind supported his ideas and attacked Kronecker\'s actions.
However, it was not enough. Cantor simply could not handle it. Stuck in a third-
rate institution, stripped of well-deserved recognition for his work and under
constant attack by Kronecker, he suffered the first of many nervous breakdowns
in 1884. In 1885 Cantor continued to extend his theory of cardinal numbers and
of order types. He extended his theory of order types so that now his previously
defined ordinal numbers became a special case. In 1895 and 1897 Cantor published
his final double treatise on sets theory. Cantor proves that if A and B are sets
with A equivalent to a subset of B and