Georg Ferdinand Ludwig Philipp 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