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

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