R 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

B equivalent to a subset of A then A and B are equivalent. This theorem

was also proved by Felix Bernstein and by Schroder.

The rest of his life was spent in and out of mental institutions and

his work nearly ceased completely. Much too late for him to really

enjoy it, his theory finally began to gain recognition by the turn of

the century. In 1904, he was awarded a medal by the Royal Society of

London and was made a member of both the London Mathematical Society

and the Society of Sciences in Gottingen. He died in a mental

institution on January 6, 1918.

Today, Cantor’s work is widely used in the many fields of mathematics.

His theory on infinite sets reset the foundation of nearly every

mathematical field and brought mathematics to its modern form.

II. Infinity

Most everyone is familiar with the infinity symbol . How many is

infinitely many? How far away is “from here to infinity”? How big is

infinity?

We can’t count to infinity. Yet we are comfortable with the idea that

there are infinitely many numbers to count with: no matter how big a

number you might come up with, someone else can come up with a bigger

one: that number plus one–or plus two, or times two. There simply is

no biggest number.

Is infinity a number? Is there anything bigger than infinity? How about

infinity plus one? What’s infinity plus infinity? What about infinity

times infinity? Children to whom the concept of infinity is brand new,

pose questions like this and don’t usually get very satisfactory

answers. For adults, these questions don’t seem to have very much

bearing on daily life, so their unsatisfactory answers don’t seem to be

a matter of concern.

At the turn of the century Cantor applied the tools of mathematical

rigor and logical deduction to questions about infinity in search of

satisfactory answers. His conclusions are paradoxical to our everyday

experience, yet they are mathematically sound. The world of our

everyday experience is finite. We can’t exactly say where the boundary

line is, but beyond the finite, in the realm of the transfinite, things

are different.

Sets and Set Theory

Cantor is the founder of the branch of mathematics called Set Theory,

which is at the foundation of much of 20th century mathematics. At the

heart of Set Theory is a hall of mirrors–the paradoxical infinity.

Georg Cantor was known to have said, “I see it, but I do not believe it,

” about one of his proofs.

The set is the mathematical object which Cantor scrutinized. He defined

a set as any collection of well-distinguished and well-defined objects

considered as a single whole. A collection of matching dishes is a set,

as well as a collection of numbers. Even a collection of seemingly

unrelated things like, television, aardvark, car, 6} is a set. They

are well-defined and can be distinguished from one another.

Sets can be large or small. They can also be finite and infinite. A

finite set has a finite number of members. No matter how many there are,

given enough time, you can count them all. Cantor’s surprising results

came when he considered sets that had an infinite number of members.

Sets such as all of the counting numbers, or all of the even numbers

are infinite sets.

In order to study infinite sets, Cantor first formalized many of the

things that are intuitive and obvious about finite sets. At first, it

seems like these formalizations are just a whole lot of trouble, a way

of making simple things complicated. Because the formalisms are clearly

correct, however, they provide a powerful tool for examining things

that are not so simple, intuitive or obvious.

Cantor needed a way to compare the sizes of sets, some method for

determining whether sets had the same number of members. If two sets

didn’t have the same number of members, he needed a method for telling

which one was larger. Of course this is simple for finite sets. You

count the members in both sets. If the number is the same, they are the

same size. If the number of members in one set is greater than the

number of members in the other, then that set is larger.

You can’t count the members in an infinite set, though, so this method

won’t work for comparing their sizes. If there are two infinite sets,

one must need some other way to tell if one is larger.

The formal notion that Cantor used for comparing sizes of sets is the

idea of a one-to-one correspondence. A one-to-one correspondence pairs

up the members of one set with the members of another. Sets which can

be matched to each other in this sense are said to have the same

cardinality. We could pair up the elements of the imaginary set

television, aardvark, car, 6} with the numbers 1,2,3,4}. It is

possible to do this so that one member of each set is paired up with

one member of the other, no member is left out, and no member has more

than one partner. Then we can be sure that the set1,2,3,4} has the

same number of members as the set television, aardvark, car, 6}.

one-to-one correspondence:

television, aardvark, car, 6}

1, 2, 3, 4}

So, what is bigger? infinity+X? infinity+infinity ? Or

infinity(infinity)? To calculate which is bigger cantor used sets and

one-to-one correspondence.

These one-to-one correspondence sets show that even though we add an

unknown variable, multiply by two, and square a set, the upper and

lower sets still remain equal. Since we will never run out of numbers

any correspondence set with two infinite values will be equal. All

these sets clearly have the same cardinality since its members can be

put in a one-to-one correspondence with each other on and on forever.

These sets are said to be countably infinite and their cardinality is

denoted by the Hebrew letter aleph with a subscript nought, .

OTHER INFINITIES

Cantor thought once you start dealing with infinities, everything is

the same size. This did not turn out to be the case. Cantor developed

an entire theory of transfinite arithmetic, the arithmetic of numbers

beyond infinity. Although the sizes of the infinite sets of counting

numbers, even numbers, odd numbers, square numbers, etc., are the same,

there are other sets, the set of numbers that can be expressed as

decimals, for instance, that are larger. Cantor’s work revealed that

there are hierarchies of ever-larger infinities. The largest one is

called the Continuum.

Some mathematicians who lived at the end of the 19th century did not

want to accept his work at all. The fact that his results were so

paradoxical was not the problem so much as the fact that he considered

infinite sets at all. At that time, some mathematicians held that

mathematics could only consider objects that could be constructed

directly from the counting numbers. You can’t list all the elements in

an infinite set, they said, so anything that you say about infinite

sets is not mathematics. The most powerful of these mathematicians was

Leopold Kronecker who even developed a theory of numbers that did not

include any negative numbers.

Although Kronecker did not persuade very many of his contemporaries to

abandon all conclusions that relied on the existence of negative

numbers, Cantor’s work was so revolutionary that Kronecker’s argument

that it “went too far” seemed plausible. Kronecker was a member of the

editorial boards of the important mathematical journals of his day, and

he used his influence to prevent much of Cantor’s work from being

published in his lifetime. Cantor did not know at the time of his death,

that not only would his ideas prevail, but that they would shape the

course of 20th century mathematics.