GEORG CANTOR

GEORG CANTOR (1845 -1918)

Georg Ferdinand Ludwig Philip Cantor was born in St. Petersburg, Russia in 1845. His parents were of Danish decent and he moved with them to Frankfurt, Germany in 1856. His father was a Jew who was converted to Protestantism and his mother was born a Catholic. Georg was a very religious Christian and had a deep interest in medieval theology which became very influential in his motivation to investigate the Infinite. His father suggested that Georg become an engineer, but Georg decided to concentrate on philosophy, physics and mathematics. He studied at Zurich, Gotingen and Berlin and received his doctorate from Berlin in 1869. Georg taught at the University of Halle from 1869 until 1905. He died in a mental hospital in Halle in 1918.

Georg Cantor is famous for his work in developing set theory. He published his first work in 1874 on set theory. This work was very controversial because it differed radically from the mathematics of that time. Today the idea of sets is widely accepted and used by almost every branch of mathematics.

Cantor worked with both infinite and finite sets. He suggested that the word infinite had two meanings. The first is a magnitude which increases beyond any indicated limit. Cantor called this the improper set because the magnitude is always finite although variable. The second meaning of infinite is that of the proper or completed infinite. This use of the word relates to the idea of real numbers. The conclusion was reached that real numbers could not be defined without reference to a completed infinite set which is what led Cantor to investigate the general theory of sets.

In 1895, Cantor defined a set as any collection into a whole M of definite and separate objects m. The separate objects m are called elements of M. Examples of a set include a set of dishes, a set of rational numbers between zero and one and a set consisting of the even prime number, the sun and Socrates. He used the following rules to further define sets:

Two sets M and N are equal if every element of M is an element of N, and vice versa.
A set M is a subset of a set N only if every element of M is an element of N.
If set M is equal to set N then it is also a subset of N.
If M is a subset of N and there is at least one element of N which is not a member of M, then M is called a proper subset.
Sets may contain one, several or no elements.
A set without elements is called the empty set. All empty sets are equal and the empty set is also a subset of any set.
Two sets M and N are equivalent if they can be put into a one-to-one correspondence; that is, if it is possible by some rule to associate each element of M with exactly one element of N, and vice versa.

EXAMPLE:

If M is the set of primes greater than two and less than twelve and N is the set of even numbers less than ten, there exists a one-to-one correspondence as shown below:

PRIME NUMBERS		EVEN NUMBERS
3	corresponds to	2
5	corresponds to	4
7	corresponds to	6
11	corresponds to	8

Please note that this does not imply that 3=2, 5=4, etc. This is simply stating that the 3 from the prime numbers set corresponds to the place where the 2 is in the even numbers set and the number of elements in set M is equal to the number of elements in N (cardinality of sets are equal).

The concept of equivalence allows us to define the concepts of finite and infinite sets. This study led to the conclusion that an infinite set is equivalent to a proper subset of itself, but a finite set is not. For example, the set of natural numbers (infinite set) is equivalent to a proper subset of itself. Examples of subsets which prove this are:

If the subset is the natural numbers greater than one then there is a one-to-one correspondence between the infinite set of all natural numbers and the subset as shown below:

ALL NATURAL #s		SUBSET OF NATURAL #s > 1
1	corresponds to	2
2	corresponds to	3
3	corresponds to	4
4	corresponds to	n + 1

if the subset is all even numbers then there is a one-to-one correspondence between the infinite set of all natural numbers and the subset as shown below:

ALL NATURAL #s		SUBSET OF EVEN #s > 1
1	corresponds to	2
2	corresponds to	4
3	corresponds to	6
4	corresponds to	n + 1

Cantor also defined a set which is equivalent to the set of all natural numbers as a denumberable set. Cantor determined that although not all infinite sets are denumberable, the set of all positive and negative rational numbers is denumberable and the set of all algebraic numbers (roots of polynomials with integer coefficients) is denumberable.

The proof which Cantor used to determine that there is a one-to-one correspondence between the natural numbers and the positive rationals is shown below. FIGURE 1 shows the ordering which was used:

1/1

2/1

3/1

4/1

5/1

6/1

7/1

...

1/2

2/2

3/2

4/2

5/2

6/2

7/2

...

1/3

2/3

3/3

4/3

5/3

6/3

7/3

...

1/4

2/4

3/4

4/4

5/4

6/4

7/4

...

1/5

2/5

3/5

4/5

5/5

6/5

7/5

...

1/6

2/6

3/6

4/6

5/6

6/6

7/6

...

1/7

2/7

3/7

4/7

5/7

6/7

7/7

...

FIGURE 1

FIGURE 2 shows the sequence which is suggested by FIGURE 1:

1/1, 2/1, 1/2, 1/3, 2/2, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 2/4, 3/3, 4/2, 5/1, 6/1, 5/2, 4/3, 3/4, ...
FIGURE 2

If each fraction is reduced to its lowest form and we cancel out any repetitions, then we are left with a sequence of positive rationals that contains each positive rational exactly once. This leaves a one-to-one correspondence of the natural numbers and the positive numbers as shown in FIGURE 3:

1		1/1,	2		2/1,	3		1/2,	4		1/3,	5		3/1,	6		4/1,	7		3/2,	8		2/3,
9		1/4,	10		1/5,	11		5/1,	12		6/1,	13		5/2,	14		4/3,	15		3/4,...

FIGURE 3

This proves that the positive rational numbers are denumerable.

One of Cantor's most important achievements was to prove that not all infinite sets are equivalent. This was important because it was believed that when one set is "infinitely big", that no set can be any larger. Cantor proved this is false by showing that the set of real numbers is not denumberable. The proof is more indirect than the proof shown above but it follows the same logic so it will not be shown here. The argument which Cantor used to prove the nondenumerability of the real numbers is called the diagonal process.

Cantor also defined a cardinal number which was used in his work on levels of infinity. The cardinal number of a set M is what M has in common with all sets equivalent to M. The concept of cardinal numbers involves only correspondence which means that cardinal numbers indicate how many members are in a set.

Cantor worked on determining how many different types of infinite sets there are. He named the cardinal number of the set of all natural numbers aleph-null (0 in symbol form) which is the smallest infinite cardinal. He defined the concept of greater than as follows: a set M is greater than a set N if and only if N is equivalent to a subset of M but M is not equivalent to any subset of N. The cardinal number of the real numbers is defined as x and is greater than 0 .

Cantor went on to prove that there are cardinals greater than 0. He did this by proving a general proposition which is known as Cantor's theorem which states: for any set M there exist sets larger than M; in particular the set of all subsets of M is larger than M. This follows from the fact that for any finite set with n elements, the number of subsets is 2**n. For example:

(a). the empty set has one subset, itself (2**0 = 1)

(b). a set with one element has two subsets (2**1 = 2)

(c). a set with two elements has four subsets (2**2 = 4)

Cantor's theorem was extended to include transfinite cardinals which are the cardinals of infinite sets. This gives an infinite hierarchy of transfinite cardinals as shown below:

0,
2**

0,
2**(2**

0),
2**(2**(2**

0))

FIGURE 4

The question than arises: Is there a transfinite cardinal which is greater than 0 but less than 2**0, ? This is called the continuum problem because it asks where the cardinal number of the continuum (real number line) belongs in the sequence of transfinite cardinals. Cantor believed that no cardinal existed which met this condition and he tried very hard to prove his conjecture. Unfortunately, his efforts caused him to suffer a mental breakdown so he was never able to resolve the continuum problem. It was proved in 1963 by Paul Cohen that the continuum problem could not be proved or refuted.

Cheryl Haefner

[Ed. note] Cantor's problem was that he felt that unless he could prove conclusively that "intermediate" transfinite cardinals did not exist, the entire foundation of Set Theory would be invalid mathematically. Cohen's work simply proved that the dilemma was not decideable, and this conclusion combined with Kurt Godel's Incompleteness Theorem essentially verified that Set Theory was consistent mathematically. One can only look back with sadness and regret that such a brilliant mind could become so tormented and obsessed with a single fixation that would lead ultimately to a breakdown.

Robert H. Orr