Aleph
by Jeff Williams
Where there is the Infinite there is joy. There is no joy in the finite.
- The Chandogya Upanishad
No one shall expel us from the paradise which Cantor has created for us.
- David Hilbert
The book, or movie, A Beautiful Mind, has recently entranced us all with a view
into the troubled thoughts of Princeton mathematician and Nobel Laureate John
Nash. He is often compared with the painter, Vincent Van Gogh, another unhappy
spirit who spent much time in and out of mental institutions but never wavered
from his quest. Many of us have a weakness, even envy, for these people, driven
to madness by their single-minded search for truth and their determination to
declare it. Georg Cantor is another example.
He was born in Russia in 1845, but lived most of his life in Germany. Although he trained with some of the foremost mathematicians of the day, and obtained his doctorate in 1869, he was unable to secure a position at any of the prestigious research universities. Disappointed, he accepted a post at Friedrich's University in the small industrial town of Halle, almost midway between the two great university cities of Gottingen and Berlin. Time and again, he applied for positions in both of these places, always to be refused.
Cantor's work with mathematical set theory, which would eventually lead to
a revolution in our understanding of the infinite, began with the question:
How do I count the number of elements (members) in a set? The answer, he concluded,
was to associate each element in turn with the so called counting numbers, 1,
2, 3, 4, etc. Thus consider the set containing a club, a diamond and a heart:
{ 
, 
, 
}. Cantor would say that this set contains 3 elements
because we can pair them off with the counting numbers in the following fashion:
But take the set of counting numbers themselves: { 1, 2, 3, 4, 5, . . . }. How many elements does this set contain?
This question is more subtle. The ellipsis ". . ." after the 5 means that the
set "goes on forever." Thus 5 will be followed by (an unwritten but understood)
6, which will be followed by (an unwritten but understood) 7, and so on. Cantor's
breakthrough came in taking this question seriously and giving it an answer.
He gave a name to the answer. Most mathematicians like to use Greek letters
to name special numbers. For example the Greek letter pi, which is written
π. Instead of Greek, Cantor used the Hebrew letter aleph, which is written
. And he denoted the answer
to the question "How many elements are there in the set of counting numbers
{ 1, 2, 3, 4, 5, . . . }?" by 0
. Notice the subscript 0. Cantor realized that the number 0
was, in some intuitive sense, "infinite." Other mathematicians were willing
to accept (indeed, it was obvious) that the total number of counting numbers
was infinite. What they were unwilling to accept was Cantor's next hypothesis.
Cantor maintained that the "infinitely large" number 0
was merely the first in a succession of infinitely large numbers, each one bigger
than the one before:
Thus 1 > 0 , and 2 > 1 , etc. Anyone
who finds this strange, or even ridiculous, can console themselves with the
knowledge that many leading mathematicians of Cantor's time also found this
ridiculous. Most notable amongst them was Leopold Kronecker who presided over
German mathematics from his chair at the University of Berlin. His ferocious
attacks against Cantor's work would eventually drive Georg Cantor into the Halle
Nervenklinic, the town's lunatic asylum.
If we think of an ordinary number and call it n (perhaps n is
7 or 92) then it is obvious that n + 1 is not equal to n. In fact
n + 1 will be greater than n. This simple fact does not hold for
Cantor's "infinitely large" number 0.
In fact 0 +
1 = 0. How
can this be? If we include an extra element (call it 
,
say) in the set of counting numbers to obtain the set { ,
1, 2, 3, . . . }, how can we say that the latter set still has the same number
of elements (namely 0)
as the original set of counting numbers { 1, 2, 3, . . . }? "Easy," said Cantor
- because we can pair off elements in both sets and have nothing left over:
This trick only works because both sets are infinite.
Another approach to understanding the fact that an infinite set can be put into pairwise correspondence with a subset of itself is by way of Hilbert's Hotel - named after one of the most influential mathematicians of the day, David Hilbert, Professor at the University of Gottingen. Hilbert liked to tell the story of a hotel with an infinite number of rooms. You arrive at the hotel one night, only to be told that it is full.
"But there are an infinite number of rooms," you object.
"True," says the manager. "But they are all full. We have an infinite number of guests."
You now think up a clever ruse. "Very well," you tell the manager. "Move the person in Room 1 to Room 2. Move the person in Room 2 to Room 3. And so on." Since there are infinitely many rooms, everyone can be accommodated. "Room 1 will then be vacant," you explain, "and I can move in."
One can add any finite number n of elements to an infinite set of size
0 without increasing
the size of the resulting set. This idea can be expressed by generalizing the
equation 0
+ 1 = 0 to
read 0 + n
= 0. Cantor
proved many other unexpected results, such as 0
+ 0 = 0,
and 0 x n =
0, and 0
x 0 = 0.
These equations lead one to wonder whether it is possible to generate answers
other than 0.
This can be done, but only after introducing the concept of a real number.
The numbers discussed so far have been counting numbers, which are whole
numbers, not decimals such as 6.3 or 0.352678896765 or the square root of 2
(which is difficult to write down because its digits go on for ever in a seemingly
haphazard fashion). The latter numbers are quite ordinary, everyday numbers
and they are usually called real numbers. Unlike the counting numbers,
which are separate from each other (in fact one unit apart), the real numbers
"flow together" forming a continuum. Cantor denoted the total number of real
numbers by the letter C. Clearly, C is infinite.
The question that Cantor wondered about was: "Is C more infinite
than 0?" Put
another way: "Is the number of real numbers greater than the number of counting
numbers?" He used his pairwise comparison method to answer this question.
Cantor began by imagining that the set of real numbers (in fact he focused just on the real numbers between zero and one) could be paired off with the counting numbers, as shown in the illustrative example below. (The real numbers are written in no particular order).
1 - 0.3139054 . . .
2 - 0.7546891 . . .
3 - 0.2485521 . . .
4 - 0.5296115 . . .
5 - 0.5214859 . . .
etc, for ever.
If this were possible, then the number of real numbers would equal the number
of counting numbers, i.e. C would equal 0.
However, Cantor showed that such a list is not possible because there would
always be real numbers that were missing. He constructed an example of this
by working down the list, choosing the first digit after the decimal point,
and then the second digit after the decimal point, and then the third, and so
on. These digits are shown above in bold type. They are 35868.
. . (Of course, the digits go on forever). He then increased each digit by one
and put back the decimal point: 0.46979. . . This number differs (in
at least one digit) from every number on the list and hence is not on the list.
If one decides to add this number to the list, the same argument can be used
to conclude that there are other numbers that are still not on the list. Hence
the original assumption that such a list can be constructed is false. Thus the
real numbers are more numerous than the counting numbers, meaning that C
> 0.
Cantor wished to publish this important result in Crelle's Journal for Pure and Applied Mathematics, one of the leading mathematics journals of the day. Unfortunately, the editorial board was dominated by professors from the University of Berlin, and by Kronecker in particular. However, Cantor reasoned that Kronecker would only read the title and abstract, looking for some mention of Cantor's objectionable infinities. If no mention was found, Kronecker would most likely pass the paper on to be refereed by a more junior faculty member. Cantor chose an unprovocative title for his paper, calling it "On a Property of the Collection of All Real Algebraic Numbers." The work that he was so proud of was tucked away in the body of the paper, hidden amongst a mass of more conventional mathematics. The trick worked. Cantor's paper was published in Crelle's Journal later that same year.
Cantor now asked the question that would, one day, take centre-stage in the
world of mathematics, that would absorb the rest of his life, and would eventually
bring him to a tragic end. Question: Given that C is an infinity
that is bigger than 0,
is C equal to 1,
or equal to one of the larger infinities in the hierarchy (perhaps 2,
or 3)? Put
simply: Is the equation
C =true or not? The declaration that this statement is true is called the Continuum Hypothesis.
Cantor oscillated to-and-fro, sometimes believing in the Continuum Hypothesis
and trying to prove it true, other times believing that the hypothesis was incorrect
and so trying to prove it false. The escalating battles with Kronecker, who
by this time was calling Cantor a charlatan and describing his work as humbug,
and Cantor's own frustrating lack of progress, led to his first nervous breakdown
in 1884. For two months, he became deeply depressed and unable to do any mathematics.
Cantor was to suffer such breakdowns throughout the rest of his life, and was
regularly admitted to the Nervenklinic in Halle. His hospitalizations became
increasingly frequent, and often occurred at times when he had been working
particularly hard on the Continuum Hypothesis. Sadly, Cantor began to drift
away from reality just as his work was starting to gain acceptance. German mathematicians
Ernst Zermelo and David Hilbert began to champion Cantor's ideas. Hilbert drew
up a list of ten problems that he considered of central importance to the development
of mathematics in the upcoming century. He presented them to the International
Mathematical Congress in Paris in 1900. First on the list was Cantor's Continuum
Hypothesis. Cantor was not present to hear himself acknowledged as the person
whose insights into the infinite had fingered the problem whose resolution was
essential before mathematics could be put onto a sound and logical footing.
The last word on the Continuum Hypothesis was to wait until 1963. Strange as it may seem, it was demonstrated to be neither provable nor disprovable. Stanford University mathematician Paul Cohen showed this to be the case by a clever argument based upon the so-called Godel incompleteness theorem.
And so mathematicians nowadays have two options. They can assume, as most do, that the Continuum Hypothesis is true, and then develop their work using a set of axioms very similar to those developed by Cantor. Or they can study what are called non-Cantorian transfinite numbers, where the Continuum Hypothesis is assumed to be false. Further details can be found at http://mathworld.wolfram.com/ and searching for aleph.
None of this was to be known to Georg Cantor. In June 1917, he entered the Halle Nervenklinik for the last time. The man who had created the first truly original mathematics since the Greeks died there on the sixth of January 1918.
His ambition to hold a professorship in one of Germany's famous universities
was never realized. Ironically, Friedrich's University in Halle has now become
one of Germany's most famous universities. It is famous because Georg Cantor
was once a professor there.