Georg Cantor (1845 - 1918)

Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg, Russia, on March 3, 1845, to Danish parents. His father was a merchant who had converted to Protestantism and his mother was a Roman Catholic, and came from a family of musicians.

As a child Cantor excelled in mathematics, but his father pressured him to train as an engineer - because the money was good.

However, Cantor’s natural abilities were so obvious that after several years of study his father finally agreed to let him study mathematics in Zurich, the capital of Switzerland. Just a year later his father died, and Cantor left Switzerland for the University of Berlin, where he specialised in physics, philosophy, and mathematics.

He later taught at the University of Halle, first as an unpaid lecturer but eventually becoming a full professor in 1879.

1872 saw Cantor start his major life’s work - the theory of sets and the concept of transfinite numbers. A transfinite number is one that is not finite - in other words it is bigger than any given finite number.

It was not until 1874 that his first paper was finally published, the same year he married Vally Guttman, with whom he would eventually have six children.

Publication in Crelle’s Journal (the famous mathematical magazine of the day) was delayed by Kronecker - his former professor at Berlin.

This was to be the start of a long and aggressive opposition by Kronecker, leading to Cantor’s appointment to the faculty at the University of Berlin being blocked and his work being publicly criticised. Richard Dedekind was one who consistently supported his ideas and his work.

From 1874 to 1879 Cantor tried to prove his work on set theory. However, the Catholic church also used his theory to try to prove the existence of God. Cantor went to great lengths to distance himself from all of this.

Cantor’s study of mathematics gained him many mathematical enemies, because it was new and revolutionary.

He shattered the existing understanding of infinity - which was largely to ignore it - and replaced it with a set of tools and concepts that established it as a mathematical entity like any other.

He showed that there were different types of infinity - in fact an infinite number of them - and then went on to explore the relationship between them.

His work can be seen as creating an arithmetic of the infinite. Today, Cantor’s work is widely accepted by the mathematical community.

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 Göttingen, in recognition of his achievements. He died on 6 January 1918.

Cantor’s mathematics

Cantor’s revolutionary idea of numbers was based on a concept of exact matching - matching (and therefore counting) one thing with exactly one other thing.

It is called one-one matching. Think of an infant’s tea party for 5 infants in which only 4 places have been set. They all cry!

Cantor realised that there are as many natural numbers as there are squares. Here is the one-one matching:

\matrix{1 \cr \big\updownarrow \cr 1} \qquad \matrix{2 \cr \big\updownarrow \cr {2^2=4}}\qquad\matrix{3 \cr \big\updownarrow \cr {3^2=9}}\qquad\matrix{4 \cr \big\updownarrow \cr {4^2=16}}\qquad\matrix{5 \cr \big\updownarrow \cr \cdots} \qquad \matrix{\cdots \cr \; \cr \;}

But now think about what this means: it says that a subset of a set has as many elements as the set itself ! Surely ridiculous - but not according to Cantor. It is just one of the strange properties of an infinite set.

There are different kinds of infinity as well. Think of a number line, say between 0 and 1 - does it contain the same number of points as there are Natural numbers? According to Cantor, it does not. Imagine otherwise, so that each number (point) on the line is written out in order like this:

1 \leftrightarrow 0.a_1a_2a_3a_4\cdots

2 \leftrightarrow 0.b_1b_2b_3b_4\cdots

3 \leftrightarrow 0.c_1c_2c_3c_4\cdots

where each natural number is matched to a number that starts with a zero and then continues with its decimal digits. If the two sets contain the same number of elements, then there are no omissions. But now consider this number:

0.a_1b_2c_3d_4

It does not occur in the list above since in the second place, for example, it is different in its decimals to every other member of the list except the second. But it must be included in the list - it is a point on the number line between 0 and 1. This shows that our construction is impossible. The matching cannot take place. The ‘infinity’ of the Natural numbers is different to the ‘infinity’ of a number line. Cantor gave them special symbols: \aleph_0 and \aleph_1 respectively. They have strange properties:

\aleph_0+7=\aleph_0

\aleph_1+\aleph_0=\aleph_1

3\aleph_1=\aleph_1