Yesterday I have seen Story of Mathematics, part 4. ( See previous posts on 1,2 and 3 ). Among other topics it was about Cantor's math on infinity. Cantor introduced an entire array of infinities. The 'smallest' infinity is the cardinality ( number of elements of a set ) of N, the set of natural numbers. The paradox that the sets {1,2,3, ... } and {10,20,30,...} have the same number of elements was shown. A nice graphic followed about how Cantor reasoned that Q, the set of all fractions, has the same cardinality as Z. It went more or less like this.

Make an infinite square of all fractions such that the first row contains all fractions with numerator 1, the second row with numerator 2, etc. Do the same for the denominators but by colomn. The square should look like:

1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...

2/1 2/2 2/3 2/4 2/5 2/6 2/7 ...

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

4/1 4/2 4/3 4/4 4/5 ...

5/1 5/2 5/3 5/4 ...

...

It is then possible to traverse this square like

1/1

2/1 1/2

3/1 2/2 1/3

4/1 3/2 2/3 1/4

The sum of the numerator and denominator is constant by row. Because this traversal includes every fraction it is possible to map the fractions in a 1-to-1 relation with the natural numbers and thus the set of fractions has the same number of elements as N. ( It is easy to include the negative fractions in the traversal, the result is the same. ) It turned out that a similar reasoning is -not- possible for the reals R and thus R has an infinity which is more infinite than the natural numbers. ( It wasn't in the documentary but I know that there is a third level of infinity which is the same as the total number of possible curves in a 2D-space ( plane ). Theoretically there is an infinity of possible infinities ( I think ).

The mathematician Hilbert and his famous problems for the 20th century was introduced. The first problem was called the Continuum Hypothesis which asked if there was a set in between Q and R. Around that time Kurt Godel proved that there are statements in every possible system of mathematics which cannot be proved. Until the 30's of the last century Europe had the world centers of mathematics in cities like Gottingen and Paris. Then the Nazis came but many scientists and artists left. In the US Princeton was established to become the next world center of science and mathematics. Among others Godel and Einstein lectured there. Needless to say both were Jewish.

Many famous mathematicians had some sort of psychological issue. Godel suffered paranoia and Cantor had a bipolar disorder. In the 30's there were no anti-depressants or ADHD type of drugs so they must have had a dark life. Yet, I don't think having a disorder is a prerequisite for being able to produce excellent math because there are many more mathematicians without a disorder. I am not even sure if the mathematical community has more psychological issues than other groups of scientists.

At Princeton there was a young mathematician Cohen who took on Hilbert's first problem. His answer wasn't what the community expected but because Godel approved his paper the community accepted that there are two types of mathematics: one where the Hypothesis is false and one where the hypothesis is false.

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