The Story of Maths - "To Infinity and Beyond" - Hilbert's First Problem

Hilbert's First Problem

The final episode considers the great unsolved problems that confronted mathematicians in the 20th century. On 8 August 1900 David Hilbert gave a historic talk at the International Congress of Mathematicians in Paris. Hilbert posed twenty-three then unsolved problems in mathematics which he believed were of the most immediate importance. Hilbert succeeded in setting the agenda for 20thC mathematics and the programme commenced with Hilbert's first problem.

Georg Cantor considered the infinite set of whole numbers 1, 2, 3 ... ∞ which he compared with the smaller set of numbers 10, 20, 30 ... ∞. Cantor showed that these two infinite sets of numbers actually had the same size as it was possible to pair each number up; 1 - 10, 2 - 20, 3 - 30 ... etc.

If fractions now are considered there are an infinite number of fractions between any of the two whole numbers, suggesting that the infinity of fractions is bigger than the infinity of whole numbers. Yet Cantor was still able to pair each such fraction to a whole number 1 - 1/₁; 2 - 2/₁; 3 - 1/₂ ... etc. through to ∞; i.e. the infinities of both fractions and whole numbers were shown to have the same size.

But when the set of all infinite decimal numbers was considered, Cantor was able to prove that this produced a bigger infinity. This was because, no matter how one tried to construct such a list, Cantor was able to provide a new decimal number that was missing from that list. Thus he showed that there were different infinities, some bigger than others.

However there was a problem that Cantor was unable to solve: Is there an infinity sitting between the smaller infinity of all the fractions and the larger infinity of the decimals? Cantor believed, in what became known as the Continuum Hypothesis, that there is no such set. This would be the first problem listed by Hilbert.