Provable Unsolvability

By Juliette Kennedy 

The continuum hypothesis was under discussion as an "undecidable statement" at the Princeton University Bicentennial Conference on "Problems of Mathematics" in 1946, the first major international gathering of mathematicians after World War II. Kurt Gödel is in the second row, fifth from left.

In 1900, David Hilbert published a list of twenty-three open questions in mathematics, ten of which he presented at the International Congress of Mathematics in Paris that year. Hilbert had a good nose for asking mathematical questions as the ones on his list went on to lead very interesting mathematical lives. Many have been solved, but some have not been, and seem to be quite difficult. In both cases, some very deep mathematics has been developed along the way. The so-called Riemann hypothesis, for example, has withstood the attack of generations of mathematicians ever since 1900 (or earlier). But the effort to solve it has led to some beautiful mathematics. Hilbert’s fifth problem turned out to assert something that couldn’t be true, though with fine tuning the “right” question—that is, the question Hilbert should have asked—was both formulated and solved. There is certainly an art to asking a good question in mathematics.

The problem known as the continuum hypothesis has had perhaps the strangest fate of all. The very first problem on the list, it is simple to state: how many points on a line are there? Strangely enough, this simple question turns out to be deeply intertwined with most of the interesting open problems in set theory, a field of mathematics with a very general focus, so general that all other mathematics can be seen as part of it, a kind of foundation on which the house of mathematics rests. Most objects in mathematics are infinite, and set theory is indeed just a theory of the infinite.