The Probability of Undecidability

There’s a lot we don’t know. There’s a lot we can’t know. But can we at least know how much we can’t know?

What fraction of mathematical statements are undecidable—that is, can be neither proved nor disproved? There are many ways to make this question precise… but it remains a bit mysterious. The best results I know appear, not in a published paper, but on MathOverflow!

In 1998, the Fields-medal winning topologist Michael Freedman conjectured that the fraction of statements that are provab...