Question: Since the 1930s, we have had to live with Godel's theorem — the apparently unshaken proof by the logician Kurt Godel that there can be no system of mathematical logic that is at once consistent (or free from contradictions) and complete (in the sense of being comprehensive). The question is whether there is, or whether we should expect, such a fracture in the logical basis on which people now look for a description of the nexus between particle physics and cosmology.
Why: The chief interest of Godel's theorem is that it is a negative answer to one of the questions in David Hilbert's celebrated list of tasks for the twentieth century, put forward at the International Mathematics Congress in Paris in 1900. Mathematicians in the succeeding century seem not to have been unduly incommoded by Godel. But if there were a comparable theorem in fundamental physics, we should have more serious difficulties. Perhaps the circumstance that string theory is getting nowhere (not fast, but slowly) should be taken as a premonition that something is amiss. The search for a Theory of Everything (latterly gone off the boil) may be logically the wild goose chase it most often seems. If science had to abandon the principle that to every event, there is a cause (or causes) , the cat would really be among the pigeons.
Moral: Godel's theorem needs seriously to be re-visited, so that the rest of us can properly appreciate what it means.