Kurt Gödel is probably the greatest mathematician most people have never heard of. Perhaps his greatest accomplishment was his incompleteness theorems, which demonstrate the inherent limitations of any formal axiomatic system – ie it is impossible to devise a system governed by a set of rules that does not contain within itself an inconsistency.

It is easy to think of examples in language, such as the sentence: ‘This is not a sentence'. There are also some famous examples in mathematics, such as Russell's Paradox, sometimes explained using the ‘barber paradox': suppose a barber shaves everybody in town, except for all those who shave themselves. But who shaves the barber? If he shaves himself, he cannot be shaving only people who don't shave themselves. But if he doesn't shave himself, then as he shaves people who don't shave themselves, he must then shave himself!