We take the real numbers for granted, but what are they really?

Nineteenth-century German mathematician Leopold Kronecker once said

God created the integers, all the rest is the work of man.

For millennia, puzzles and paradoxes have forced mathematicians to continually rethink their ideas of what proofs actually are. Jon Walthoe explains the tricks involved and how great thinkers like Pythagoras, Newton and Gödel tackled the problems.