Kittens & Gorillas 155 The mathematical reason for it being a false paradox is that some series converge and some do not. If I move progressively closer and closer to something in smaller and smaller time intervals then I may indeed reach it. On the other hand, some series never converge. I will never reach infinity how ever many steps I take. The Barber Paradox Now, for a slightly harder paradox, let’s suppose there is a town with just one barber. In this town, every man keeps himself clean-shaven by either shaving himself or going to the barber; the barber shaves all the men in town who do not shave themselves. All this seems perfectly logical, until we pose the question: who shaves the barber? This question results in a paradox because, according to the statement above, he can either be shaven by himself or the barber, which is he. However, neither of these possibilities is valid! This is because if the barber shaves himself, then the barber must not shave himself and if the barber does not shave himself, then the barber must shave himself. You might think this paradox an oddity but, using this simple idea, Bertrand Russell changed the course of mathematical history and it is the fundamental paradox used to show computers are Turing limited. The Russell Paradox In the late 19 century, mathematicians began to think about the nature of numbers. What is a number? It is certainly not an object we can hold. I can't hold a two, unless it’s the brass number plate, for my front door. And, in that case I am holding one number plate, so I am not holding the idea of two, but rather the idea of one: one brass plate in the shape of a two. The ‘idea of a number is to say something about the things I have in my hand: two apples, two oranges and two brass number plates. These are all sets of two things and ‘two’ is the collection of all these sets. In 1890, Gottlob Frege completed his theory of sets. The project had taken him five years. Unfortunately, just before send