Bertran Russell is considered one of the greatest philosophers, but his works in mathematics are, if possible, one of the most important contribution to mathematical thinking. In 1903 was published Principles of Mathematics, where ‘types theory’, the numbers as ‘class of classes’ and the ‘Russell’s paradox’ were the most important results.
Russell’s paradox was the reasoning that broke up his hope to get the foundation of mathematics on logical terms. The objet of investigation was the set theory. And Russell thought that all type of sets must belong to a set, but Russell’s reasoning drove to a paradox. He discovered that there are sets which do not belong to any set proven that logic is inconsistent: the set of all sets which do not contain themselves must contain itself and mustn’t contain itself at the same time.
Russell explained that through a little tale: Once upon a time a barber living in a wonderful emirate, but one day the emir was informed about the low number of barbers in the emirate. By solve that problem, the emir declared: ‘I, the emir of ….. say that the barbers only will shave the men which cannot shave themselves’. In a little town lived only that barber, and he went to see the emir because of his problem: I am the only barber in my little town. If I shave myself, then I can shave by myself, so I must not be savhed by the barber…but I am the barber. If I do not shave myself, then I must be shaved by a barber…but I am the barber. ………inconsistent.