Proper classes

Abstract

The modern problem of infinity was first raised by Aristotle who held (at least on the popular interpretation) that infinite sets exist potentially (i.e. one more number can always be counted, one more division can always be made in a line segment) but not actually (i.e. the numbers or divisions cannot all exist at one time). In fact, Aristotle not only held that completed infinities never actually exist, but also that they are impossible, that is, that the assumption that they do exist leads to contradictions. To see this, consider Aristotle's view that mathematical entities depend for their existence on the existence of primary substance in which they inhere, coupled with his view that there can be no infinite body. From these it follows that if there were actual completed infinities, infinite bodies would both exist and not exist. The details here are not so important as the idea that if a completed infinite is assumed to exist a contradiction follows. This negative attitude towards completed infinities flourished for centuries. By the mid-1500's, the German mathematician Stifel (who apparently also invented Pascal's triangle) was moved to condemn irrational numbers simply by their association with the completed infinite: … just as an infinite number is not a number, so an irrational number is not a true number, but lies hidden in a kind of cloud of infinity. About a hundred years later, Galileo further discredited the completed infinite by pointing out that line segments of different lengths can be brought into one-to-one correspondence by projection, as can the natural numbers and the perfect squares by assigning each number to its square.