Real numbers less than 1 haphazardly arranged on the floors of a series of rooms. Georg Cantor, the founder of set theory, famously asserted that infinite sets will have the same size (or cardinality) if they can be formed in a one-to-one correspondence the the set of natural numbers, that is they are countable ??thus there are as many odd numbers as odd and even numbers, or fractions or prime numbers. He then proved that some sets, such as reals less than 1, are not countable, and therefore constitute a greater infinity than the whole of natural numbers. By this definition there are an infinite number of uncountable infinite sets. Computer generated image.