 Galileo Galilei epitomized the paradoxical situation we run into when trying to generalize counting from the finite to the infinite. In Two New Sciences (1638; Galilei 1974), he presented a paradox of infinity. On the one hand, there is an intuition that there are fewer square numbers than natural numbers, since the first collection is properly contained in the second (the former has some but not all of the latter’s members). On the other hand, there is an intuition that there are the same number of squares and natural numbers, since there is a one-to-one correspondence—a bijection—between the natural numbers and their squares. Galileo’s own conclusion, following Oresme and Albert of Saxony who had discussed similar issues in the 14th century, was to claim that one cannot apply the relations of equality, greater than, and smaller than to infinite collections. Much subsequent theorizing about infinity can be regarded as respecting one intuition at the expense of the other.