 Cantor defined two sets as having the same cardinal number if there was a bijection between them. This is of course simply common sense in the finite realm; his originality lay in extending it to the infinite realm and refusing to shy from the apparently paradoxical results. In the infinite realm, cardinal and ordinal number are not isomorphic notions as they are in the finite realm: a well-ordering of order type \(\omega\) (say, the usual order on the natural numbers) and a well-ordering of order type \(\omega + \omega\) (say, the order on the natural numbers which puts all odd numbers before all even numbers and puts the sets of odd and even numbers in their usual order) represent different ordinal numbers but their fields (being the same set!) are certainly of the same size. Such “paradoxes” as the apparent equinumerousness of the natural numbers and the perfect squares (noted by Galileo) and the one-to-one correspondence between the points on concentric circles of different radii, noted since the Middle Ages, were viewed as matter-of-fact evidence for equinumerousness of particular infinite sets by Cantor.