 But suppose that one holds that some collection (the points in a line, say) is dense, hence ‘unlimited’, or infinite. The first prong of Zeno’s attack purports to show that because it contains a definite number of elements it is also ‘limited’, or finite. Can this contradiction be escaped? The assumption that any definite number is finite seems intuitive, but we now know, thanks to the work of Cantor in the Nineteenth century, how to understand infinite numbers in a way that makes them just as definite as finite numbers. The central element of this theory of the ‘transfinite numbers’ is a precise definition of when two infinite collections are the same size, and when one is bigger than the other. With such a definition in hand it is then possible to order the infinite numbers just as the finite numbers are ordered: for example, there are different, definite infinite numbers of fractions and geometric points in a line, even though both are dense. (See Further Reading below for references to introductions to these mathematical ideas, and their history.) So contrary to Zeno’s assumption, it is meaningful to compare infinite collections with respect to the number of their elements, to say whether two have more than, or fewer than, or ‘as many as’ each other: there are, for instance, more decimal numbers than whole numbers, but as many even numbers as whole numbers. So mathematically, Zeno’s reasoning is unsound when he says that because a collection has a definite number, it must be finite, and the first subargument is fallacious. (Though of course that only shows that infinite collections are mathematically consistent, not that any physically exist.) 