 The study of the transfinite ordinals directed Cantor’s attention towards ordered sets, and in particular well-ordered sets. A set \(S\) is well-ordered by a relation &lt; iff &lt; is a total order and every subset of \(S\) has a least element in the &lt;-ordering. (The real numbers are not well-ordered in their usual order: just consider an open interval. Meanwhile, \(\mathbf{N}\) is the simplest infinite well-ordered set.) Cantor argued that the transfinite ordinals truly deserve the name of numbers, because they express the “type of order” of any possible well-ordered set. Notice also that it was easy for Cantor to indicate how to reorder the natural numbers so as to make them correspond to the order types \(\omega+1\), \(\omega+2\), …, \(\omega \cdot 2\), …, \(\omega \cdot n\), …, \(\omega^2\), …, \(\omega^{\omega}\), … and so on. (For instance, reordering \(\mathbf{N}\) in the form: 2, 4, 6, …, 5, 15, 25, 35, …, 1, 3, 7, 9, … we obtain a set that has order type \(\omega\cdot 3\).)