 The main result of this section is another proof of the Cantor-Bernstein theorem: “This proof is essentially the same as Bernstein’s published originally by Borel [1898: Note 1, pp. 102–7]” (PM I, 589). In this proof the one-one relation between the sets \(\alpha\) and \(\beta\) is constructed from the powers of two relations R that maps \(\alpha\) into \(\beta\), and S that maps \(\beta\) into \(\alpha\). The one-one mapping is constructed in stages. First all of \(\alpha\) is mapped onto \(\beta\) by R. Those elements in \(\beta\) not in the range of R are mapped onto \(\alpha\) by S. But some elements in the range of S will have already been mapped by R. They need to be shifted to a new image in \(\beta\), again by R. This process is iterated through all of the powers of R, and then it is shown that the resulting relation is one to one from \(\alpha\) onto \(\beta\). See Hinkis (2013) for a history of the many different proofs of this theorem.