Lesson 8

It remains to show that this fragment (denoted \(S\)) contains the entire set \(A\). If \(S \ne A\), take the element \(a = \varphi(S)\), which does not belong to \(S\), and add it to \(S\), placing it above all existing elements of \(S\). The resulting ordered set \(S'\) (the sum of \(S\) and the singleton \(\{a\}\)) is clearly well-ordered. Moreover, the correctness condition remains satisfied (for the new element \(a\) this holds by construction, and for the other elements it follows from the correctness of \(S\)). Thus, we have constructed a larger correct fragment, contradicting the maximality of \(S\). This argument completes the proof of Zermelo’s Theorem.

Finally, one can see the following corollary. Corollary. For any two sets, one is equinumerous to a subset of the other.

The concept of a well-ordered set was introduced by Cantor in his 1883 work; in his culminating papers of 1895–1897, he provided a proof that any two well-ordered sets are comparable (i.e., one is isomorphic to an initial segment of the other). Statements regarding the possibility of well-ordering any set using the Axiom of Choice appear repeatedly in Cantor’s writings, yet he never supplied a coherent proof. Such a proof was first given only in 1904 by the German mathematician E. Zermelo.